Quantum Physics - S. Gasiorowicz

' Soalta. i@Stephen Gasiorowicz |Be|UniversityofMinnesota |}quantum physics a =EP £8. EN. mstiToni. Gs “A | John Wiley andSons, ne { |newyork»london +sydney +toronto : |preface This book isintended toseve asaninttoduction toquantum physics. Tnwecing it,Thave kept several guidelines inmind 1.Fis, iishelpful forthe development ofinition inanynewfed of study coseatwithabaseofdeailed knowledge about simple systems. [have therefore worked outanumber ofproblems ingreat deal, sothattheinsight thus obeained canbeused formete complex systems 2.Every aspect ofquantum mechanics hasbeen helpful inunderstanding some physical phenomenon. Ihave therefore laidgreat sttes onapplications at every sage ofthedevelopment ofthesubject. Although noateaofquantum physics istotally developed, myintention iscobridge thegapbetween‘modephysicscourseandthemoreformaldevelopment ofquantummechanics.‘Thus, many applications arediscussed, and[have stressed onderofmagnitude estimatesandtheimportance ofnumbers 3.Inkeeping with thelevel ofthebook, themathematical structure has been kepeassimple aspossible, Newconcepts, suchasoperitors, andnew ‘mathematical cools necessarily make their appeatance. Ihave dealt with the former more byanalogy thanbyprecise definition, andIhaveminimized theseofnew tools insofar aspossible. loapproaching quantum cheory,Ichosetostatwithwavemechanics and theSchtBdinger equation. Although thestate-vecior approach gets atthe essential structure ofquantum mechanics more apdly,expetieace hasshown thattheuseofmote familite tools, suchasierenial equations, makes the theory more accesible andthecorrespondence with casial physics more wanspares. “Thebook probably contains alitlemore material thancancomfortably be covered inoneyea. Thebasic mateal canbecovered inoneacademic quarter. viii Preface Iconsists ofChapters 106,8,nd9,inwhich themotivation foraquantumtheory,theSchrédinget equation,andthegeneralframework ofwavemechanicsarecovered, Anumber ofsimple problems aresolved inChapter 5,andtheir relevance tophysical phenomena isdiscussed. The generalization tomany [patties andtothreedimensions isdeveloped. Thesecond-quarter mateial deals directly withatomic physics problems andusessomewhat mote sophiscicared tools, Hete wediscuss operator methods (Chapter 7),angular momentum (Chapter 11),thehydrogen atom (Chapter 22),operators, matrices, andspin (Chapter 14),theaddition ofangulac momenta (Chapter 15),time-independent perturbation theory (Chapeer 16),andtherealhyérogea acom (Chapter 17). ‘Thismaterial prepares thestudent tocopewithalargevasievy ofproblems that arediscussed during chethird andlastquarter. These problems include theinter.actionofchargedparticleswithamagneticfield(Chaptet13),cheheliumarom(Chapter18),problemsintheradiationofatoms andrelated topics (Chapters 22 and23),collisiontheory(Chapcer24),andtheabsorpcion ofradiationinmacter(Chapter 25).Thismaterial issupplemented byamorequalitative discussion of thestruceure ofatoms andmolecules (Chaprers 19(021).Thelastchapter onelementary particlesandcheitspmmetsies servesthedualpurposeofdescribing‘some oftherecent advances onthatfrontier ofphysics andofshowing howthe basic ideasofquantum theory bavefound applicability iachedomain ofvery shore distances. Several topics arisenaturally asdigressions inthedevelopment ofthe subject matte, Instead oflengthening some long chapters, Ihave placed this ‘matctial inaseparate “Special Topics” section. ‘There, relativistic kinematics, theequivalence principle, theWKB approximation, adetailed treatment of lifetimes, Tinewidths andscactering resonances, andtheYukawa theory of nuclear forces atediscussed. For the same reason, abrief introduction tothe Fourier incegral, theDisac delta function, andsome formal material dealing swithoperators have been placed inmathematical appendices attheendofthe ook, Tamindebtedtomycolleagues atcheUniversity ofMinnesora, especiallyBenjamin Bayman andDonald Geffen, formany discussions onthesubject of ‘quantum mechanics. 1amgrateful toEugen Merzbacher, whoteadthemanu: script andmade many helpful suggestions forimprovements. Ialsothank my students intheintroductory quantum mechanics course thatItught forseveral years. Theit evident interest inthesubject ledmetothewriting ofthesupple ‘menal notes that later became this book. Stephen Gasiorowice |contents Chapter 1TheLimits ofClassical Physics 1 Black Body Radiation: chelawsofWien andRayleigh-Jeans; chePlanckformule.ThePhotoelectic Efect.TheCampionEffect.BlectronDiffaction, TheBobrAtom:thepostulates;experimental‘consequences; theCorrespondence Principle. TheWave-Panidle Problem. Chapter 2.Wave Packets andtheUncertainty Relations 27 ‘Thegaussian wave packer; thepropagation ofpackers; group velocity; theDeBroglie ration, TheUncereainty Relations:‘measurement ofpositionofanelection;thetwo-slicexperiment;the"realy" ofonbits intheBohs atom; theenergy.time ‘uacertangy relation; theuses oftherelations forsximerical estimates, Chapter 3TheSchrédinger Wave Equation “5 . ‘Theficeparce equation. Theprobability interpretation, Flux 3 conservation, Expectation values. Themomentum operator. The ; realty ofexpectation values. Theequation fora particle ina,pocential x Contents Chapter 4Eigenfunctions andEigenvalues 37 ‘Theenergy eigenvalue equation. The particle inabox:igenfunctions andeigenvalues; octhogonalty ofeigenfunctions;theexpansionpostulateandtheinterpretation ofcheexpansion‘coeficients, Party. Momencum eigenfunctions; unnormalizable states; degeneracy andsimultaneous eigenfunctions. Chapter 5One-Dimensional Potentials 5 ‘Thepotential sep; reflection andtransmission coefficients. The potential wellandbound states. Thepotential barrier; unnelling; ‘old emission; cunnelling chrough thinfilms; alpha decay. One- dimensional models ofmolecules andthedets-fanction potential. ‘TheKronig-Penney model, Theharmonic oscillator Chapter 6TheGeneral Structure ofWave Mechanics am Eigenfunctions andtheexpansion theorem; analogy withvector spaces. Linear operators; hemitian operacots; completeness; degeneracy; complete setsofcommuting observables. Thetuncertaingy telecions.Theclassicalieofquantumtheory. Chapter 7Operator Methods inQuantum Mechanics 127 ‘Theharmonicoxcillatorpeoblem:raisingandloweringoperators;cigenstates andeigenvalues. Theintespreation ofthewave functiona8probability amplitude. Thetime-development ofasystemin .terms ofoperators; theSchrédinger andtheHeisenberg pictures. . Chapter 8N-Particle Systems un ‘The ScheBdinger equation forN-paricle systems. Momentum conservation; seperation ofcenter ofmassmotion; reduced mass. ‘eentcal panicles; symmeuy under their interchange. The Pal Principle. Fermionsandbosonsinabox;cheFermienergy. Chapter 9‘The Schrésdinger Equation iaThree Dimensions 155 Separation ofcenter-of-mass motion; invariance under roeations:theseparationofangular momentum. Theradial equation. Fermi cefergy forthree-dimensional box. Chapter 10Angular Momentum 167 ‘Theexpression forL3;elgebraicmethodforsolvingtheL,andL*eigenvalue problesn;usingandloweringoperators;Legendrefonctions, Contents xi fGuper 1TheRadial Equation 179 : Behavioratcheorigin;behaviorfrlarger.Thefreparticle; sphericalBesselfunctionsincomingendoutgoingsphericalwaves; phaseshifts.Thesquarewell:boundsates;deepwel;hellstructure; contiauum solutions (Chapter 12The Hydrogen Atom 195 ‘implication ofthe radial equation. Quantum numbers, degeneracy. Wave functions andrelations co“orbits.” Chapter 13Toteraction ofElectrons with Electromagnetic Field 209 ‘Maxwell equations. Coupling ofelectrons tovector potential Equationforelecuonin«uniformmagneticfeld.Thenoemal ‘Zeemaneffec. Elcwon motion ia4uniform magnetic field;iuustationofconespondence principle.Hixquantization; theBohm-Aharanoy Effect. Chapter 14Operators, Matrices, andSpin 27 Max representation ofharmonic oscillator operators. Matsrepeesencation ofangolamomentum 1operstors.Spin¥matrices;spinors,Theprecesionofspininamagnercfedsparamagnetic resonance. ‘Chapter15TheAdditionofAngularMomenta 243 ‘Theaddition ofewospin 5;singlet andwiper eigenfunctions. Spio-otbital angular momentum addition. TheExcesion Principle and angular momeaeam stares. Chapter 16Time Independent Perturbation Theory 255 First-order energy sift. Second-order perturbation theory.Degenerateperturbation theory.TheStarkEfec:absenceoflinear Shiftforground state; electri dipole moments; second-order shiftLinearSiskeffectfoe»=2sates, Chapter 17‘The Real Hydrogen Atom an Relativistic massconections.Spin-orit coupling.AnomalousZeeman Effect; Hyperfine interaction, *Chapter 18TheHelium Atom 283 extapproximation. First-order shiftdueto¢eepulsion, Thefase : ‘excited sas. Exchange energy. TheRitevariational principle, ‘Aeroioniastion. 4 Coowenss iit So Special Topics 459 1. Relativistic Kinematics 461 TL, The Equivalence Principle 465 Ml, TheWentzel-Kramers-Beillouin Approximation 469 IV. Lifetimes, Line Widths, and Resonances 43 V. The Yukawa Theory 481 Appendices 487‘A.TheFourierIntegalandDelaFunctions 489B. Operators 495 References 501 Physical Constants 507 Index 509 xi Contents Chapter 19.The Structure ofAtoms 299 “Thevariational principle andtheHarerce equations. Theperiodic table. Qualitazive discussion ofconsequences ofshell structure in ‘Chapter 20,Molecules 313 Approximate Schridinger equation; electronic, vibrational, and rotational motion, The Hs" molecale: variational cial wave fonction; spins ofnuclei andspecta; specific heats ofmolecules. Chapter 21Molecular Structure 327 ‘TheH,molecule.Molecularorbitals.Bonds.Qualcativedescription ‘ofsome simple molecules; hybrid orbital Chapeer 22.TheRadiation ofAtoms amt ‘Time dependent pertubation theory. Theelectromagnetic inceracions; semiclassical description; phase space; theGolden Rule. Marix element calculation; selection rules; the 2?—15 twansition rate. The effecs ofspin, Ghupter 25Selected Topics inRadiative Transitions 365 Lifetime andlinewidth; collision broadening; Doppler sift. ‘Méssbaver Effect. Induced absorption andemission. Thelast. ‘upter 24Collision Theory 379 Collision eross section; optic! theorem: inelastic cross sections Black disc seatering. Scattering 3°low cnexgics; resooane saatering foesquate well; effective ange formula. Spin dependence inneucron-peocoo scattering. TheBoen approximation. Scattering‘ofideoicalparicles.CoherentseateringardtheBragg,conditions, Chapter 25.The Absorption ofRadiation iaMater 409 ‘The photoelectric effect; angular dependence; energy dependence. ‘Compton scattering. Antiparticls, andpaiproduction. Chapter 26°Elementary Particles andTheir Symmetries 423 Electrons andpositrons; posironium anditsdecay modes; charge conjugation. Baryons, antibaryons, andmesons. Tsotopi spin Conservation. Theproblem ofA*decay andproduction; associated producdon; thesmangeness quantum aumber, slecion rales, nicary symmetty thediscovery ofthe G-,ehe quark model. Parityponcouservation inKdecay:generaltests.TheK°—R°system, ‘|chapter1 jTheLimits ofClassical Physics ‘The endofthenineteenth century andthebeginning ofthetwentiethwinceSanpnAaisofperinealaaogcone‘oualy incompatible with classical physics, Thedevelopment ofthese concepes, ee eee aay ofadil Prmes sodflan expen ed Boye eee ays Ouroben ais caper ne da deSteado'sssoeaewsBndeigh,ropeseeconepeseet dan, ahs nthae conety nluke hetanson ae alurunoere nade Wesewasp, trpe (properties ofradiative, shewaveproperties ofmater, andthequantization ofphic ea tienen inte promenade bere A.Black Body Radiation . When «body ietd, tse codine, Inelim theight eviged gen ov hewe sper offequntca ws opel distbutionthardependsbochonthefrequencyof,equivalently, onthewavelengehOfthelightb;andonthetemperature, OnemaydefineaquantityE(A,7),the caine pt, sce cory ened sewerdeog 3psat es peoa ‘time. Theoretical rescarch inthefield ofthermal radiation began in1859 with: thewok ofKichol, ho shovel hsaru ponte oft te powcr Ba theinriy 4,deed stheSrc ofiacide mat ‘wavelength dchatisabsorbedbythebody,isthesameforallbodies.Kirchhoff considered two emitting and absorbing panilel plates and showed from the egal condom ditiecgy chive! mst totheeotgy sed (foreach 2),thattheratios B/A must bethesame forthecwoplates. Soon “A eng an fh Seon ofun yy beun a tepn Se chy ee Sane eet we ; j 2 Quansuim Physics thereafcer, heobserved that for&black hady, defined asasurface that totlly absorbs allradiation that fills onit,s0that A=1,thefunction EQ), T)is« univers! fanction. Inoxder tostudy thisfunction icisnecessary coobtain thebestpossiblesourceofblackbodyradiation.Apracticalsolutiontothisproblemistocoa-sider theadiaton ercerging from &snall holeinanenclosure heated 0atem-peratureT.Giveneheimperfections inthesuefaceoftheinsideofthecavity,{eisclearthatanyradiationfallingontheholewillhavenochanceofemergingagsin. Thus thesurface presented bytheholeisveryacarly “coaly absorbing,” ‘andconsequentytheradiationcomingfromitisindeed“blackbodyradiation.” Providedtheholeissmallenough,thisradiationwillbethesameasthatwhich fallsonthewalls ofehecavity. Iistherefore necessary tounderstand thedist-butionofradiationinsideacavieywhosewallsareatatemperatureT. Kirchhoff showed chat thesecond lawofthetmodyeamics requires thac theradiation inthecavity beisotropic, thati,thatchefluxbeindependent of direction; thacitbehornogeneous, thais,thesame atallpoints; andshatitbe ‘hesame inallcavities atchesame temperatize—all ofthis foreachwavelength * ‘The emissive power may, bysimple geomettic arguments, beshown 10be connected with theenergy density (A, T)inside thecavity. Therelation is 10,17)=SAD ay “Theenergy density isthequantity oftheoretical interest, andforther under standing ofitcame in1894 from thewotk ofWien, who, again using very ‘genera! arguments,’ showed thartheencrgy density hadtobeofthe form 1,T)= fOT) 2) ‘wich fstill anunknowa function of«single vatitble. If,asisconvenient, one deals instead with theenergy deasicy asafunction offrequency, u(x,T),then it follows from the fact that a Ha,T)=ald,n|o =Fan as) 2These mates sredicated inmany textbooks onmodern physics and sic Physics, References canbe found athe ead ofhis charter.*wien considered 4pesfeclyreBecting sphere cavity concctng adiabatically “Thesetbiton ofthe enemy a81farction of netobecased bythe Doppler shit cnweletion. SeeChap VinF-K. Richemyer, BH.Keanrd, and JN.Cooper Ire decent Madey Phi, Meow Hl, New York, 196 ‘TheLimits ofClassical Physics 3 4 wars rare te7r= tae on re 1srK eh f* F Fig.1-1, Experimental veriication ofEq.1-2intheform w(X,7)/7* =auoiversalfunctionofXT. thar the Wien law seads aly,T)=og(z) a4) ‘Theimplications ofthislaw,which wasconfirmed experimentally (Fig.1.1),aepwofold: 1.Given thespecial distribution ofblack body radiation atonetem- erature, thedistibution atanyothertemperature canbefound withthehelp‘oftheexpressions given above 2.Ifthefunctionfls)—or, equivalently, thefunctiong(s)—has amasi- ‘mumforsomevalueofx>0,thenthewaveleagth hawatwhichtheenergyAensity, andhence theemissive power, hasitsmaximum value, hastheform ‘ doe=F as) where &isaunivers! constant. Wienusedamodel(ofnointerest, excepttothehistorian) topredict« foxforg(+/T). Thefore was 80/T) =Grae (6) ‘nd,remarkably enough, thisform, containing twoadjustable parameters, fc thehighfrequency (lowwavelength) dataverywell.Theformula isnoe,how: 4 Quantum Physics E ] \ ware an \ orn Veterans so00 NN WWeen 3000599mHORTww inwaeHHeg nk ‘engin ke Fig.1.2. (4)Distiborion ofpower radiated by4black bodyatvatious cempers ures. @)Compasison ofdataat1600°K withPlanck formula andRayleigh-Jeans formala. ‘ever,inaccordwithsomeverygeneralnotionsofclassicalphysics.Rayleigh,in 1900, derived the result ar? 1,0) = ar an where &isBoltzmann's coostant, k=1.38% 10-¥ erg/deg andcisthevelocityoflight,c=3.00X10"cm/sec.Theingredients thatwentintothedeivationwere(1)thedassical lwofequipartition ofenergy, according towhich the average energy perdegree offteedom foradynamical system inequilibcium is,inthiscontext,‘AT,and(2)checalculation ofthe number ofmodes (i. degrees offreedom)forelectromagnetic radiationwithfrequencyincheinterval(6,»++4),confined inacavity «Theequipation lawpeedice tha:theenergy perdegse offeedom is#T/2. Forsanoucilatorandthemoderofthedecuomagneic eldaresimpletronicosclacors— 1Contibtion of£T/2fromthekinetic eneigy ismatched by+tikeconbuion fromche potential eee, aiving AT.pemilceathisresaleain,anddrivetnChaprer28,Theomerofmodesis “bev/e,foet implied byfactor of2because wansvese eectromageeic wave Co tespond totwo-dimensional hrmonic oxo. ‘TheLimits ofClassical Physics 5 TheBarlighJansJaw(1-7)Jeansmade2minorconuibuconwoits derivation) toesnotagree withexperiment athighfrequencies, where theWien formula works, though itdoesfirtheexperimental curve atlowfrequencies __(Fig.1.2).TheRayleigh-Jeans lawcannot, ongeneral grounds, becore, since thetoralenergy density (integrated overallfrequencies) ispredicred 10be infinite! ‘in1900, MaxPlanck found «formula byaningenious interpolation berween thehigh-frequency Wien formula andthelow-frequency Rayleigh- Jeans law. The formula is Bh «: 1) =F aie] a8) Kwhereb,Planck'scontant,isanadjustableparameterwhosenumericalvaluewas©foundtobeb=6.63X10-*ergsec.Thislawapproaches cheRayleighJeans|form when »~+0,andreduces t0 4 1.0)=acue pears 4 aacour G9) Jwhenthefequency islarge,or,moreaccurately, wienby3>KT.Ifwetewice Ftheformulaas«productofthenumberofmodes[weobtainthisfrom(1-7)by dividingtheenergydensitybyAT]andanotherfactorthatcanbeinterpretedas *theaverageenergyperdegreeoffreedom anthy ; MOD8wir ‘i Smt. e/AT.sae G10) +. Weseechastheclassicalequiparticion lawisalteredwheneverthefrequenciesare notsmallcomparedwithAT/h,Thisalterationintheequipariion lawshows £thatthemodes haveanavetage energy thatdepends ontheitfrequency, and“tharthehighfrequencymodeshaveaverysmallaverageenergy.Thiseffective38CavoffremovesthedifcultyofcheRayleigh-Jeans densityformula:thecoal:‘energyin«cavityofunivolumeisnolongerinfinite.Wehaveer)=SR, g_rhany(o/RT)ho) ‘Sea). wary ct aeofa oan 6 Quann Physics ‘hein canbeevalael, andthe resultstheSeinolanexpesion forthe tal nnon energy peruitvolume un =ar ag with«= 756% 10° eget dept dived such ee, except forthe wrest coast infon ontheass ofthemeryuamial reasoning. A departure from thepureequiparticion lawwasnotentitely unexpected: oneetincaceofitwastheDelongPetelawofspeaichas,scoringto‘which theproduct oftheatomic (ormolecular) weight andthespecific heatis2 caeralsolidgeedepurarsfomtheDulogPeipedicionwe Siecle eaiy as1602" Thee dears laced thattespecie best read atone esperar “Theangulifed mcens ofhisformu drove Pack toseu foris cxiin sadohn twomonths hefound thathecould derive icbyassuming eee enagy ued witheachode ofthedecompose fddiaot Eyada ith verge value FT)botasannee mailofsme ‘minimum quantum ofenergy &.Under thesecircumstances acalculation ofthe Terps engy acne! thachmode, asing theBotsann pobaily ‘Gebaion fcsyem ofequim ottempertre T, He)=5 cay) led B= 5me) eo ©flste narPateEeeeos 1s +Asotin 0teequipeton kvteeny OFNels (nd ie of sone i ee rc) ahe sonny PMT e Se eee ieauth clara ald oevedanta et ieibhhoicoeToeratefortal ae ene aN Near none se rae thee “hoax onlay Sock Ee et ya aape TheLimis ofCasal Phys 7 ve sgbe| ye | =s| 5 =a ou) FThisagreeswit(1-20)providedwemaketheidentification 3 b=h (1-415)FanddonotchangethenumberofmodesPS Planck argued thatforsome unknown teason cheatoms inthewalls oftheONcavityemittedradiationin“quanta” withenergyabr(w=1,2,3,..),but{Consistency demanded, asestablished byEinstein afewyearslater,thatslevtre-mage radiationbeavedatifemiaof4olsonofegypseos© boomy bo Theenergy casted perquantum isextremely small, Forlight intheoptical ange, with, sy,2=6000A, £_665%10"X30x10" 7 .Wehin IO a53K weg sothatthenumberof lightquants oftiswavelength, mized by«1o0-watt source, sy, is 100X108 NmJOM 3X10%quan/sec 33X1-8=3%10"quanca/ Withsomany quanta presen, iispethapt notsuprising thatweJoaotex: perience theputce nature ofightdite; wesaleethaton»mctoscopic Fale nodeviations from classical optics ateexpeced, Neveheles, Planck's imerpretation ofhisfoxmula rally changes ourpicture ofradiation B.The Photoelectric Effect Assuceesfal asthe Planck formula was,checonclusion from itofthe<qantumnateofradiationishardlycompelling. AnimporantcoatsisstionC0itsaceepance ame from thework ofAlbert Einstein, who ia1909 wed the "For ee Geguocy» thee maybeanyteil numberof quit pees end hence copy nse onthewie hemi =Oka 8 Quantum Physics ‘conceptofthequantumnatureoflighttoexplainsomepeculiarpropertiesofmetals, when these ateiradiared with visible andultraviolet light. Tn1887,thephocoelecric effect wasdiscovered byHertz, who,while cogaged inhisfamous experiments onelectromagnetic waves, found thatthe lengeh ofthespark induced inthesecondary circuit wasreduced when the ‘terminals ofthesparkgupwereshielded fromtheulravolet lightcoming fromthesparkintheprimaryctcuit.Hisobservations attractedmuchinterestandthefollowing fscrswereesablshed byfurther experiments: 1.When polished metal plates areirmadiated, theymayemicelectrons"? they donotemit positive ions. 2.Whether theplates emielectrons depends onthewavelength ofthe light.Ingeneral chetewillbeathreshold thacvaries frommetal rometal: only Tightwitha frequency greater thanagiventhreshold frequency willproduce a photoelectric curent. 5.Themagnitude ofthecurrent,whenitexists,isproportional totheincensity ofthelight source. 4.Theenergyofthephotoelectrons isindependent oftheineasicyofthe lightsource butvaties linearly withthefrequency oftheincident light. ‘Although theexistence ofthephotoelectric effect canbeunderstood “within theframework ofclassical electromagnetic theory, since icwasknownthactherewereelectronsinmetals,andonecouldimaginethemtobeaccelerated bbyabsorption ofradiation,thefeequency-dependence oftheeffectisnotcompre Densiblewithinthatframework. Theenergycarriedbyanelectromagnetic waveisproportional totheintensity ofchesource, andfrequency hasnothing todo withic.Furthetmore, aclassical explanation ofthe effect, which would haveco involve theconcentration oftheenergy deposited onsingle photoelectrons,‘wouldcarrywithitanimpliedtimedelaybecweentheartvaloftheradiationandchedeparture oftheelectron, thedelaybeing longer when theintensity is decreased, Infact, noauchtimedelays wereeverobserved, atleastnonelonger than 10-*sec,even with incident radiation ofvery lowintensity instein considered theradiation 10consist ofacollection ofquanta of energy br,where »isthefrequency ofthelight. Theabsorption ofasingle ‘quantum byanelectron—a process thatmaytakelesstimechantheupper limitquoced above—increases theelectron energy byanamount br.Someof J theenergy mustbeexpended toseparate theelectron ftomthemetal. This amoune, W(called theworkfunction), might beexpected tovatyfrommetal to metal, butshould notdepend ontheelectron encrgy. Therestisavailabe for theelectron kinetic energy, sothatonthebasisofthis picture oneexpects that ‘This wasetblae byane/mmenvoreent “TheLimits ofCasical Physics 9 a Foe a 20 a Eos q 3es _— bos Be. os Fs: eg % a : 10we! . a a Rae Fig.1-3. Phoroelctic fectdatashowingaplotofreading« BRpore necenary tostopclecron fowfomameal (ith),GPRorcquivaleny cleczonkineienergy,asfunctionofequencyBEEPoftheincidenelight.Theslopeoftheineish B thefollowing relationbetween electronvelocity #andlightfrequency » - fet=hy a6) should hold. The threshold effect and thelinear relation berween electron FKinetic energy andthefrequency arecontained inthisformula. Thepropor. *—dionaicy ofthecurrent andthesource intensity canalsobeunderstood intems 10 Quantum Physics oftheselightquanca,otphotons,astheycametobecalled:amoreintenselightSourceemitsmorephotons,andtheseinturcanliberatemoreelectrons.Millikan catied outextensive experiments andestablished thecomectness oftheEinstein formula (Fig.1.3).WhatMillkan’s andthecalis experiments proved wasthacsometimes lightbehaves likecollection ofparticles, andthatchese“particles” canactindividually, sothatitispossible tocontemplacetheexistenceofasinglephotonandaskwhatispropertiesare,Aby-productoftheseexperiments wasinformation sboutmecals. IewasfoundthatWwasoftheorderofseveralelecuronvolts(1eV=1.6X10"erg),andthiscouldbecorrelated withother properties ofthemetals C.The Compton Effect “Theexperiment thatprovided chemostdirectevidence forthepercidle nature ofradiation istheso-called Compton effect. Compton discovered tha ‘adiation ofagivenwavelength (intheX-ray region) sentthrough ametal foilwasscattered inamanner notconsistent withclassical radiation theory.‘Accordingtoclassicaltheory,themechasism fortheeffeceisthere-adiation oflighebyelectrons setintoforced oscillations bytheincident sadiation, andthis leads 0theprediction ofintensity observed atanangle @thatvaries as (1+cost6),anddoesnotdepend onthewavelength ofthe incident radiation, Gompton found thactheradiation scattered chrough 4givenangleactuallyconsistsoftwocomponents: onewhosewavelength isthesameasthatofcheJncidentradiation,theotherofwavelength shiftedrelacivecotheincidentwave~Jengthbyanamountthatdependsontheangle(Fig.1.4).Comptonwasableto‘expliin the"modified" component bytreating theincoming radiation asabeam fofphotons ofenergy br,withindividual photons scattering elastically off individual electrons. [aanelastic collision, momentum aswellasenergy must beconserved, andwemustfistassign amomentum tothephoton. Byanalogy ‘withrelaivstic particle kinematics weargue that p=bie aay “Theargument ischavitfollows fromtherelativistic relation berween energy and ‘momentum B=let?+GOT (1-18) ‘wheremstherestmassoftheparticle,tharthevelocityatdhismomentum is a ae aoe .bE Geetpay (es) Foraphotonchisisalways¢andhencethephotonrestmatemustbezee,Thustherelation (1-18) becomes Emp (120) j ‘TheLimitsofClassicalPhysics11ni. Fig. 14. The spectrum ofradiation sat tered bycarbon, showing theunmodified line2€07078Aonthelefandtheshiftedlineat0.7314Aomtheright.Theformeris thewavelength ofthe primary radiation, hich yields (-17) when wesubstitute E=Je.Onemayalsocerve (1-20) fromconsideration oftheenergy andmomentum ofanelectromagnetic wave,buteheanalogy argument issimpler. Consider, now,«photon withinital momentum p,incident upon an slectron atrest.Afterthecollision, thephoton momeaturn isp’andtheelecton secoils withmomentum P,Conservation ofmomentum yields (Fig.1.5) pep+P (21) from which itfollows that Pim (p— p= pt pt app! (22) Energy conservation ceads : Io me=hal+(net+ree 25) where mistheelectron restmass. Hence A$ Pre =(hn—BS+ety f=he~ta")Decthn—he!)+met 4Ontheotherhand(1-22) mayberewritten inthefork we(Y(t ae 12 Quantum Physics ¥, x ‘Scattered° photon oN preg oan ote Fig. 1-5. Kinemasis forCompron eft that is, Pic=(hw—hy')® +2(hw) (ho’) (1—cos8) (124) where ischephoton saceting ange, Thos hr'(1 —cos8)=met(v—¥’) orequivalently vane Racoon (23) “Themeasurements ofthemodified component age verywelwiththe above prediccon, Theuamodified lineispresumably duetothesaerig bythe‘wholeatom;ifisreplacedbythemassoftheatom,theshiftinthewavelengthEvery small since aatom ismany thousends times mote massive Ganan flecn, Thequantity b/me haschedimensions ofleged. Wicalled che Compton wavelength ofthe election, anditemagoitude is wax 1m (26) “Measurements ofcheelectron recoil were also made, andchese areinagreement swiththetheory. twasfurthermore destined bygoodimeresolution cine Tence experimen, tietheoutgoing, phoron andtherecoil electron appeat Ninulaneouly, Thre noquestion ofthecontecress ofthe incerpretation of thecolison asanotdinary “bllard ball” typeofcolon, thais,ofthe Fricke benaviot ofthephoton. Since stdiation asohiswave propertiesretcanis icftence enddifeaction, weeight expect some conceptual dliicales These exe adweshal discuss them atthe endofthe chapeet, 4 “The Limics ofClassical Physics 13 D.Electron Diffraction In1923DeBroglie,guidedbytheanalogyofFermat'sprincipleinoptics, andtheleastaction principle inmechanics, wasledtosuggest chatthedual wwave-particle natureofradiation shouldhaveitscounterpart inadualparticle. ©.wavenatureofmatter,Thusparticlesshouldhavewavepropertiesundercertain fcircumstances, andDeBroglie suggesced anexpression forthewavelength associated withtheparticle." This isgiven by j A a-+ 4.27) ? (27) fwherebisPlanck'sconstant,aadpisthemomentum oftheparticle.DeBrogie's GBBwork accraced much attention, andmany people suggested thatverification Ecould beobtained byobserving electon diftaction." Theexperimental observa- AEP sionofciseffece occured inexperiments ofDavisson andGetmer, whofound GE. tminthescattering ofelecrons byacrystal surface, thete wespreferential BE scancsing incerain directions. HE Figuee 1.6isasimplified picture ofwhat happens. fnthescattering of Fiswaves by2peviodic seructute, there willbeaphase diference between waves ffcoming fromadjacent scattering “planes,” whose magnitude isgiven by G@2x/9) 22sin6.Thete willbeconstructive interference whenever thisphase Bcdifference isequal to2er,whete misaninceger, thatis,when BEBTiciocertrence permoberedinelecuonsein byDavisonandGeer Fig could becorrelated withthe above formula, provided theassociation (1-27) wasBEmade.Thisverification constituted amajorstepinthedevelopment ofwavebymechanics [2 Theparticle diftaction experiments have since been catried outwith Bemoteculas beams ofhydrogen andhelium, andwith slow neutrons. Neucron fedisiaction isparticularly usefulinthestudyofcrystal structure. TogetaroughBeideaofthekindofenergiesneededforthediffractionexperiments, wenoteFEthatthe crystal spacings ateoftheorderofAngstoms. Thegrating constant in Behe Duviston-Germer experiment, iawhich nickel wasused was 213A Henceisoftheorderof10-*em,sothatp=B/ASY66X10"gmcmn/sec. fFThus forelectrons thekinetic energy ispt/2m, =(66 X10-™)}/ j"Chape: 2conans «iscssion ofwave packets inwhich theDeBroglie relation ‘emerges as vey plush ese .“ThehistryoftheveiatonofDeBroglie’ conjecrre canbefdinJue, The Conrad Dement Quantom Mecho 14 Quantum Physics < RY a ee NY oN oe \ Fig. 1.6. Schematic drawing, ofelectron seaering georey (2X0910-%)©25X10"ergs,andforneutronsthekinsicenergyisPriam =(meme) XAelecton energy) %(1/1840) X2.5X0-Hergs&1.3%10°?ergsIntermsofthe more convenient electron volt, these energies areapproximately 160eVand0.08eV,respectively. ‘Onamactoscopic scale, thewave aspects ofparticles arebeyond our ability toobserve them, Adroplet 0.1mminsie,moving at10cm/sec will haveaDeBrogliewavelengthofb=.6.6X10-P/4X10-*1.6X10-*cm, Sincethe“see”ofaproton isabout 10°em,clay chereisnowayinwhich thewavepropesties ofanobject ofdimensions significantly larger than10cmcanbeobserved.Asfortheparticlepropertiesofradiation,iisthesmallnessof2thatdecermingstheclassicalpropertiesinthesensechatthedualaspectsbecomeapparentonlywheatheproductofmomentum anddimensionisoftheorder ofb:Weshalleethatthe formalism ofquantum mechanics describes the Sitution very well, E.The Bohr Atom Experimentscartedoutin1908byGeigerandMarsdenonthexatering ofaparticles bythinfoilsshowed signiicane large angle scattering, torally Inconsistent withexpectations based ontheThomson model oftheatom, cconding towhich elections wereembedded in2continuous distribution of poutine charge. Rutherford propoxed 4newmodel chataccounted forthe dace “TheLimits ofCassia Physics 13, allofthepositivechargeandessentiallyallofthemassoftheatomwerecon-‘eouatedinaregionthatissmallcomparedwiththedimensions oftheatom,thatis,inthenucleus ofthe atom. The electros,attiacted tothenuckus with 4 1force, taveled iaplanetary osbits about it.Alchough themodel explained particle scaering quanciaively, itfaced owoinsuperable dificulces. Since it implied apetiodie motion forthe electrons, itcould notaccount forthe speera ‘ofradiation fomaroms, which didnothavetheexpected harmonic structure«f(cf.2vibrating sing), butinsted hadthestructure L 14 te cone. (4-4 A.sven (l-) ow whete rand 1wete integers. Italsolacked 1mechanism fosailing atoms: fanelcton ina citculaofelipiconbic isconstantlyaccelerating,andaccording Coelectromagnetic theory, should beradiating. Theconstant lossofenergy ‘would, withinaveryshoretime,(oftheoderof10"sec)leadtothecollapse BBoftseatom,withtheelectronsplungingintothenucleusJustewoyearsafterthismodelwasproposed, NielsBobrin1913advanced‘aseries ofpostulates, which, while sharply breaking withclassical theory, “explained chespecial structure andbypassed thestability problem, Bott Lproposed tha E3 1.Theelectrons move inorbits restricted bytherequirement thatthe Ff sogutar momentum beaaintegral muleple ofB/2x, thatis,forcircular orbits of PAP nds, eeclecton velocity #isresected by a ah. a oreo (1-30) acd furthermore cheelectrons intheseomitsdonotradiate inspiteoftheir BA acceleration. They were saidcobeinstationary states. Bi 2.Blecrons canmakeddcontiavous tanstions fromoaeallowed obittoGissvother,andthechangeinenergy,E~2wllappeararadiationwithfrequency ee Bk .e aa 31) FeAnacommayabsorb radiation byhaving itselectons make«canstion toa Eihigher energy orbit. a ‘Theconsequences ofthesepostulates areverysimply deduced forone-clecuonacomssuchashydrogen,singlyionizedhelium,nndsoom,ifwedealwith checircular orbits.'* Ifthenuclear charge isZand thatoftheelectron is ©Whenellipticalorbitsare“allowed, amuchricherstructureemerges.Thiswillbedinced laChee 1 16 Quancum Physics =e,andifcheradiusoftheobitisr,then,cakingthenuclearmasscobeinfinite,wwebalancecheCoulombforceagainstthecentrifugalforce Ze om a0 (1-32) “whencombinedwith(1-30)leadsto2eZ ee (1-33) and 1wht ate a3) Theenergy is 1Ben Bagar—== (135) ‘which,bypostulate(2)immediately leadscochegeneralform(1-29)(Fig.7).Before evaluating these quantities toobtain anidesoftheitmagnitude,‘wewillintroducesomenotationsthatwillbeveryuseful.Firstofall, itis4/2 ratherthanbthatappeatsinmostformulasinquantummechanics. Wethereforedefine AB==103sX10xgec (136) ‘Tokeeptheexpressions fortheenergy simple, weshalldealwiththeangular frequency wrachec then »,where one (37 ‘Thos (1-31) reads E-E on (138) ‘Similatly,thequantumofradiationcarriesenergy Boho (139) Ieisconvenient tointroduce the“reduced wavelength” Ale rade (40) s0thatcheDeBroglie elation reads »-t (tn) : ‘TheLimitsofClassic!Physics 17 FP -astev| 73 b / 3 / os. ap / iESEN~~ ae at wstiTury, oFFk [S_Big.1-7.SpecarumfochydrogenatomasderivedfromBoheatomic BF©modelTheexsrenceofthequantumnumbers!emergefromadiscussion F. ofclic! orbits. Thelines connecting energy levels represect theGomi 2“TheBohrangular momentum suantzation condition reads a mor=nh(w=1,2,3,...) (a2)Ptisslsoveryconvenienttointroducethedimensionless “inestructureconstant” Ga 1 ° 8fe”pres a 18 Quantum Physics hich wewillapprosimate by1/137. Intetms ofchese quantities wefinddhe Inch simpler expressions em oa eon "Zam a) and B=—hae (1-45) [Novicethattheradius,whichhasthedimensionsoflenge,iwreninesOfme,theredacedComptonwavelength oftheelectron,aadthattheenetgy$swuiten intexms ofmeInallatomic calculations weshall expeess ourresultintermsofmc/mc,h/t,andmeforenergy,length,time,andmomentum,respecively. Angular momenta wilalways appear ssmultiples offTetusnowcalculatesomeofchequatiiesthatemergefomcheBohrtheory. Weeelate mn 051 X108eV 051 Mev & 4es9x oem fom isxio (1.46)Bimaxwise . and thus obtain (a)theradius ofthelowest (w=1)Bohr orbit is 137k_053 aonWKOK a (b)thebindingenecgyoftheelectroninthelowestBohrorbit,thatis,theenergy requted topucicin#statewith B=O(cottesponding tor=©)is E=4m! (Za)? =13.62"eV (1-48) “Thus, forexample, acansition fom the w=1saetothen=2stateinhyéro- _gen (Z=1)corresponds toachangeinenergyof13.6(1—3)eV=10.2eV. JTfrequency oftheeaitted radiation cnbecalculated byconverting thisaco cexgs,butitsmoreconvenient coworkthisoutintheform MAD Dacea ah “ge ax ot 15 X10" tad/sec 2 4TheLimits ofCase Physice 19 Equivalently ; «tek de ae eR eo dat me os=12004 F which lies inthe ultraviolet. * ‘Thesuccess oftheBohrtheory withhydrogenlike atoms gavegreat imps tofurther research onthe "Boke tom” Insite ofsomeexesordoany 4.sthicvementsby Boi andother, itwasetthatthetheory naspronsona%fesidnothing aboutwhenthecectons wouldmaketheijumps,asete quantization rolewesresticed toperiodic stems «mote gene serene BySommereld andWison, fpdy=oh 49) oe herepisthemomentumn associa withthecoordinate 4,wasofa0belpin 3)eating problems other thanthose associated withatomic levels ofhydrogen "Brom theBoh theory emerged: 1.Theeorepondee prin, whic, nesenc tates thatclassical physics>/fesultsshouldbecontained aslimitingcasesofquantum mechanical results‘Thelimicshould bereached whenthe“quantum numbers” atelarge,forex- {-aaple, forlarge »intheBohr atom, Once 4content theory ofquanton pesos wsonce, stomsialy cone ealphysnit,butchepicile wasveryhlpal ipucing theoretical gure, adle |Heisenberg tothepoinefomwhichheouldmaehespaneesptoguia Eymechanics, Toillustratehowthecottespondence principleissatisfiedbythe “50Bohratomicmodel,considerthefrequency oftheradiationemittedwhenan“electron makesa"jump"fromtheorbitwithquantumnumber#+1tothei15,Bebicwithquantum numbern,whenmisverylarge.Thisis«gooddomaintoask Fpl tecassical tit,sincetheangular momenta afindeed muchsper gfte06Cascally anclecuonmovinginacrearonimhseloiyvouldbe {GEEcxpeced coraat withthefrequency oftmotion tet 3 = BatLame_Lalnct 1 a OFoer Death ah ont (1-50) {Ontheotherhan,thefrequency ofthe adi associated withthetastonis,according to(1-31), ol tm ftot "See Rove (6), Nitur, North Holand Pubsing, Amen, 967 20 Quantum Physics which approaches veforn>>1.Notethatthisissignificant result, singeics ‘onlythefrequency associated withan1+1—+1transition thatcoeresponds (0 thefundamental classical frequency. Theradiation associated with thejumpfr-+2»hasnoclassicalcounterpart eveninthelarge1limit.WeshallseeinChapter 22thatthereareno#+2» transitions for“circular orbits” in quantum mechanics." 2,Thequantization ofangular momentum heldinothersituations aswel{esapplication toellipticorbitsgaveamorecompletepictureofthespecturmofbydrogenlike atoms, anditwasdiecly observed intheexperiments ofStem and Getlach'* in1922. F.The Wave-Particle Problem “The factthat adiaton exhibits both wave and particle properties raises a deepconceptual difficulty, ascanbeseenfromthefollowing considerations. 1.Ourdiscussion ofthephotoelectric effect, inparticular thecorrelation ofthenumber ofelectrons emitved withcheintensity oftherediation, strongly suggests thattheintensity ofelectromagnetic radiation isproportional tothe ‘number ofphotons emitted bythesource. Letusnowconsider 2Gedanken- “experiment” inwhich radiation isdiffracted by2rwo-slit system. Imagine that theintensity ofthesousce isreduced tochepoint where, oatheaverage, one photon perhourartives atchescreen. Notethatwehavetodealwithentirephotons:astheComptoneffectaswellasthephotoelectric effectshow,itisnotpossible tosplitaphoton intopartswithfeequency wbutenergy essthanfu ‘Thedectease ofintensity incheincident radiation should notaffec: checlassical Jiffaction patter, since, inefect, weareonlystretching outthetienescaleon ‘which thetransmission from thesource tothephotographic plate ofalarge number ofphotons tikes plice. Photons thatcometotheplateanhourapart earlycannotbecortelated,andwemaychereforethinkaboutthisprocessone photon atatime.Aphoton, asaparticle, willpresumably gothtough oneslitor theother, IfweaddtoourGedankenexperiment apparatus «small monitor thac "Such eansitions canoccur forepi orbits (oo:considered hese), andchitis consistent withthe conespondence pipe "These matters aedscosed ianyextbook inmodern physics (eereferences atheendofthischavee. WAGulanbmerperiment (hough exptimest) isovethaxmaybeimagine, has, covethaticonsistent wihtheKnown Inwsofpiss, evenchough itaynotbetech: tically feasible Thus,mesoring theaccelerion dueogravy onhesurface ofchesnBaCedankenespeiment, wrengeesuingtheDopplersiftofsuriasseefom2facetipmoving wthtwicetheveloc ofHetinonsense. InChae: 2wesalsee Howcuefal ovemot bevoinsist onconssency withthehws ofphysks inseing up= Geobasenerseriment { ‘heLimits ofCasal Physics 21 1ellswswhetherchephotonwentthroughslic“1”orslit2,"wecandivideche+Photons intoewoclasses, associated withthecwosis.Forthefistcass,we ‘couldhavecloséddownslit2,sincethephotondidnotgothroughit,forche »second classwecould haveclosed down slit1.Wemight thusexpect thatchefpattemonthephotographic plateshouldbechesameifwerepeatedtheexperi-mentwithoneslitclosedforhalfchetime,andtheotherslitclosedfortheotherhalfofthetime,Ths,however,cannotbe,sincethesecondexperimentdoesnot BBgiveanincerference pattern.Thusthereisaninconsistency thatwillbetracedto theassumption chatchepresence ofthemonitor thattellsuswhich aithe Fphoton went through does notaffect theexperiment. When wediscass the Heienerg ancerainty principle, weshallsethattheation ofthe monitor desoys theinterference pattem, s0thatthereisnoinconsistency. Atthissageitos fuficent topointoutchatwhentheteisnomonitor,exchphotonset4s2 Bfwave, andicdoesnocmakesensetoaskwhich slicthephoton wentthtoughREPresumably, wecanstilspeakofanaverageintensityofradiationateachsie©4,Pthismastmeanhacfrindividulphotonswecanonlyspeakofaprobabilyof25)Boing chrough onesiofsnorber. 32.Thenotionofprobability must againbeinvokedinunderstandingthe passageofpolarized radiation theough ananalyzer. Asiswellkrown,abeaenof ageMediation ofintensity Ipwillbeattenuated to1»cos*a,where «istheangleFPberween theaxisofthepolarizer andthatoftheanalyzer. IntemsofsinglePe phoronsthacateindiible suchanattenvation isonlyexplaiouble ifwestateag#Bie* «givenphocon willeithergothrough orbeblocked bythesystem, witha ‘\Gpebability ofwansmission governed bytheconstruction oftheapparatus,spiis,bytheanglea.Gye >:Inchesameway,considerradiation fromadistantstar.Thestatische EAPsource of»spherical waveofelecromagnetic fieldexcitation, spreading withvelocity .locexms ofindividual photons icisnotsensible tothiak ofthephoton asspreadthinlyoverasphereofradiuscf(whete£isthetimesincethe pwhoton wasemitted), sincethecollapse ofthatphoton toasinglepointonafBhotographic plate,otontheretinaoftheeye,wouldviolatecommonsense,ifit Were"really"happening. Wemayhoweverinterpetthesphericaldistribution axBpiving ustheprobability ofnding«photonatagivensolidangle 4,Sometimes itispossible tointerpret agiven experiment bothin ffarticandinwavelanguage,bocthennonclassical aspectcrepsinelsewherePDicke andWirtke™ haveproposed thefollowing Gedankenexperiment (Fig.4.8)Consideracylindricalbitdcagewiththebarsspacedregulatly,andspacing 7 R enh, B gtBM.Dicke and. P.Wie, intadecinteQuatomMacunis,Adcnoo.Wee, eating, Mats 190 22 Quancum Physics / / Fig.1.8,EndviewofDicke-Wittke “cage”showing‘equally spaced barsandgeometrical quantities con- seceed with it ‘where Ristheradius ofthecylinder andNisthenumber ofbers.Consider ‘adiation emitted fromasource placed ontheaxisofthecylinder. Thebarsact asadiffraction grating. Ifchebeamemerges atanangle @withtheoriginal direction, wehive maximum intensity iftheangle andchewavelength are related by asinQ=mh(HALA) chats, 2aR sind re (152) ‘Wecouldalsointerprettheintensitypeakbyassumingthatheparticlesscatteredthroughanangle@offthebarsofthebitdcage. Themomentum transfered to thecageispsin@and hence cheangular momentum transferred cothecageis L=pRsin@ 53) IfwenowmacetheDeBroglieassociation, p=2xf/hweobxain 2xiNe=NT.Rsin0=Nok 4 Lmee Reind=Ni (54) thatis,angular momentum isquantized! ThefactorNiassociated withthefact thatchebiedcagelooks thesammewhen itisrocated chrough anangle 2m/N, as ‘will become clear later ‘TheLimits ofClassical Physics 23 j,11925themoderntheoryofquantummechanicssarcedwiththeworkof Heisenberg, Botn, Jordan, Schrédinger andDirac, Thistheory providesawayof reconciling alloftheconflicting concepts atthecostofmaking usabandon a ‘ertsin amount ofclassical thinking. Teisoneofhejoysofbeing »student of {physics cobeablecoappreciate thisbeautiful theory andthemonumental sudvances inourunderstanding theproperties ofmarce thatthe theory enabed pus tomake. Problems 1,Prove thetelation (1-1)between theenergy density inacavity andtheHemissivepower[HinTodo50,lookatheigure.Theshadedvolumeclement ® /re b/¥ aray :a 2isofmagnitude r*drsin6ddd=dVwhereristhedistancetotheorigin(at|theapertare ofareadA), isthe angle withthevertical and@itheazimuthal Angle about theperpendicular axisthrough cheopening, Theenergy contained inthevolumeelementisdVmultiplied bytheenergydensity.Theradiationis =sowopic, sothatwha:emerges igiven bythesolidangle dAcos@/Arr* mite F.pliedbytheenergy. Thisisobeintegrated overtheangle #and@andiftheBhpflowofradiationintimeAriswanted,overdrfrom0joctthedistancefom whichtheradiation willexcape inthegiven timeintetal} 2.Use(1-1)and(1-12) co"obtain aformulafor therotalrateofradiation perunieareaofablack body. Assume thatthesunradiates 282black body. 24 Quam Physics ‘YouaregiventheradiusofthesunRo=7X10"cm,theaverage distance of,thesuntoheearthdo=1.5X10"cm,andthesolarconstan,theamountofnergy falling ontheeathwienthesunisoverhead 1.4X10 ergs/cm* sc. Usethisinformation coestimate thesurface temperature ofthesun. 5.Given (19), calculate theenergy density inawavelengeh interval 1X.Useyourexpression tocalculate thevalueof=Roaforwhich this ‘dens ismaximal. ShowchatNousisoftheformB/T,calculate b,anduseyour estimate ofthesun's sucace temperature t0calculate hoaforsolartadiation {itiIncalculating byouwillneedchesolution xoftheequation (5—x)=SesSolvethisgraphically otby@successive approximation method, inwhich youfirswritex=5—6,with1] 4,Howmuchofthesun'senergyisradiatedintherangeofwavelengths ‘4ocoXJo00A?tisetheTesimated iaproblem1.Pottheenergydensityon ‘raph paper toobtain thenumerical result 5,‘There issomeexpetimentil evidence thattheuniverse contains black bodyradiation corresponding toanequilibrium temperature of3°K.Calculate theenergy ofaphoron whose wavelength isNmcorresponding tothistem- peraure. 6.Ultravioletlightofwavelength3500Afallsonapotassiumsurface. “Themeximumenergyofthephotoelectrons i1.6€V.Whatistheworkfunction ofpoussium? 7.Themaximum energy ofphotoclecttons fromaluminum is2.3eVfor radiation of2000A and0.90eVforradiation of3130A.Usethisdatatocalculace Planck's constant andthewodk function ofaluminum. 8.A100 MeVphoton collides withaproton thatisatrest.What isthe smaximam possible energy lossforthephoton? 9,A100keVphocon collides withanelectron atcst.Itisscattered through 90°.Whatitsenergy aftethecollision? Whatichekinetic energy inEVoftheelectronafterthecollision,andwhatisthedirectionoftscoil? 10,Anelecton ofenergy 100MeVcollides withaphoton ofwavelength 510"&(conesponding totheuniversalbackgroundofblackbodyeedation). ‘What ishemaximmurn energy losssuffered bytheelectron? 11,AbeamofXraysisscattered byelecuons atrest.Whatischeenergyof theXraysifchewavelength oftheXraysscattered ar60°tothebeamaxisis 0.035 A? 12.Anitrogen aucleus (mass&14Xproton mass)emits2photon ofenergy62MeV.thenucleusiiiillya¢es,whacistherecoilenergyofthenucleus ineV? 13.What istheDeBroglie wavelength of(a)a1eVelection, (6)a TheLimi ofCasi Physics 28 10MeVproton,(c)«100MeVclecton?(cation!setherelativisticenergyformula), (A)athermal neutron? (defined as&neutron whose kinetic energy 8347/2 with T=300°R) 14,Considercrystalwithplanarspacing3.2A.Whatorderofmagnitude ofenergies would oneneed for(a)electrons, (b)helium nuclei (mass =4X proton mass) toobserve upto3inerference sxe? 15.Thesmallest separation teslwable byamicroscope isoftheorder of magnicade ofthewavelength used. What energy elecons would oneeed in ‘anelectcon microscope toresolve separations of(a)150A,(b)5A?16.Ioneassumesthatin2stationarysateofthehydrogenatomchelection fisintoacircular orbit wihanintegral number ofwavelengths, one careproducetheresusoftheBohrtheory.Werkthisout ql 17.‘Thedistance between adjacent planes inacrystalaretobemeasured, 1Xraysofwavelength0.5Aaredeeetedatanangleof5",whatisthespacing? FAcwha angle wilthesecond maximum cut? 18,UsetheBoh quantization rules tocalculate theenergy levels foca harmonic oscillator, forwhich cheenergy is#1/2m -+mui/2, chati,he forceismu%r, Restrict yourself tocitcular orbits. What istheanalog oftheRydbergformula?Showthatthecomespondence pncpleissatisforallsalves ofthequantum number usedinquaotixing theangular momentum E19, UsetheBohr quantization sues toealculate theenergy states fora ati given by vin=vi(2) Ewith&verylarge.Sketchtheformofthepotentialandshowthattheenergy Frales apploach Ey~Gt 20.Thepower, thati,theenergy radiated byanaceltated charge ¢is dusialy given bythe fola 3 pata38 apis [where aistheacclention. Iaaciculr obie4=+4/r,Calculate thepower ‘diated byanclecton inaBoheontcharacterized bychequantum aumber fWhen aisverylarge, thisshould gee withaproper quantum mechanical result according cothecomrespondence principle 21.Thedeaty rateforanelecuon iaanotit maybedefined tobethe Power radiated, P,divided bytheenesgy emitted inthedecay. UsetheBohttheoryexpressionfortheenergyradiated,endtheexpyssionforPfompeoblem20tocalculate the“contespondence” value ofthedetay rtewhen theelection makes aasiion ftomoebietoorbit —1.What ischevalue ofthis decay racewhen =27(This wllnocagree exacly with thetrvequantum theory 26 Quancum Physics resul,sincethecorrespondence principle willnotholdforsuchsmalvaluesof thequantum pumber) Whatisthedecayratewhenchecanstion isfromanDebitto-anomitw—m?Whaisthelifetime=(decaytae)"?22,Thelassicalenergyofaplanerotatorisgivenby Be wy where Lischeangular momentum and1isthe moment ofinertia. Apply che Bohrquantization ruestoobtain theenergy levelsofcherocato, IftheBobr frequency condition isassumed fortheradiation intransitions fromsates labeled Bytostatebeled bym,showthat(a)thecortespondence principle holds, and(b)caticimplies chaonlyuansitions An=-k1should occur. 125.Molecules somesimes behave likerotators. Ifroacional spectra are ‘characterized byradiation ofwavelength oforder107A,andthisisusedto ‘estimate interatomic distances inamolecule likeHs,what kindofseparaions (ic.A)aceobtained? References F.K.Richtmyer, E,H.Kennatd, andJ.N-Cooper, Inradaction #»Modera Physics, McGraw-Hill, New York, 1960. Robert Mattn Eisberg, Fundamental: ofModern Physics, Wiley, NewYork, 1961 ‘Arthur Beiser,Peripetves ofMadern Physi, McGraw-Hill, NewYork, 1969. ‘JohnD.McGervey, ntradaction 0Modern Physic, Academic Press,NewYork, wn. RobereB. Leighton, Priniples ofModern Physics, McGraw-Hill, NewYork, 1959. Martin Karplus andRichard N.Porter, Atoms andMkener, W.A.Benjamin, ‘New York, 1970 Eyvind H.Wichmann, Quantum Physi, McGraw-Hill, NewYork, 1969. Richard P.Feynman, Robert B,Leighton, andMatthew Sands, TheFeyeman ‘Lectres onPhys, Addison-Wesley, Reading, Mas., 1963. “Theftsefivebooksonthislitcoverthemaintopicsof«sanderdmodemphysics couse, wthvariations iolevelandemphasis, sochtcheyshould allbeomsultedforanortootheoreticalcreatmentofthesubject.Wichmann’s bookfrovides anuncenveneional introduction toquantum cheory. Itsess altheImportantpoints,rangesoveraverywidefieldofquiliaciveapplicasions, andprnvides anewperspective tothereaderwhoalready hassomebackground inFlemubjeet TheFowman Lacars cannoc becharacterized inanysimple way.Theyavebrillant andshould beeadbyeverystudenc, undergradsate and[graduate Instructors aleady koowthat,andeadthemagreatdea chapter 2 Wave Packets and theUncertainty Relations Quotum mecasiy provides uswith anundentnding ofallofthe phenotenn ducal iaCher 1Tesindapenable othe wedensandiag of Toms, molecule, omic nue andsaga ofthese, Wewlspeach the Sry ofquneum mechanics dough teeucdnger equation andtespp,eteinteetation oftssoatonsTheesnowayofderingthisequationFrom casa! pyc, ice esse therea fea pyc cn ony beguessed, which iswhat Schrodinger did,following theeater insights of Derole Wewilmoat hegue somewhat diferent bysong howone righ yf econcle thewave tedpuie prope ofeenonn T'sdifficuletothinkofconfigurations ofparticlesthatsomehow simulate ive behavior. Thisswhythediftaction experiments ofFresnel andYoung ledtoiheumcamous accepofthewateheyofightOntheterhadis .| possible toimagine configurations ofwavesthatarevetylocalized.(Aclapof thunderisanexample ofasuperposition ofwavesleadingtoaneffeclocalized inne)Such totised "weve ches cane scieved byspeponing wes, Fith dtant fegucocic aswel way, sohttheyinfer wah ent cer |ost comlecly oui of given sata elon, Te tech tool foe [doing thisinvolve Fourier integrals, andAppendix Asummarizes themforthe teat who fami mh Tone siesta no does no ist ontate |mac age fm Sencamp, consider thefnction defined by fis)=fahgh)om (aay 28 Quantum Physics ‘Thesealpareoff(x)isgivenbyf”degli)coske,andthisislinearsuper-postonofwavesofwavelength b=21/h,sinceforagiven&chewavetepo-Teesitselfwhenxchangestox+2e/kGhoose hen 2) “Theinegnlcanbedone:with=A—bywehave fo)=fdhgb)6Hoe ete[7 aeta edn wherein thelststepwehavecompleted squues. Iisjustified colee— Tia/2e) =andsulKeeptheintegra alongthetalaxis?Makingvseof oedeo= 0 f Ai 3) weobtain 0)=fEeaeect esf= I 24) “Thefactoreis known as“phase factor,” since|]? =1.Thustheabso: lucesquare offs) pair=Beem es “This function showsapeakingchatcanbeverypronouacedwhenaischotento beamallTeepresents functionlocalizedaboutx=0,with+widthofthetudet 24/2a, sincewhen 2=-£-V/2a, thefunction drops offro1/¢ofspeak Saluc,Thewidth inespace isconlated witdatinspace, Thesquat of¢@) Joafunction peaked abouthywithwideh2/4/22,Thereisareciprocityhere: efanctom svongly localized inxisbroadinhandvicevert,Theproduct of the ewo "widths i 2ahar~taaVin=4 e6) t-teadr fail withbeery ofopen vibes wlhavenowou com cinciog hme ch Wave Packers 29 ‘Theactual value ofthenumerical constant isnotimportant; what matters isthatitisindependent ofa.ThisisgeneralpropertyoffunctionsthacareFouriertransforms ofeach other (Fig. 2.1).Werepresent itbytheformula Arak 2O(1) 0) WhereAxandA&arethe“widths”oftheewodistributions, andweimplyby©O(1)thatthisisanumbertharmaydependonthefonctionsthatwearedealingwith,bucisnotsignificantly smallerthan1.113imposibletomakeBethxandAksmall,Tisisageneralfeatureofwavepackers,butweshallsoonseethatithassome verydeep implications forquantum mechanics. ‘ InEq,2-1weconsidered afunction fl)theismade upofacontinuous superposition ofsimple waves «#*.How willsuchawave packet propagate in ry <$ ° + se Fig. 2-1, Relation becween wave packet and itsFourier transform for square-shaped wave packet 30 Quantum Physics ime?Theanswer tothatdepends onhowsheinva wavesprope. Ia seraeeshallwateforchesimpleplaneware(30calledbecae inlhas8 cal variation inx,Buttin of2)hefxm ane es) Herew=2nvistheangulaefequency.Thequantitybisrelacedtothewae-[eatbyA3e/hsothatwetywatefrtesimplewaveanother form sate oe) irweareconsideingthepropagationofighwaveavacuum,hentheeiUap coon betwee’ vund1/b,snl, »=«/,80th thespewave becomes: itwenowakethesuperposition, withample (8ofhesesimple waves, we aentimeh fla)=fdh(MHP =fixa) (210) “Thisisdhesameshapethatwesatedfiom,excepttatinsteadofbeing lcained sae Ripmowlocaed atf= 0,Thsawavepacketoflightwaves‘opagten itu dserion witveloc ¢,tevelocity ofBight.Srahowever, ateconcerued withwavesthaearesupposed codescribe partie andmemayoe,heer, equhatw=A,Ingene wlbe& function of£,sothat sie)=[aegaeorses eu Forthetimebeing,wedonoeknowwhathefrmof(8is,butweshaltyto sclamine fonstheequiemen thafx)resemble arelymoving casial pileTorus consider awavepacks thatisstrongly localized inspace, abour2 valuefeThiswould comespond coachoice like(22)witharlarge. IeiscrueTei wlboeptsene anfa)sharply localized inxspace, butoueal aninbeease, tadweatearly silying comake inceligent guesses. Sincerheintegral in(2-11)willcenteracound =by,weexpand (4)aboutfy sndassume chat(A)isnotaveryapidly varying function of&.Thuswewrite t ay lg pyF8 ¢ ty=iy+et), 5aH0(2), ‘Thenusingthefom(2-2)fodeinitenes, andwriting k—y=FweBe fxs)=devorfaeottWetrnedar teteileen) (2-43) 7 Wave Packers an P Aside fiomthephase factor infront, chexand#coordinates appearinaformthat suonglysuggests thatthe velocityofpropagation ofthepacketthegroup vec. ist den=(4), ow) [)Thus, defining ee LfPe(Ge), -# ev) we have This isjusttheincegtal thatledco(2-4),sothatreplacing xbyx—#yfanda Coby a+ ir,weger Gxt)=etteoteon ("ergo tateranfen=errno(5) sad theabsolute square ofthis function is This repcseats& wave packet whose pak iaveling withveloc t,butie does nohave«dete width: thequay thatwas at=0nowcomes Ft (B'6/a), thai, tepacket rein, Sincethewidth isproporsonel to 4 (0+)"=va(i4™) theraeofspreading willbe smalifisage,shatis,iFehepacket isspatially lage tobegin with. Themostimportant result shatif(2-11) isvorepresene aparticle with ‘momentum pand knee energy p/2m, thenwemust require chat os . naBet 19) . *Thisiscertainlyinagreementwithwhatwefoundinthespecialcaseoflightpropa- son, wise =ErThesun el nepeel pesinenshe eeekayaker eeobewoepusfrwavsinioomans octneea 5 wees(4)eo 52 Quan Pryscs IFwefuermakedeasocitonthat B= he (9) suggested bythequnatom sation formltion, 0shat ontanh (2-20) thenconsistency demands chatwemaketh ssocition ame .b=x& (2-21) fistderived inasomewhat similar waybyDeBroglie Tercin opihe expsion (211) eambeSowien inthefor 1 sp)etree ¥ vd=Fagfoon en) “Thewave packer ¥(x)isageneral solution oftheparil differential equation BD5[apa neers wet [rane 1 PFpornoFalouker __ Bed)-- Bes (23) provided, wehvedoneabove, wedescribe themotion ofthe “patie” in pantalicregion,whereB=/2m.Icshisequation,andgenerationaera festale moving intpeal, tarepens theimportant abstraction fromthearguments outlined above. Itshould bestressed thatthe wenn repens gene thetwasnojostcaton ontebassofcsi sis Yoru placement ofwbyEmo forthe eplacerent ofthe wave oumber £byp/f.TrnacethetaofthespreadingofthewavepacksIfwe‘consider aGausian packet (2-17), weseethatnomacter howlargeis,derewill Seni cn thespreding wilbecome notes. Thisfiocondicion Necicmnc, which shows verycea thrnucle, frexample acarey aytaken changed dunge paid of310!yas(10sc).Wealset eee shar aenouns ofpoke, hited atinChapter ya oeeraeingly seers oagoowing probly thtCeparle fiom whee twas fend a= 0 Ba eraporantguative observations thatwemade inour Uncertainty Relations 33 wave packer discussion isthe reciprocity relation between thewidths inx-and space Akar zt (2-24) Tfwemultiply thisbyAanduseA=p,weobtaintheHeisenberg uncertineyrelations dpaxzh (228) Sincethewidthrepresents aregion inwhich aparticle islikeytobeinx-space ot : inmomentum space, (2-25) states tharifweeytoconstruct ahighly localized wavepacket inx-space, thenicisimpossible woassociate awell-defined momen-‘cumwichit,incontrastwithwhatitakenforgrantedinclassicalphysics.Bythe f.same token, wave packer characterized byamomentum defined within nattow limits muscbe spatially verybroad. Thislimitation isonethatisimposedonthe lassidescription, which insists onbeing abletospecify bothpostion and momentum. Inquantum physics position andmomentum, justlikeparticle bchavior andwaveaspects ofasystem, arecomplementary properties ofthe system, andthetheory doesnotadmit thepossibilty ofanexperimentinwhich bothcouldbeestablished simultancously. Thesmallness offguarantees that onlyformicroscopic systems willtheusualnotions ofclassical physics fail.For f¢ ample,foradustparticleofmass10~*gmmovingwithavelocityof10!- Fem/see withanuncertainty intheproduct ofonepartinamillion impliespAp~ 10+ andchusAx~10-*ce,whichis10-7timessmallerchantheradivs J (of&proton!ThisisnotsoforanelectoninaBobsorbit.ItwetakeAp~p~ *mca/a,thenAx~fin/mca,ofthe order ofmagnitude oftheradii oftheorbits, TnwhatfollowswewilldiscussaoumberofGedankenexperiments in Fwhich wewillshowindetail howthewave-parcicle duality acs€0conspite Prohibit aviolation ofcheeelation (2-25) (2)Measurement ofpotions ofanelectron. Consider theexperimental setup . inFig. 22,whose purpose istomeasure chepostion ofanelectron. Theelec.tonsareinbeamhavingwell-defined momentum p,andmovinginthepositive |*direction,Themicroscope (lens+screen)istobeusedtoseewheretheelectwonislocated byobserving chelightthaisseattered offtheclecuon. Weshine Highealong chenegacive x-axis; 2particular electron willsetter aputiculat Photon, andthelatterrecoils through themicroscope. Theresolution ofthe microscope, thais,theprecision with which theelecron canbelocabized is known from optics. eis rN . one 26) where disthewavelength ofthelight. Iewouldappear thatbymaking dsmallenough, andbymaking sinlarge,Ax‘anbemadeassmalasdesired.This,wewillnowshow,canonlybedoneatthe 34 Quanum Physic semen 7Ny AN an an a hy \ / \ ¥\|\ \ |e,7 i \\/ — eT en Fig.2-2.Schematic dawing oftheHeteabarg miaonepe forte Fmesuement ofeon putin. expenseoflosingifornaionabouttes-componet oftheeltonmomen:Seeorton erytle hatwhategies onthesteenbxhid dheTentseairraidea photons taegotdereBeste theyseed oftheAceeast ction ofthephowon atersatering isandetenmined within che Tegsobended Sytheapetre Hencethemapaitude oftherecoilmomeatamofteeconisuocerinby : apn2!sine am Hence by spean~2Maneomtt oan Canwegetaround thisdifficulty? Afterall, thedirection ofchephoton is .Carcted wihnemomeneu, aadifwecouldsomehow measur therecoiloftheseer,wecouldspaythephoton(andbeaceleeton)momentumbee.Teeterns weingode themioscope usparofthe “observed” syste, we srePengabout Toston. ineismomentum istobespecied. Dutthereeonytoomustoytheuncertaintyrelation,anditsmomen8ccases pede pouionwilbelescetemneTea!"dasaobservation Peaswilaleaybeaclwiththeindeterminacy (0)Theses apne InCaper1vesuggestedthatheinveferencepaces, pened inthepsage ofanstint eeough twositswaslogy seeacy Scone photon, atheify ene foreons which ease local Unceninty Reitons 35 f_incompuible withourbeingablecoknowwhichslictheelectron wenttheough,4ssuchkaowledge would implythatthe pattern isasuperposition ofclecooegcoming, fomoneslitortheother.This,however, cantor giveaninterne pacer. Wemayusetheunceruincy relation toshowthat2"moniot” shee‘denies theslsofpassage willdestroy theinterference pattern, Letthesheebeseparated by2distince aandletchedistance fromtheslestothestewbe [Thecondition forconstructive interference i rN sin8=a (2-29) ; $0thatthedistance berween adjacent maxima onthescreenisdsinys—; 4sia8s=d/a.Consider,nowamonitorthatdetermines thepostionofan electronjustbehindthescreencoanaccuracy Ay<a/2,chats,ittellsvswhich estheelection wentthrough (Fig.23).Indoingso,itmuseimpente thelectron 2momentum intheydirection whoseamount isimprecise ch ap>A 220) Hence i Me2h aaypoapoia Suchanuncetintyintroducesanindeterminacy inthepostonoftheelectionatthestem,whosemagaizue is24/e,ttheverylensThis,however apes | / FFig.2.3. Therwosis experiment withmonitor 36 Quantum Physics thanthespacing berween maxima, s0veconclude chaaworking monic willTnecattheimeference paver, andtheeis90logicalcontsdicion, Con-Menta,wecould,ofcouseatguethatlogicconsxencydemandedthat pyby>he (2-32) (6)Thera ofobitsintheBabato.soodinChapter theBohratomic mnode!denswithorbitswhoseradiiaregivenbyRy=Fite, Thasan “experiment designed tomeasure theoutlines of»givenorbitmustbesuchthataFrotion measutement oftheelection inthearomisdonewithanaccarcy 2iin Ax@Ry—ReSe +(233) ‘Thisimpliesanuncontrollable momentum cansferrotheelectron chatisofimagnitale Ap>>mea/2x. Thisimpliesanuncertainty intheencigyofthe <decton ofmagnicude PP metoe_Liateel 034) shais,muchlargerthan-the binding oftheelectron iatheorbit,Thussuch&concutementaselyasnot,wllkicktheelectionoutofthe otbit, sothat no ‘suchmapping, oftheorbieispossibleTd,Tbeerytimencrtiny relation. TFwetaketheelation (2-25)and vice itindheform pap Axm 5 =P semayinterpret chefstfactora8ameasureoftheuncertainintheenergyof thesystem, andthesecoad factor,Ax/2,a8ameasure ofA,anwncertainty iicsJocaltebilcy intime.Thissoggess theenergy-time uncertainty relation DES ZA (2-39) Sacharelationmightalsobededucedfromtheformofthewavepacker(2-22)GoceEand1appeat inthesirereciprocal elation a8pandx,anditi80tagesed bythetheoryofreais, sincespaceandtime,tkemomenta andcretgyaetuimacly connected witheachotber®Actually, innonrelativistic Slum mechanics, spaceandtimeplaasomewhatdiferentrole,ndwhetas Sentjal beabletoderive(2-25)ftomcheformalism ofquantum mechanics !Thisaotueof(2-35),Nevertheless theeneegy-time uncertaincy relation ia5 | Thc spatofthequalitative seuctare ofquantuz: mechanics a8(2-25) “neck (ay#)and(Ble»)aefora thtafr among shemales wnder Uncerainey Relaions 37 Iaspiteofhisfandamental concrbutions tochedevelopment ofquantummechanics, Einstein always fleuneasy houitsimplication. andaheSolvayCongress of930°hesuggested aGedankenexperiment thatapparently avoidedLhe limitations suggested by(2-35). Einstein suggested that«boxcontainingsadiation haveashutter concolled byaclock within thebox.Theshuctes ‘mechanism couldbeatranged suchthiholeisopened forenasbitailyshortimeds,Theenergy ofthephoton escaping fromtheboxcould bedetermined veryaccutzely byweighing theboxbefore andaftercheopening oftheshuter, Bohr’srebuttaloftheargument i«beauifulillustaion oftherequ. ‘mentthataGedankenexperiment mustconform tothelawsofphysics.Taking intoconsidertion theappartus showainFig.2-4,Bohrmadethefollowingpoints: 1.Aweighing impliesthereadingof«scalepointerwithanaccuracy Ax: ‘Thisimplies anuncerangyinthemomencam oftheboxpivenby2p=WeE__2.Hachangeofmassdvistobedetected,theweighingmusttakeatimeT,chatislongenough sothatthe impulse duetothechange iamace,thai74m(g=acceleration duetogravity) ismuchlargerthanAp,thatis, aTAm>f/x (236) 3.Thewellesablished egeivalene frincip! implies thatachange inthe vertical position Axinagraviaional Geldimplies achange inthenateofcheclock, given by at_pax7s (237) FThis yields a “ary gf: T” gdm thats, AmeAT=a8AT>f (239) ‘Thisshows thastheenergy-time uncertancy elacon ismaintained ~Theuncertainty celatons maybeusedtomakeoughmurmeical estimates {inmicroscopic physics. Letusillustrate chiswithseveral examples, chefistof “SchbenenaybyNicBob,“DiscusionyeFst0nEpsemologieatProbl inAmie Physi” whichappl nimi? PopseeaesReJohn Wiley &Sons(1958). . "Theeuirlece princi idscused itheSpeciTopicsecon2attheendof thishookTeimang ithecontent thatthe pra oisResadeed eyeee 38 Quamum Physics a je) = S. te & SS|besLeesSee SS ig.24,usec drovingofninxpsient designed sowFe ry aon,Repinced foeNisBok,onicPotTei tap JokaWily0958,bypeision ofNorthHolind Pebtating Company, Amato hichistheRydoge om.1wesayhteleans pooiseceseen smnncem, then,ifrisisadacoordinate prow (2-39) pee-£ Senor Bi Uncerainty Relations 39 : _B oe “3-5 (2-40) The minimum value oftheenctgy iobtained from 4 2 Re 3 or wt am Batis aa wokay rake a (241) WP andchecoesponding valveofEis Ee=-;mice? (2-42) ‘Thefacethatweobtained theexactvalueofthe energy is,ofcourse, «swindle, sincewecouldequallywellhavewrittenr~binsteadof(2:39),andwewould Bthenhiveobtained «diferent result. Thevalue ofEwould, however, have G—difered fromthecorece valueonlyby«nuinetical constant, ndthegeneralBorderofmagnicudewouldstilbavebeenthesame.Themainpointthatinp,concast coclassical cheory, theenergy ishounded from below because ofthe +uncertainty principle: anincrease inthe(negative) poteatil energy, obtained bybydecreasing r,chatis,localizing theelection coser tothe nucleus cuties with Hi,icthenecessity forincreasing thekinetic energy. Fe Asanother cxumple, consider theproblem ofnuclear forces. These haveBeftherangeoftheorderofonefermi,chatis,10-"cm,Thisimpliesthatp~hr Bg 30°gmcm/sec. Thekinetic energy corresponding tothismomentum i +3 7 sags .‘_ : Int~pacigPKee (2-43) j.whete Aisthenucleon (proton ofnewton) mass,whichis.6X10-¥gm,ESincethepotential chatgivesrisetothebindiag mustmorethie.compensateforthiswerequire that IP]~3X10-%ergs~20Mev (2.44) Aptio, thisisonly4cough onderofmagnicude, buticdoesindicate chatthe Potential encrgy istobemeasured inMeVrater thanineV,asinstom ‘Yetanotherillustration comesfromtheYukawamesontheoryofnuclear forces.In1935Yukawa proposed thatthenuclap forcearsescirough the[emission ofanewquaotum, thep-meson (alsocaledpion), byoneofthe ‘uccons, anditsabsorption bythe other." Ithemassofthequantum idenoted "Tis idiscussed beefy intheSpeci Topics section 5ontheYukawa then. 40 Quanam Physics bywsthenitsemission ineroduces ancarey imbslance AE~yf,which as Caitakeplacefortime3°~B/E~Rin. Therange comespondiag t= paride eaveling forthstimeisoftheorder ofAT~F/ucIfwetakeforthefange n=1.4% 10 cm,thenweSandthat,fe_1X3 108we axe ~0Mev en) When thepionwasGnlly discovered, itwasfound thatthisestimate was rematkably accu, sincefrthepionct 140MEV, Tnsmary outtenetve atempe towedwave andparticle properties consent mithexponent inhenosaie way,hasledusCoan unceasing se'the description oftfomic phenomena atthecsc level, odhisun- Ccrzinty isbothnecessary foraconsistent description of(Gedaaken) experi tment, tndinaccord with what isobserved Problems 1.Consider awavepacket defined by(2-1)withg()givenby g{k) =0 k<-K =N -Kck <x =o K<k (a)Findtheformf(x).(b)Find chevalue ofNforwhich +flap a (6How isthiselated tothe choice ofNforwhich flaigor=3 ~~s7(a)ShowthatareasonabledefinitionofSxforyouranswerto(2yes akax>a independen ofthevalueofK. 2Given that x W=5%5 Uncerainty Relations 41 ‘aleulate theformoff(x).Again, plotthetwofunctions andshowchat Abar> 1 independent ofyourchoice ofa. 3,ConsidertheproblemofthespreadingofGaussianwavepacketfora ficeparticle, where therelation ie : = holds.Use(2-17)toealeulate thefactional change inthesizeofthewavepacker inonesecond, if (@)thepackerrepresents anelecron, withthewavepackethavingasize (of10cm;10-cm, ©)thepackecrepresents anobjectofmass1geandhassize1cm, q 1.willbeconvenieac toexpress thewidthinunitsoffi/me,whetemistheamassofthepatile represented bythepecker |ofanc:4beamofelecuons iscobefiredoveradistance of10*km,Ifthesize oftheinitialpackeris1mm,whatwillbetssizeuponaftival,ifitskinetic U1,energyis(a)13.6eV,(6)100MeV? (Cention. Therelation between kinetic encigy andmomentum isnot‘always KE. =p*/2m!) >Therelationbetweenthewavelength andthefrequency inawave guide isgiven by 4 de Vie What isthe group velocity ofsuchwaves? “ema FOsunfcetension wavesinshallow water,thefelation betweenfrequency andwavelength isgivenby ~ phereTsthesurfacetensionandpchedensity.Whatisthegroupvelocityofthewaves,anditsrelation tothephasevelocity, defined toberr—-ds?For‘ravisy waves(deepwater), therelation isgivenby ye(4)y" 27 Gn, Whar arethegroup andphase velocities? 42 Quancurn Physics 1.Usetheuncertainty relation toestimate theground stateenergy ofaharmonic oscillator. Theenergy isgiven by aS matepaEm 8,Usethevalue ofche“lifetime” pfanelecton inan#=2Bobrobit, calculated inProblem 21ofChapter 1,t0estimate theuncertaincy inche nergyofthe#=2energylevel.Howdocsiccomparewiththeenergyofthat level? 9.Nuclei, typically ofsize10cm,frequently emicelectrons, with typicalenergies of1-10MeV.Usetheuncertainty principle toshowthatCheccrons ofeneigy 1MeVcouldnorbecontained intheucleus before the decay. 10,Theapparatus sketched belowappears coallowaviolation ofche uncertainty relation. Thelateral locacion canbedetermined withaccuracy[ay~a,andtheqansvetse momentum oftheincident beamcanbemadeasSmallaspossible bymaking /atbiarly large.Analyze theappanitus indeel,pointoutthehiddenassumptions madeintheabove,andshowthateheun- certainty relation isnotviolated. Stetganeone nn : ied References ‘Wavepacketsaediscussedinanumberoftextbooks,Mostusefultehlevel §.Borowite,Fundamentals ofWareMechanis, W.A.Benjamin,Inc.,NewYork,1967, { ‘Uncertainty Relations 4a J-1-Powel!andB.Casemuno,QuantMechanis,Adison Wesley,Rendng, Mas, 36 SD. Bohm, Quantum Ther, Pretce-Hll, Englewood Clif,NJ,1951 4Allcextbooks omquantum mechanics necessarily dealwiththeuncertainty: telations. Verythorough discussions canbefoundinthebookbyBohmcitedie above,andinW.Heisenberg, ThePhysicalPrinciple:oftheQuantumTheory, © Dover Pablientiona, Ines 1938 #_iscssons ofthe ncertainytelatons mayalobefoundinanyofthemoee4)2advanced bookslistedattheendofthisbook. eg 13 | |chapter3 q | The Schrodinger Wave Equation A ’ InGuper 2weobsine! aparal ifcentileqution sed by« FNavepacker that,within cemain approximations, described afeclymoving“jie.” From thispont weane tasnga QWs) A ns) 4 amOe omOxt oy |asthe cone egution forthedscipton of«feepanicle, Tvening theSequencethatedto(2-23),weseethatthemostgeneralsolutionofthis equation 1 ¥08)=Fe[ooo 62) {The reason forchenotmaization factorinfontofcheintegralappeatin(3.26}) Beforetoningothecialpoiofierpeing hemoet ae Vix9)ofthisequation,wedrawattentiontothefactthatcheequationisoffst ‘orderinthetime-derivacive, Thisimpliesthaoncetheintialvalueof,namely,¥(<0), isgiven, itsvalues atallochertimescanbefound, Thisisevident frog thefrmoftheequation ssseenbytdi compact 4 5ihMen) Wes+Os)=Hes)+tmet (G3) «from theformofthemosegeneralsolution. Given¥(x0),thefonction 4(f)maybefoundfrom(3-2),with#=0; fi ves)=Fh [ap9g) em o4) maybeinverted,andonce@(f)isknown,thesolutionisknownforallvilues aqForadiscretemesh,a(x,#)/0¥mustbereplacedby(W(x,¢+as)—Hef)|/8,wie 46 Quantum Physics oftNotethatchesisao“uncertingy’" inthediferetiat equation: oncethesetatteofthewavepacket isspecifed-and thereare,0ft,nofestitions voylco)_then thatwavepacker iscompletely speciied aallaterces.Taseuchingforaminerpretaionfotyx,)wemustbearinmind(1)that ‘v(x,i)isingeneral acomplex function [e.g(2-16)}, and(2)chattheFunctionTote,1islargewheretheparle issupposed <obe,andsmallelewhete, Teree Socated withithefeature ofspreading, discussed inChapter 2. “Thesuggestion ofMaxBornthat Pls, de=[Wee1de 65) efitheproailiysa theparticle decribed bythewavefumcion Ye,8)mayJoondbetwenand3+dxfme205out0providethecoectincrpeestionsthe wavefunction. Theprobability density P(x,¢)isreal,islargewherethe Ganicteisupposed tobeandisspreading doetnotimplythatpacarvrei speuding allcansisthatasCiegoesbyoneisJesslikelyco Findtheparticle where oneputiat =0. Torthisinterpretation rohold,wemustrequire thet [lrmodne oo sincethepaticlemustbesomewhere. Ilinearequationlik(51),thesolutionJen) aybemulpled byaconstant, andistlremains asolution. Thus{BgrencethesolutionsYs1)eadassoffonctionsthatatesuareintegrableWeshalscebelow thatiisenough corequirg chat [Laweor< * on hacisteinitalstatewerefunctions mustbesquare isgrable. Whaninfinite ineegrcion ineral eismeansthatYe,0)mustgoC0eroainfty ateasafists2where«canbeacbitay soull,bcmustbepostive, Weshal alsorequze thatthewavefunctions (x,)becontinuous inx.‘Since|¥4x,|?isthephysically significant quancicy, iwould appear that thephasofthesolution oftheequation issomehow wnimportant, Thatiswrong! Sincetheequation (3-1)islinear,if¥a(x,#)andy2(x,#)aresolutions, ois HesA)aay, +cbs) 68) hereayandayateasbitary complex numbers, Clenlysheabsolote squareof‘Yee,2)ia(8)willdependcrucially ontherelativephasesoftheewopars.A |Weoephysical wayofsingthsitonotechata3inasia optics,theineTeeseperaideerme bythephaseeelation berween thetwopatsofthesecre associated withthetwoatsioatwositexpetiment. Ici,of Tousse, tnthatanoverall pate factor canbeignored. 4 TheSchrddinger WaveEquation 47 C‘Wenowshowthatthecondition (3-6),imposed ats—0,holdstrueforal B times, WeneedEq.3-1andiscomplex conjugate 2) HDamOEO_ BOvren a9) P Now is 2ryMy yhe BP yaye “Bs =i(#6-av®) ag ~aieceeanae 4--2ftae) 4 —_EsGOe”eY, 16 wedeinechefby 4 ioe)=<B(ye2bOH Fa ionFe(eHy) 10) Be weseeat a a 2 , when Dien =o con) B tocegracng, wefindthat a : Sf rn=— fa?Hena 6. a Sf rea[a2 inno jaa) BR sinceforsquareinegeabe fanctions, j(%#)venishes atinfinity. Jncidenally,hadweallowed discontinuities in4s),wewouldhavebeenledtodekaane, tion®singulris intheflux,andhenceintheprobability density,which J,Noscepaable inaphysilly dservable quay,“hereaion(511)saconservation lwexpressestheicethat«change E)inthedeasiy inaregioninxiscompensted byaneechange infsinteteeHexion are +42alern==faio Tae) ~10,1) 69) *SeeAppeniiAfdcoofdelafuntion 48 Quantum Physics “Thedefnition ofP(x2),ie#andtheconservation lawaemaintsind ifthe quaton (3-1) ischoged £0 uted) __BDY) ; ihuoomOx?+VX)Hx,8G14) providedthatV(x)smal.Thisisimportant,sincewewilegueachae1) Priehroogee equation frepace inporental V(x),Thegeneralization weGhee dimension isstighsorward, Bq,314becomes a cr £(e =2)a a odce ao $Vb7,2)¥en) hati, aH) BeyMSO=—Sova+rode 649) andthegeneralization of(3-11) teads 2rte.)+vin)=0 (16 where Pee.) =MeN 61 and ite.)=Fee. rv.)—Weve} G8) ste.=Fore. wHle.9—WAGHe) Giventheprobability Jeasity Px, expecation valuesoffunctionsof=raybecalealtedIngeneral,wehave? yo=fasePe=[arennavan ei “Thisonlyasmeaning fcheincega! converges, Theexpression docsnothelptsifwewantcocalculate theexpectation valueofthemomentum, sincewedowerittowhowtomitemomentumintermsofx.Wecythefollowing:since hss, pom e 620) forafindiscrete pace”wihproba9.501theean veiney icetbesecsnO=Bhi q ‘TheSchwingerWaveHguition 49‘ weshall write qawtant )) : == 2genaWe,) 21) 4 This yields a wmf a 1OFj )afer ve) ABH Nowethasthereisnods/drundertheincre sign.Theonlyquaatity hatEg weswitciney(s,0;andwnhayrunonthgiesoseclo EB)withcme.Making weof(1)andtscompley conjugate, wehave 4 a PY a 4(=[a(Bawve2) f Now § OM" 2 (Hy) HO; ae (e)ae¥~On“ox 4 2for a 2Ob .4 -E(%9)-Lewer™ .4 2yr)4yeyya -ae) +detae BEHencetheincegrand hastheform 3 (ov 1—oy)eaea‘$0that ; & ;(=favre ®2very ex f._tincetheinregral ofthedesivatives vanishes frsquare integrable functions. Thissuggests theethemomentum isrepresented bytheoerases q Ao onyox (3-23) andchat,mocegenerally , ? fh 10=favos(%2400 oe 50 Quantum Physics ‘Armed withthisrepresentation wecannowdiscuss thephysical sig-' ilicance of6),whichappears in(3-2).Fist,iissufficient toconsider thatequation at2=0since(9)doesnothaveanytimedependence, With L penn [E ae vo)=ggfpenen=VE[ame weid,wsngtheinversion formula for«Fost ince, hat 1Hie)=Fagfevene | sacs, 1 sipaik . #0)=Seg[ave 29 Now favnun- [200Taeverome [ 1 *(“ipsyh - -[averFafaewe =fevwe=1 G-26) “Thiseuisknown a8Para hoe intemahemaia erate, Kestates Beefafenton wnotmalaed to15018 tsFouter eansfoun. ‘New conider orefavert ae ad 1 A(x) tel=favnrrgfavo =Jdpp)pa*(p) 3-27) hiselecoger with(6.26), stongly suggests eat4(9)shoud beincepretedaschewavefunction inmomentum space,with|6(p)|*yielding cheey eonforngtepewihmore .Whenxs)84ree ay wemaydebe 6,0) by TheScddiger Wave Egution $1 E. Wx,1)=vale 2,4) (3-28) |The facethatingeneral9p,4)hasavimedependence doesnoechange(3-26),6-27),oFtsinterpretation. Lestthereaderthikchatinspite ofthissyametryFberween x-andpspace, p=(W/)(0/2x) isanopertor, ind=isnov,weaoe,hanintanoperCo,Leappensroe+purlsingeaoe space,burlwewant alee Gi) immomentum spies thenSenjshow bymethods verynla othe ones atedshove thar a tno=farsra.ns(a2)an.0 6.29) Inotberwot, theopeioe hastherepesattion 2 eo : Ey(3-30) inmoneata sce ‘Wewilndthatopto paycenaraeinquttum rehanis, and wewilllowly len«gr deatabout them. Attspotmewlcsay a 1Inconst tooriary number, oper donotalvapscommute, bi weSe (4m) =aB— aa oan pythen BDicyaeROMY Tp.x]ve)=7agMHA x7Ow y a . =FWx,4) (G32) Fh is,weave hecma lain 5 [ax]=7 (3-33) Thsedsoanambiguity inwascbing hssfnctonfp)toopener form,andweshalladope therulechatfs,p)besymmetzed iaxand The Hp +ps) Peet ope+mt) o30 tod 0.0 FLaerwewiletaitheckBFcommatvoandptsandsend theunceraint eations connecting ches woaes 52 Quan Physics 2.Theappearance ofthe opertor wihisJ,might leadwstowont bouthetayoftheexpecaton valueofp.Wecn,howevet, checkthefat thatpist. Wehave a hoyw=or=faveaM-[xvo(- ae) ot 2,oAfae %+¥y) =Afad ww 35) =o provided thewavefancon vanishes atinfinity, which idoesforasquat Resale fancin. Sometimes onhasocasion tas Fntions thaafnot sme imegble Butthtavecoin perioicy condions, orexample, Vx)=Hx+1) 3-36) Ifone restricts onself toworking intheregion 0<x<L,then fi/idjdeis still 4hermitian operator, since in(3-35), . Ape or EfaSwreHen A=Fini ivor0 G37) ‘Aopetor whose expectation valefralldmsible wavefunction ielis and erin seo, ssots, ike isaheritan operon Weconclude thischapter byning hatheequation Qh Med ae Ost ay,withtheidentification (8/)(2/2x) =Puybewrittenintheform 4 240) _Ponyey, : ifOo leom) (3-38) ‘Theopeitoron theightisjustheenergy frafeepace wegene | thiscospsi inpore, wewate one. [Pe |b,2 . ay Law+PO)]Hed G39) “sone mateo background onopto cael Append B # TMesehodags WaveEouson 53 ot, moreexplicitly BE) TOMY Lrsee PEO END reYe)(40) ‘Tisequation, generalising (51),ithebaieequation ofnnmlatite quent5mehani,anditwasfrstproposedbySchrdinger TheSchodinges oecenn FSobvainedabove,canalsobeweteninthefox 4 ED ary, (41) (whore 11stheenergyeperatr, H'scommonly calledtheHamilonian, because iF_i8amopeatorversionofthedascalmechanicalHamionianfunction,tne’p S45 isabermitian operator, soisandtherefore soi 4 u=F ym 2)4ifV(x)isatealpotential.4 Jnsummary: 1.Thetimedependence ofwavefunctionsisgivenbythefstorder pani dierevial equation a nDi465,Be STwhere Histheopertr p/2m +Vs). 2.Wavefunctions ateresticted tosquat integrable functions, 3.Theprobability density forfinding theparticle atxis P= |? 4.Thefunction 6p,1)defined by 4 pelt He=Te[amen isthewaefunction inmomentum space,andthepbabiiy denyfrfindingtheparticlewithmomencum pisg(p,5.Themomentum pandthepostion xaeoperas, thati,theyatesiamtis dardiferfromnumbers beste ofthe lackofomenunyonyInx-space, themomentum operator takestheform -42POF oe 4 *Fromnowonwewilldropthesubscript¢pOnpy.Wewilluseitonlywhenthereis Angerofcotisionwthxeubeseerheeer 54 Quantum Physics andinpspace, chexoperator akestheform a zeae Py bothconsistent withthefundamental commutation relation forxwith ‘ bal=> Wearenowreadyfor«quantitative discussion ofquantum mechanics. Wehaveabandoned thenotion ofawavepacketa8represencing aparle.Thisresineashelpful tousinmaking theScidinger equation plausible, but08seefor) antitspeobubisc interpretation catelluswherecheparticle i,vvithourthepaticlebeingthoughtofas“madeupostofwaves.” Problems .1.Use(32)and(5-4)towetethesolutionofthefreparticleScrBdinger equation intheform Mea)=fde!Kx,YE,0) ‘Obra acepresenaton forKlx,x)inthefomofanintegral, andevaluate theintegral. Show chat R(x,;0) =a= *) 2,Showthattheconsrvatiod Jaw(3-12) holdswhenYx4)«solution oftheScotinger equation withapotential V(x),(3-14), provided tharVs) iseal. 5,Suppose hat(siscomplex. Obtainanexpression fordP(Brand spaf5,9,Fxaaoin, heeststbeage,Wedosi asabout VO? 4,Consider theKlein-Goedon equation OME) MD (HY09= 4MD_+(4)wwa-e showthatthereisconservation lawoftheform(5-1)givenchats #)has the form pw (ye ye “TheScbdingee Wave Equation 55 What isthe formofP(x,1)?Canyougiveanargument forwhytheKleine Gordon equation isnot«goodcandidate foraone-particle equation (Le,an alcernatve fortheSchrodinger equation)? q 5.Given that ; w=(Byewe be calculace we)7 (b)VG)—P=ae 6.Calculce themomentum spacewavefunctionforthesystemdescribed bythewave fonction inproblem 5.Useitocalculate @Gy _ ©)VG) —GP=apaleulacethevalueofAxApusingtheabove,andtheresuleofproblem3(b). 7.Given thewave fonction N Ho)= WO=sya : (@)Calculare Nneeded tonormalize (x) (©)Usetheabove wave function tocalcula (a).What values ofead toconvergent integrals? (©)Galeulate (p*)directly, andusingthemomentum spacewavefunction. (@) Use thedefinitions ax= Vie) = p= VE) OF «0calculate Ax4pforthisproblem 8.Show thactheoperator elation holds. Theoperator isdefined tobe aS ait (ine.Calculate"xe fp)whecefp)isanyfonctionofp,andusethefepresentation x=shd/dp] 9.Consider thefunctions #8)ofthe angular variable 8ressictel cothe inwceeal <r <0S x 56 Quanen Posihewavefunctions satsthecondition(e)=#(—a),showdeeoperon path. ; ade isberi, wo.Consider 6),themomentum space wavefonctionofapace Ahia eee foxpone sales ofp,whatcondition must&()synced thasbeahemitanperce?Wie(30) 4 =|chapter4 ' Eigenfunctions andEigenvalues : »ee Letusconsider thetime-dependent Schrédinger equation obeained ia. Chapter 3, Bee) _HME Levay % Lao Sm aetO) Mey) (41) qandatcemptcdsolveitbyreducingitcopaiofordinarydifferentialequationsieoe matic, Wa : Had) =Te) ey wich pis hae Of a) ed‘ inuey -[-oedet+Yow]ne) P Dividing byu(x)Tle)weget 4 Ufa _(882m)(adsfdet)+Vs)af) , ( ihTo7 a) (43) Thiscanonlybesited ifoohsidesarult const, which weal“Thesolution of aT). iha7 ET) (4-4) TO=Crem (45) wher Cis. onse. Theothe eqetin i Wedels) weae beet Vax) =Ewe) (4-6) ‘Thisequation ffequely called thetineindia Shige sguationChass aly int fomthtof(00) Boe a eae 58 Quancam Physics development ofyx); Eq.46isaneigenvalue equation. Toexplain whatthis means, wemustretufn (0thenotion ofanoperator, which wasbriefly mea- tioned butnotdefined inthelastchapter. “Mosegenerally, anoperator acting onafunction maps itintosnother fonction. Letusconsider some examples Ofis) =fs) + oft) =LP Of) =fx" +0) Oftx) =lafled/ae . Ofte) =aftad/ae —2fe) Offs) =fe) a [Alloftheseexamples share theproperty thatgiven afunction /(s),thereis& tulethatdetermines Of(%) forus.There isaspecial cassofoperators called Tncaroperators (wedenote theseoperators by1.todistinguish themfromche {general operators 0).These havetheproperty that Ufils) +AGE) =LAG) +LAG) (a) and,witheanarbitrarycomplexnumber, Lefix) =ebfi) (49) “Thus, inouriseonly cheasttwoarelinear operates. "Alinear operator willmaponefunction intoanother, asintheexample. af) tte)=LE?—afte) Itisinstructive tothink ofthefanctions asanalogous tovectors inathtee- dimensional space. Theaction ofanoperator istotransform &vector into another vector. Inthespecial casethatthevectors aceallofuaitlength, a0 ‘operator willtransform onepoint onaunitsphere intoanother. Anoperator, inthisspecial (bueveryrelevant) example, maybearotation about anaxis (Fig.41).Lettheoperator bearotation of,sx,30°about chez-axis. Ieiseasytovisualizewhathappenstovariousvectorsunderchisoperation,Thesewillbe‘wovectots tharhaveaspecial property: theunitvectors fothenorthandsouth poleswillbemapped intothemselves under therotation. Thisis@special ‘example ofanoperacor equation like(4-6), which maybewritten as Hur(s) =Eax(o) (1-10) thee aealoair opens fotwhich (69)iselaced byLefts) =etfte. Eipenfonctons andFigeoales 59 i es Fig.4-1. Anillszaion ofheopto ratingallyetorsby30°withshevectoriinontheun tphere: forvectors onthe eq (fos) stan iemedate latte (B-"Band aeeleDenes ‘Thisequation satesthatH,theHamiltonian peor actingon4specicas "offunction, willgivebuckthefunctiontacitisacingom,mutedby2constant, Theconstant iscalled theegeralee. Thesolction athe eaurondepends onandwehivetherefore labled itwithanEThesolution sceg alethesignin, conesponding totheeigen E,ofthepanes UhWesalseetaeeigenalues canformacontinuum orbedisc J Thesolarion (42)isofthe formvats)«1 Since(61)islinear sation, «sum ofsolutions ofthe above form,withpenmisble values ot jialsosolution.Tharthemostgenersolutionof(1) von=(E+fa)0wa)een (ay whereC(B)isanarbicaryfunctionofteeigenvalues, andthesumextendsovethediscrete valuesofE,theintegral ovrthecoutinvous mugofeet 60 Quaccum Physis values, ‘Theeigenvalues oftheoperator Hatecalled cheenergy eigenvalues, tsissuggested bytheform of =Pes 16) a ne£2+vo) @) Befoce discussing 4verysimple butinstructive example, wenotethatthe feporation ofthe equation would failfthepotential Vdepended expicy ontine,Werwlsceaterthawhenthisisthecase,enezgyisnorconstantofthe A.TheEigenvalue Problem foraParticle inaBox Weconsider Ba,4-6with Va)=o sl<e : == elsewhere (443) “Thisimplies thatthe waveFunction enustvanish for|x|>thatis a)=w(—2) =0 4) Inside the box Puls), IMEa)4Ey)=0 3) Firstwenotice thatifE<0,chen (418) cakes thefrm PHD_als)=0 16) wich¢=2m|E|/P. Themostgener solution ilinear combination of¢* nde" anddereisnowayofsatisfying theboundary conditions (4-14). Thus theenergy eigenvalues must bepositive. Wewrite Be= (417) sothatthe equation (4-15) takes theform FH) ate)=a)+pals)=0 (4s) ‘whosesolutionsatesinbxandcosdx:Theboundayconditionsimplythiforthesinesolution, which wedenote bya"(x), fame 123, (419) a2 iy sothar :Boeoe (4-20)am “GE iseasy tocheck taethenormalized solution is : HO)=sin (421) {he cosine solution, denoted byuf) mustbesuchthat ; fea (e-De n=12.3. (42) E thacis big in—(o/sa ag=Bacal tas) 5 The normalized solution istherefore . 1 fo= a) ax.0)JcosBa0 (424) FWeseechatche(4)signsrefercotheeven/odd propery under thereflection : ‘Thesolutions havethepropety that a [anit P00=[aloe 0)=a : [iaaicrene 16)=0 (423) thats, heysatily whataecalederthonermaiy condom, Sincethesolutions ate real,thecompler conjugation isnoereilly necessuy, butitinseted forton, sistency withfuture usage.Thesateofloweseenergy,thegroundsatisrepeseatedbyf(s),and itsenergy is Z EP=ze (426) ‘Thesolutions aereal.Iecherefore follows thae @)=0 (427) | Thincam bedonebytectcatcuaton, otbyasymmetry argument: foranyoneofthesolutions,whichareel,(p)isofcheform(i)XGane,Since(p)‘ustberal,theintegral, iovolving onlycalfunctions, musteanichs equ,Fenty,theincegralinvolves#productoftwoevenoftwooddfunctcwswih 62 Quancam Physics dxinseredbeeweenther.Thecotalintegrandisthusanoddfunctionof«, ‘ind upon integration overasymunetticintervalmustyield«vanishingintegral ‘Wecancalculace (*)forthe vatious solutions. Infact,sinceinside the boxpt=2m, wehave (pt)=2mE* (4-28) ‘Notice that. 2aV G~ tanh> (429). isconsent wichtheuncertainty telation.* Wealsonotetharchelarger the fenumber ofnodes inasolution, chehigher isitsenergy (Fig.4.2).Thisiswnder-standable, sincethekineticenergyislargerforasolutionwithalargercurvature,measure ofwhichis 24/de?. Specifically BL gsrte Bf ach ® fall?-pfeos-anl*edfale islagewhen thefunction hasalotofvariation init B.The Expansion Postulate ‘Anarbitrary function 94x),satisfying theboundary conditions Ws)=¥-4)=0,canbeconstructed fromoursolutions.Itwillbeasuperpositionofall ofther He)=EAMG +APT 430) “Theorhonormality relations canbeusedtodetermine thecoefcenss 4:*Withthehelpof(4-25)weeancalculate,forexample, fbab) | -5[~/Pepa) de+A?fipee) «| -49 sochat AS=fbral) x) (431) isageefearhatforighereigenfunctions xpgrowswithhegenus, Figenfunctions andEigenvalues 63 i ~ He 1ta 5-8 ie ae eee Ping Fig.4-2, Eigensolutions forpaticle iabox. ‘Asin ourdiscussion ofthefreewavepacker, wecancalculate thetimedevelop- ‘ment ofthis arbitrary inical packer. Since exchofthesolutions a(x) acquires thetimedependence ¢~i#:0™ see(4-11)], wehavequitegenerally He) =SLAP AMG) amon 4AL) OHM (432) Togetanideaofthephysicalmeaningofthecoefficients A"),wecalculatetheexpectation value oftheenergy inanarbitrary state. Since inside theboxH= P/2m, andoutside theboxnothing contributes, andsince Hs) =B(x) (4-33) wwehave, using theorhonotmality relations (4-25), 64 Qeancum Physics , an=favre 109) / -faf{S, [arearonrs acne ]} -x{¥[erarairo +erarareo]] =Sah apes lari 439), Tnexactly thesame wayweshow that favove = implies that EAM AP) a (439) Equition 434, together withthesomalizaton condition (4-58), stronglysuggestschat|AS?|?beinurpretedassheprabability thatameasurement oftheenergyforthearbiorarytateyeldsthevalueE<*?,NotethatonlythevaluesES"?atepossiblefortheenergy,sothat@givenmeasurement canonlyyieldoneofthevalues E orwaepackee willchemeasurement always yieldanenergy Bf?(on eigenvalue}? Cleatly thiswillbe0onlywhen ASP bus 430) thatis,when Y(x) =af?G0), theeigenfunction corespondiog totheeigenvalue EL. This leads ustoaveryimportant conclusion: ‘Suppose thetwehave@generalpackerdescribedbyY(x).Ifanenergymeasure-meat iscarried out, only aneigenvaluc oftheHamilconian operator Hcan resul, with probabilcy PCEn) =|fdeaat(x) Ha)|* (437) (here wehave lfcoffthe(2)labe! forgenerality). Furthermarr, afterthemeas urementthathasyieldedtheeigenvalue Eq,thestateofthestemisdescribedbytheaigenfeion tals), since otherwise 2repetition ofthemeasurement would not necessarily givethesame result, andFeproducibilty ofameasurement for& fiven system isessential forthemeasurement tohaveanymeaning. TheseStatementsatenotpecilasfotheproblemofaparticleinabox.Theyholdformoregeneral systems [with 2V(3)}, andalsoforherman operators other than theHamiltonians, a8willbeseenagsin andaguin, andthese statements leat theheart ofquantumn mechanics a igenfncsions andPigenaues 63 a C.Parity F “Theeigenfunctions fortheparticleinboxweredividedicotwoclassesthose eveninx,denoted with«(+)andchoseoddinx,denoted with().IfwesirewithawavepackerY(3)chaiseveniax,s4y,thenin(4-30)alltheAC? 4‘mustvanish. Equation (4-32) thenshows thatthepacket remains eveninfot Fall time.Thesameholdsfor«packet chatisinitially odd‘Thusforourbox,which wassymmetsically centered about x=6,wefindthat“evenness” snd © “oddness" aretimeindependent. Sinceanyconstant ofthemotion isofinterest 10us,wewillformalize thediscussion somewhatq ‘Wedothisbyintroducing thepartyaerator,whoseruleofopertionis | torellece x+—x.Thus foranypacker yx),wehave PYtx) =via) (438);Foranevenpacketwehave a PEPE)=Px) (439) {and foranoddpacker . 2 POG) =O'R) (4-40) k.Theseewoequations areeigenvalue equations, andwhatwehaveshownisthat[evenfunctionsazeeigenfunctions ofPwitheigenvalue +1,whileoddfunctions[we cigenfuntsions ofPwitheigenvalue —1.Intheproblem ofthepatie insbox,thefunctions w(x) arenotonlyeigenfunctions ofHysheyaresimul.samessyeiges fantsofP.‘Theeigenvalues +1aretheonlypossibleones,Supposewehave Pads) =dae) (4-41) Applying Pagsin, wewould get Pru(x)=Pau(x)=Nu(x) (4-42) HoweverPid)=2(3),sincetworeflections shouldnotchangeanything. Hence\=4,chicis,N=261.Anarbitraryfunctiony(x)canalwaysbewrites G88 asum ofaeven andanodd funecion Ho)=Siva) +(21+ Bee) —2) «) thats, justewiththe eigenfunctions ofHdiscussed inourexample, anyfunc.tioncanbeexpanded intermsoftheeigenfunctions ofchsnewoperate, This100i2generalfeatureofhermitianoperators:theeigenfunction ofoyhrwiionpert aresaiformacomple xt,inrmsofwhichanyfuntion taexpend.Weleaveittothereader t0showthat(P)isrealforanystateYo)whackimplies tha:eheoperator isherman 65 Quantum Physics 1 7 | y ' ) \'l 1 oY , Ly Fig.43. Boxforwhic thee tno aymecy te selitons. “Theexplicit appeance ofevenness andoddness cameabout because we centeredtheborst?=0.Hadwetaken1olebetweenOand2,nothing Soul hvechanged, nodtherewoul stilbesyniey under elections abotx=a,Suchsymmetrywould,however,bemuchlessapparent.Thelessontobeicamned heeisc iseting up»quantum medankal problem ooeshuld tleays poyatenson tothesymmetis intheHamiltonian, andchoose the Courlintes inwayhatexh thesymmetier mos expiily. Iftheboxstoreuoeen(Fg#3),noamountofchangingcooriateswosldbingabouSyme.TheimporanfacshatthemntbenheHamitoion*Thismay beseenmote cleatly byasking under whatcircumstances anevenfunction will feamin even foraltime, Le Hx,0) ==x0)=w(x) (4-44) ‘Thetime developnent igiven by AD—yc (46) Ifweopete withPon thisequcon, wesee 2 ayPHxd) =PHOMxs) (4-46) Uades thespeci itcumsances that ; PHYxs) =HPWx) (4-47) +whendingwithheowwecomerhemspsfpoesia, he es ee onapf funy conden ned he Rano, 4 Eigenfunctons andEigecvalues 67 ‘which holds when His even underx—+—x,chats,whenV(x)isanevenfunc tion (since d/ds* iseven), wehave 2 , iS(PHA) =HEPA) 648) Hence :; VO =AF Pos) (4-49) Rand a VOCs) =4—P)4x2) (450) Myseparately obeytheSchridinger equation, anddonotmix,iftheinitialstateis5 even(orodd).Thecontltion forthe ime-ndependence ofparityonlyholdsif (PH—HP)Hes)=0 (sn) forallpossible states, chati,ifcheoperatores Pand Hcommute PH) =0 (452) PTsimporaneconditionwillbesentobequisegeneral:anyapeazorthatdeesBiwothaveanexpici imedependence andthatcommutes withthe Hamiltonian Hisa is.sontent ofthemotion. Inparticular, ifthe potential changes withtime, thati,we 4 hhave Vs), then theenesgy itself isnoc@constantofthemotion,justa3ia classicalmechanics.NotethatwhenVdependson1,theseparationofthe Pequation intoanequation forthetimedependence andanenctgy eigenvalue Fequation isnocposible F._D. Momentum Eigenfunction andtheFree Particle ‘Ourdiscussion ofpartyshowedthatitisnotonlytheenergyoperator Hf tachaseigenfunctions andeigenvalues. Letusnowsolve theeigenvalue equation forthemoment operator Denials) =paste) (453) Since pop=(6/i)(d/as), thisreads 4 dass) ip HO) Fate) 439) F Thesolution tothisequation is wala) =Comm (459) withCaconstant tobedetermined bynormalization, sndtheeigenvalue pral,so chattheeigenfunction does nocblow upateither +©or =.Thisisthe 68 Quantum Physics. onlyconstzint onp:wesythatPuphasacomtnunas spectrum. Wemight, by analogy with(4-25), expect thattheeigenfunctions obeyorthonormality con~ : Aitions Weseethat : few(2)als)=lofaceonsion . =24|C|*68(p —p') (4.56). With the choice WO)=Ty (sn) (4-56) reads: fdetle)uals)=aCe=p') (458) “Thisdiffers from(4-25) onlyinthattheKroenecker ban,appropriate fordiscrete indices isreplaced byaDirac deltafunction &(p—p’)forchecontinuows indices. “Thestatement tharanywavepacket (3),maybeexpanded interms ofa completesetofeigenfunctions canalsobeestablished here,Theanalogof(4-30)‘mus akeintoaccount thatwearesumming overacontinuous index p,sochat . wwe write v=[9005 (9) According totheinterpretation implicit in(4-37), |@(P)|*, where =faeSY vo gives theprobability thatameasurement ofthemomentum foranarbitary : packet (3)yields theeigenvalue p.Inthiswaywejustify checonjeccze made about $(p) inChapter 3(cf.Eq.3-30). Letusnowturntotheficepasticle Hamiltonian, When V(x) iszero . cveiywhere, theenergy eigenvalue equation reads PW) payin) =dé‘Mulx)=0 (4-61) where I=2mBjf, Thesolutions acee#*and«~%, orlinear combinations of these, forexample, cosAxandsinkx,Thetrouble withallofchemistharthey sentsquareitepabesince"dsjAeBe?divergesfoallues of Aand B e Eigenfunctions andEigenvalues 69 i ‘Thereatethreewaysofgettingaroundthisdifficulty " (2)Wemayconsidertheproblemdefinedby(4-61)asthelimitingcaseof &+partie inabox,withthewallsreceding toinfinity, thatis,4>@.Inthis limicthesolutions (4-21) and(4-24), evenaside fromthenormelization factors Eve willbecome wivial, unless becomes verylarge,50that e . Tres (4-62) Be becomes finiteWecanthenneglecttheinthe(#~4intheevensolutions F424), andobtain thesolutions a 1 1 Jin cose 46ve Ve o) fWemaykeepche1//afactors:theywlldropoutoftheanswertoanyphysicalf_questionthatwemayaskaboutthesystemTtissometimesusefultokeepthem, GaB _since cheie presenceinafinalresultindicatesthatanertorhasbeenmade, :2 (b)Wemayworkwithwave packets. Asolution oftheform 3 ix)=ete (4-64) is4specialaseof(439)with 49—)=Vehap—fit) (4-65) HEFta:is,anindinlypeakedmomentumspacedistibuion. Seppotewereplacethislimiking @(p)byaverysharply peaked function \/2eh x(p~Rl).TheaFee willbereplaced by -“ftyeig) (4-66) Fwhichitplanewave,i,muliplied byavetybroadfunction ofx,Wemay ‘makethisfonction 30broadthatitisestentislly constant overthetegion ofPhysical icetest, Theuncertainty inthemomentum willaowbeofthe orderof Fmagnitude f/(size ofx-packet), andifthedenominator isofmacroscopic sist,thisuncertiney isnegligible. Wedhussatisfy themathematical equitements ‘without changing anyofthephysics. Thewavepacket descripion iactalyF.theonehueiscloses towhaccallyhappens physically, sinceanywayofpe. ; 1% alo eep+Gn,andaentpanicsiaeretedinyaleof«thaae« ft faction of "Acwetionthals notmeanegfl physical isonethtdepends ontheexienc of thewall. Forexample, "How logwill keforawavepices aeeetecathead‘tunco=OP"naquestiontatwecaiywsoxppseatycient 70 Quancum Physics pating theinital sae,forexample, Gringonelectron gun,cannever, inpractice, i caeareanexactmomentum eigenstate.(6)Thedifficulty stems fomthefatthatforawavefunctionike«,the particleisnotconfinedtoanyregionofspace,sothatcheprobability offinding,itanywhere iszero.Ifwedonotaskquestions thatinvolve theprobability offindingtheparceinanyfnitetegion, noproblemsarise.Onewayofavoidingthenormalization dificulty istodealwiththeprobability current, offlee =EL pegHO HO .i=AlyTe geHO 46) discussed atchebeginning ofChapter 3.Forawave function Cele, cheux is[C|?p/m;forthewavefonction C7", theBuxis—|C|* p/m. Ifwenore thatforaone-dimensional problem, thefluxofparticles withadensity of 1particle/om, moving withvelocity »=p/misjust»—thac isthenumber ‘rostingapointx=xopersecond—weseethat|C|¥representsthedensityof parties percm.Thus (4-57) represents particles withadensity 1/2xf percm. {Inthee dimensions, with pie) =Corr (4-68) thefaxwillbe|C|*p/m, andthiscorresponds toaflowofparticles, wich density |C|percm?crossing aunitareaperpendicular top,when theparsiles aremoving with velocityv=p/m(Fig.44). ‘Theenergy cigeavalue equation (4-61) hastwoindependent solutions,‘and-#;equivalently, thepairofrealsolutionscosAxandsinBxisasoindependent. Whichever paitwechoose, wenotice thatincontrast totheprob- Jemofaparticleinabox,therearesvesolutionsthathavethesameenergy associated with them. This isanexample ofsomething that happens quitefrequeatly: heremaybemorethanoneindependenteigenfunctionthatcorrespondstthe sameeienvaladof ahermitianoperator.Wenthisoccurs,wehaveadegeneracy. Inthetwocasesthatwehaveabove,thetwosolutionsareorthogonal: f”aleeetef”deeemo [asiabecost=0 oy fork+0.Icisalwayspossbietomakelinearcombinations suchthathisistrue.‘Such Tineat combinations are,ofcourse, orthogonal toeigenfunctions chat ‘correspond todifferet values oftheeigenvalue, forexample, theencrgy.* ‘What distinguishes. chewo degenerate eigenfunctions? Fortheset “See Appendix B, igenfunctions andEigenvalues 71 3 Farts ot Paris oon cory n inn cond j inno ery Fig.4-4. Therelation between velocity ofparcicles andflux,thatis,sumber ofpartiescrossingaunitareaperpendicular tovelocity,perunittime (e%, oi),thedifference isthattheyareeigenfunctions ofthemomentum |,opertor aeenEFpoteahgists (- PopcitemFelt=hb (40) conespondingt0diferenteigenvalues ofthemomentum. Similadly thepair (00sdx,sinAx)areeigenfunctionsofthepartyoperator,correspondingto ©diferent eigenvalues 2 Pcoskx=cosbx F Pisinkx=—sinby Cor) Inbothcases,whasdiferentates thedegenerate cigenfusictions isthatcheyaresimultaneous eigenfunctions oftnother hetmitian operator. Boththeoperators PepaodPhavetheproperty thattheycommute withtheHamiltonian p,,*/2min thisproblem. Weshallshowlaterthatthisisanecessary condition forthe cxisteace ofsimultaneous eigenfunctions. Forexample poptndPdonotcom. mute,[since(/i)(d/dx) changes signunderx~»—x),andtherefore theeigenfunctionsofoneoftheoperatorscannotallbesimultaneous eigenfunctions of the other. ‘Wehavelearnedanenormous amountabourquantum mechanics from, [thecwosimple problems thatwehave considered. Weshallreturn tothese smaccetsinlaterchaptersandgeneralizethem.InChapter3wewillagainconsider somevery simple problems, buthistime wewillconceatate notonthe mathematical features, burrather onthephysical systems thateheyafsimplemodels of, 72 Quam Phys Problems 1.Yount given theflowing operators 4 @)Os)=Wx) (b)Ox)=“Ezve) (©)Osx) =WW) (a) Oafx) =ae? () .©om244 ©omerf”acer) ‘Which ofthese alae operon?2.Salvethecigeoveproblem oats) =Xe) Wat ales ofthe cgenale endrosquat integrable eigenfunctions? (Hint, Differentiate both sides oftheequation with respect tox.)3,allethefllowing commutator @ (0s,On} o) 101,Ox} ‘Theprocedure iscocalculate (A,B) byexpressing A(BY) —B(Ap) intheform a. 4,Giese ax=VGH) fortheu(x)givenby(4-21)and(4-24).Using(p")givenby(4-28)calculace apa Tecan thfothehighstatescheueincasewith5.Solve theSchrédinger equation foraparticle inaboxwith sides at r= and»=Lwithsheboundaryconditiontat vo) =KL) Whataretheeigenvalues andthenorinalized eigenfunctions?6Apicisinthegroundateof«boxwitsidesat=4,Veysuddenlythesidesoftheboxaremovedtox=6(b>a).Whatstheproba-bility thattheparticle willbefound intheground stateforthenewpotential?WhatstheprobabilityhatwlbefounditheSuseexcitedsete?Tnheercat, thesnpe sewer as sinpe explanation. Whats7.Apatiekaownoblociintheefaboxwthsidesat = alia alas rn theellfsideareequal plobbl, what wave E igenfunctions andEigenvalues 73 gfunction describes theparticle ats=02Willtheparticle remain localized atInte times? CGalculte cheprobability hatanenergymeasurement yieldstheground state energy; cheenctgy oftefistexcited state 8.Aparticle isintheground sate ofaboxwith sides atx=Oandx=L, +Suddenlythewallsoftheboxatemovedto-t©,respectively, sothatthepartici isfee.What istheprobability thaeehe paticle hasmomenrem inthe ange(0,p+4p)?Aftertheremoval ofthewalls,cheenergy oftheparticle is P/2m,whichneednotbeequaltotheground stateenergy, Canyougivean E>explanation fortheappatenclackofenetgyconservation?9.Repeattheabovecalculationforaparticleinitiallyinthembeigensate. ‘Showthatthe cortesponding probabil isgiven by 4 Bate1=(=1)*cospL/h—-. BLEU —(on/OFF ‘Sketchthedistribution, Showthtieconformswiththeunceraintytelson,and °Fdhactheresuleisinagreement withthecorrespondence principle whennislarge. 10,Aparticle infreespaceisinitially inawavepacker described by 4va)=()oo (@)Whatistheprobability chatitsmomentum isintheringe(p,p+dp)? (©)Whatistheexpectation valueoftheenergy? Canyongivearoughsegment,basedonthe"sineofthewavefunctionai!theuncertaintyprincipe,"for whytheanswer should beroughly whacicis? (0)Thewavefunction for«particleisgivenby We) =Ae Bete What uxdoes thisrepresent? EG2) Wharishefxassociated with«particledescibed bythewevefunction Ho) =ala) where #(x) isarealfunction? 13,Consider theeigenfunctions foraboxwithsidesatx=a.Without ‘workingoutheintegral, provethatheexpectation valueofthequantity AP+aap +pe vanishes forlltheeigenfunctions. 74 Quancum Physics 14,Prove thatthepasty operator, defined by P¥@) =W-9) isaexmitian opentr, Alsoprovetatthecigenfonctions ofP,conespoM™totheeigenvalues +1and—1areorthogonal References [Adetaileddiscussion oftheproperties ofsecondorderdifferential equations 4srelaced toquantum mechanics maybefound inJ.L.Powell andB.Crasemann, Quantum Mechania, Addison-Wesley, Ioc.,Reiding, Mass, 1961, aadD.S. Saxon, Elementary Quantum Mechanics, Holden-Day, SanFrancisco (1968). ‘Seealsoanyofthemoreadvanced textbooks listedattheendofthebook. :|chapter5 »One-Dimensional Potentials : |. Haewesolesome simple poems ofone-dimensional oton, They ©aeofincest Becse they thse some noma efecssod beacSayphysisteaonasefelyonedimensionalvenughweIne ELathedimensional woe F@ A.ThePoteatial Seep Forthisproblem wetake (Fig. 5-1)theform ofV(x) tobe Vix) =0 x<O0 =VW x>0 (5-1) |theSelinger equation Aas) : me dee+VC)ws)=Buln) (5-2) =o * Fig. 51. Thepoten sep 9s 76 Quantum Physics .. takestheform NX PA) 2 yial)= :AD4FEEVola(x)=0 63) ‘We write, asusual amErtBP 6-4) snd wesso incoduce 2an(B— Ve) . ¥ ¢ 55) “Themost general solution of(5-3) forx<0,where V(3) =Oi ax) =eit+Revit (6-6) “Thiscommesponds tofluxmoving inthe postivexdirection,ofmagnitude pas4= ete Rees iktte~aR9 —complex conjuge] ik= IRD 6 ‘Wemayviewe*withBuxfik/m asa incoming ware Ifthere were90poceata,wwecouldchooseeasthesolutionforall,sothatweataibuteRothepresence (ofthepotential. Thispotential gives risetoaelected wave, RM", witha reflected fluxfk]R|*/m. Forx>0,wewetethesolution . as) =Teer 68) ‘Themost general solution farx>0isalinearcombination ofe**and«~“",but :4term involving thelatter would describe awave coming from ++ inthe negative diction, andwith the “experiment” thatwehave setup,theonly ‘waveontherightcanbeaaansmited wave,Thefuxconesponding to(58)is johire 65) : Since there isnotimedependence intheproblem, theconservation law(3-11)[impliesthat(x)iindependent ofx.HencetheSuxontheleftmustbeequalro thefuxontherigh, chatis,weexpect chat Ho—\niy=Bre (10) See. & One-Dimensional Potenisls 77i ‘Thecontinuity ofthewavefunction implies that 1+R=T (1) (ER obtained bymatching thetwosolutioas atx=0.Inspiteofthefactthatthe {_poential indaconiovous, theslopeofthewaveuneionissocontinuouseo|nbesenbyiateuating (3)fiom~etoFe(with«ably coalad 5positive) andusing theconciuity ofthe wave functon, de (de) ft dnF ().-G@).-fess -fbe2[re-*a)=0 (12) ‘Wenote,forfuture reference, thatiftheporential contains atermlikeVoblx—a) thennegation ofthe equation fom#"—eeo« 4«giver ‘dus (du ampete ) Pe(2),.-@)- cs a a a Vowa) (5-13) ¥ ‘Thecontinuity ofthederivative forourpotential implies that Eis iA(L—R)=iq (5-14) j.)Wecantherefore solveforRandTtoobtain paint te 2 T=ite (5-15) From tisweancule therefeced andenamine fares 5 Bkik(;=gs. th(b=9 7ates tq fkAkg Mins = Aig: wt eae (16) We notethefollowing: 1.Incontas toclasical mechanics, according towhich«paniclegoing ‘overapotentialstepwouldslowdown(toconserveenergy)butwouldneverbereflected,herewedohave«certainfractionoftheincidentpariclesrelected, 78 Quantum Physics . “This is,ofcourse, aconsequence ofthewave properties oftheparticle; partial ‘reflectionoflightfromaninterfacebetweentwomediais2familiarphenomenon. 2With thehelpof(16) weeneasly check thatthe conseraion law (6-10 isindeed suis,ForE>Vothati,fot¢—bfrombelow,thetiooftheefleted ‘fluxtotheincident flux, thatis,|R|®approaches zero, Thisagrees wich intuition,‘whichtellsuschatarveryhighenergies,thepresenceofthestepisbutasmallFercurbationomtheropigaionofthewae‘4,IftheenergyEislessthanVo,thengbecomesimaginary.Ifwenote that now thesolution forx>0mustbeoftheform ax)=Tele (6-17) 50asnottoblowupat+©,weseethatnow k=Atl)(t=ddl) iRh= del) 5.18) Wealtesr tear) ow ‘Thus,asinclassical mechanics, thereisnowtotalreflection. Note,however, that 2k T= (5-19)atig O19 does otwith, andaprtofthe wave penetrates intotheforbidden tegion.‘Thispenecration phenomenon againischaracteristic ofwaves,andweshallsee2idecerthafepermis"fonnlag”troughbarsthatwouldtotalyblockparisin«csscaldescription."Terisn0Bxotherigh,since(=)vanishes fra teal solution even iftheconficen infont ofi taken cobe comple. B. The Potential Well ‘Wenextconsider thepotential (Fig. 5-2) Vi) =0 xine aM -eexde =0 cx (9-20) Weaguin wice. +22 :ee 2) and 22m(E+ Vo)cn (5-22) P neDimensocal Forni 79 i} Fg. 52. ‘Tepov wel We canimmediately writedown thesolutions i Wee eRe nce 4 Ma) =AOE BOM gence a(x)=Tete ace (5-23) ‘hese cespond tan incoming Huxf/m fromthele«elected fx fib|R|?/m andatransmitted fluxfée|T|*/m tocheright, Inside thewell.there (are waves going inbothdirections because ofthereflections atbothdiscon. tinuites atsta,According toBuxconservation weshould get i i: Fanirl=gaiiaiy=Bir ay %Matchiogwavefunctionsanddesvativesgivesthefourequations Reem Aces Bem ie —Rel) wigci —Be) Bem Bem =Tem (Ae—Bet)=ikTom 6-25) [Ale alee yids thesus Rnjente ——_(— #)sin240: Wicohtga He Bin aya 6292gcos24a—Hg?sin2ga » AinifB2>Vetere pactily 00telecon, since4~#24 andas 70,chemnsmision goestor, The ianemofspecmnthe sprcasedasin2g=0,thaifohecorespveby ren Beart eepeas... oa 80 Quantum Physics thereisnoreflection.Tissactually&modeofwhathappensinthescateringof lowenergy electrons (0.1€V)bynoble gesatoms,forexample,neonandargon,inwhich thee isanomalously large tansmission. Theeffect, nseobserved by Ramsaver andTownsend, isdescribed asatransmission resonance. A.more accurate discussion must, ofcouse, involve thtee-dimensional considerations. Tnwavelanguage, theeffect isduecoadestructive interference between the wave reflected at2=andthewaveteflecedonce,twice,thrice...,a¢the tiigex=Theresonance condition 2ga=wx,which maybewriten inthe form a4a-#-* (528) ra isjsttheonethatdescribes theFabry-Perot interferometer aaddition totheabove solusions forE>0,there ae,remarkably, s80solutionsfor <0providedthepotentialisnegative,thatis,Ve>On(5-20).Teywillcurourtobediscrete.Letwswrite mE igoe 6-29) ‘Thesolutions outside thewell chaarebounded atinfinity are wa Ger x<na a= Gen ace 630) Since wearedealing withrealfunctions, iismoreconvenient towccethesolu- tion inside thewel intheform us)=Acosext Bsinge —a<xca 631) Nowe that aneen le>0 (532) “Matching solutions andderivatives attheedgesx=-bayields Goer =A.cosga~Bsings AGHwe(Asinga+B00844) Gort=AcosgetBasings acy eet=9A singa—Bie0s gé) 633) ‘These may becombined toyield agA8inge=Boos98 "4AcongaBsinga __Asinga+Bcosge 1feosqa—Bsings oa) 4 One-Dimensional Potentials at i | aye tm|| ow||Be soto] | sctrtom|| ' i t gFig.5-3. Solutions fordiscrete spectrum inattractive potential well. GE TogethertheseimplychatAB=0,chaithesolutionsateeitherevenin Fix(B=0)oroddinx(A=0),asituationencountered inchecaseoftheinfinite FZbox.ThewavefunctionsareroughlyoftheshapeshowninFig.5-3.Theground «HEEsate,witha0nodes,iseven.Thisisageneralpropertyofsimplesystems.The us conditions thatdetermine thecnergy arefrom (5-34) a =qungs evensolutions a x=—qcotgs —oddsolutions (5-33) Lecusexamine these separately 4 (@)The even solutions Ewin thenotation po 2nd +a rae ‘ ya 6:36) L,thefrstoftherelations (5-55) reads 4 Va-F— =uny 637) Ifweplottanyand-/—7*/yasfunctionsofy(Fig.5-4),thepointsofinter- gsection determine theeigenvalues. These formadiscrete setThelargerbis.theFfarther thecurvesfor/X—7'/ygo,thatis,wherthepatental isdeeperand/orfbroader, therearemorebound state. Thefigure alsoshows thatnomacter how small is,cherwilllays beatleastoiebound state: Thsischaracteristic of ‘one-dimensional arcactive potentials, andismotcrueforthtee-JimensionelPotentials,whichbehavemuchmoreliketheodd-solution problemthatwewil 62 Quantum Physics ons °Wt *Wi? EaSale oFTee ” Fig.5-4.Location ofdiscrete eigeavalues frevenwoltions iasquare wel.Theflagcarves represent tanthefling curves areV/%=3yfordfleenevalues on discuss below. As>becomes large, theeigenalucs tendtobecome equally spaced iny,withtheintersection points givenapproximately by y=Qet+ ie 7=0,1,2,... (5-38) ‘Thisisjustcheeigenvalue condition fortheevensolutions oftheinfinite bos, anidchiisamight beexpected, sinceforthedep-pin sesinthe potential, thefactcaeiisworeall innitely deepdoesnotmatter verymuch. (6)The oddsolutions: Here theeigenvalue condition reads ViceaL es (5-39) 7 Since~cot y=tan(x/2+9),theplotinFig.3isthesameasinFig.5.4withtheengeatCurvesshiftedby+/2.Thelarge\behavismoreolessthesame, with (538) replaced by yur PEL 6-40) ,3 One-Dimensional Potencals 83 aie rr , HPRig.5-5.Location ofdiscretecigneviues foroddsolutions insquarewell.TheE_ssingcurvesrepresent—coeyithefallingcurvesareV/h—y/yfoxdiferent values Bis, “ofNore chathee isnoeigenvalue for\<(#/2) Flacontrast rotheevensolutions, therewillonlybeanintersection ifVk=7A BE>0,tha is,if 2nVra? a 3 eo (s-41) d‘Theodesolutionsallvanishatx=0,andhencethebound-stae problemE;forcheodasolutionswillbechesameasforthepotentialwelshowninFig5-6,‘sinceintheater, hecondition 4(0)=0would beimposed. Weshallsethat ‘tuchconditions areimposed onwavefunctions inthethee-dimensional world a ig.5-6. Equivalent potencial foroddsolutionsofsquarewellBundsaveproblem, 84 Quantum Physics C.The Potential Barrier We now consider V@)=0 0x<na =Vy -acxce =o ace (6-2) ‘Wewillimiourdiscussion toenergies suchthatE<Vo,thatis,energies such chatnopenettation ofthebarrier would occurinclassical physics (Fig.5-7). Inside thebastier wehave theequation Gils)|2m SS+SSE—Ve)ix)= BO+FEEYea)=0 thatis ula) .Te7eH) =0 6-43) “Thegeneral solution aa)=Aart Bee [alce 44) istobematched onto ua) =e Rete x<8 =Te x>@ (5-45) ‘Actuallyweneednotgochzoughthetoubleofsolvingthissincetheresultscanberead offfrom (5-26) with thesubstitution gic =iVOmi —B 6-46) “Thus,forexample,sityPe Tee2hxcosh2xa—i(4*—©)sinh2ea Gan ve } Fig.57. Poreial batter, Energy issuch that's lasscal particle would becoully reflected byche barrier. E OneDimensional Pocenals 85 Kandthisimpliesthac 3 (r= ee1)=GeeFtinh?29+GP oa)F?Thereisransmission,eventhoughtheenergylesbelowthetopofchebare. GP Thisisawavephenomenon, andinquantum mechanis iisalsooneexhbieed Mp bypanicles. Thissumrlig of«parte tough «basic ifrequently eo. Bg councered, andweslldiscus some applications. Weasonotethatwher ces Ge lage, cheaio oftansmited Huxcoincident Huxi 4 Thx(ea)e 6-49)BeTrisbecomesan exuemlysecsicvefunctionofthewidthofthebate,andoftheamount bwhich thebatcr exceeds theincident energy, since : “[atVeal (50) In gener sheburies thatoccuinphysical phenomena atenotsquare, +BP sno discuss someapplications, wemustfestobsin aneppronimate enprecion By, forthewansmission coefficient 'T|?through aniegulrly shaped barrier. The BRE properwaycodothis,giventhefactthatthereis00exactSolutionavalfor PEmospocentls,isthroughtheWentee:Ktamery Ballou(WKE)appeonh Bfmation technique! Our discussion willbe lessmathematica! Fe observe that(549) conta ofaproduct oftwotem, thesecondOfwhichibyftthemoreimportantone.Ifwewaite E2{be)(e2) . j logl |? ~24(26)42tog2A q eT aoe)+28Geeh F/wesetharunder most ctcurmstances cheiseteam domiaates thesecond foe FRE. 2yseasobie siteofusTheproceduce weadope iscowent smooth, curved gbatier asajuxtaposition ofsquare butiers (Fig.5-8).Sines canamission co- feficiens atemultiplicative? when cheyatesmal(ineffec withmostofthe fz Fefleced, thecansmission though exchsliceisnindependent, improbable fever), wemaywt,apprxinately loglTI* ~Etog Tana? : ~2 Daw "Sethe WKB approxima inSpeci Toi secon 3 "Thissemen olycometfortheornponantcxpneat par,8cabe seenfomseat dosing shesith wlensprsinasll eae eras ‘Sofia ITP 86 Quantum Physice ig.5-8. Approximation ofsmooth barer byajstapsi- tionofsquare poten bases. 72f4VGn/iVR—EL 650) Ihe partial batters,Axisthewidthand(«)cheaveragevalueof«forthat barier. Inthelsstepalimitofinfinitely narrow baccers wastaken, Icsclear fomtheexpression thattheapproximation isleasaccurate nearthe“turning points” where dheenergy andpotential aeney equal, sincethere(3-49) isnot 2good approximation t0(5-48). Itisalsoimporaant thatV(x)beaslowly‘wryingfunctionofx,sinceotherwisetheapproximation of«curvedbarterby&feackofsquare onesisonlypossible ifchelaterarenasow, andthereagain (5-49) isapootapproximation. Aproper meatment, using theWKB approxi-mationincludesadiscussion ofthebehaviorneartheeumingpoints.Formost purposes, iissilafairapproximation cowrite |1=efeeVR (52) withtheintegration overtheregion inwhich thesquare rooisreal. D.Tunneling Phenomena “Thephenomenon ofparticle cunodling isquitecommon inatomic and rucear physics, andwediscuss cwoexamples atthspoint.“@)Considerelectonsinametal,Asnotedinourdiscussionofthephoto:electric effect inChapter 1,these electons areeldinamen by«potenti, which, cofirstapproximation, maybedescribed byaboxoffinite depth, as ShowainFig,5-50.Theelectronsateactullysackedupinenergylevelsthatare very dease, since theboxisverywide,Itisapropertyofelectronstharnomore thintwoofthem canoccupy anygiven energy level; chusforthe lowest energy fate ofchemeta, alltheevel uptoacertain energy, called theFermi energy this progeny ofelecas isdec? bythePal excasion pipe, which wil becnc’ feOper OneDimensinsl Potenials 87 " Ne : @ 7 Fig.59.(6)Becronic enegy levelsimmeal. BpistheFerienergyandisthe workfunction. (8)Poteval altered byanexternal decoey (whichdepends onthedensity officeelecuons) arefiled,Whentheempertureisabove0°K,afewelectrons aethermally exiedto higher levels, buteven4 |Yoom tempericue, cheaumber issmall. Thedifecence between theFormnergyandtheropofthewelliswhatiequiedtobringantlectonoutit'stheworkfenton discussed inconnection withthephotoeleri fect, Pectonsaaberemovedbytransferingenergytothem,etherbyphotos,ofbyhextingthem.Theycanalsoberemovedbytheapplicationofanexternalleceld&SoldemisiooccursbecausetheexternalfieldchangesthepotentalseenbyanEelecronfromWto(W~ex)(Fig.5-9),iftheelectronisatthetopofthe"sea™ oflevels. Theeansmsson coefficients [Tt=e-2fseeome ~ann 639) ‘Since yit=(A+Bat ‘ [204+a0 7 this leads to [Tt=ees RAs w70s 650) E TheFowler-Nordheim formule,a(554)iscalled,describestheemissiononly. ‘qualitatively. Oneeffect, which iseasilyincluded, istheadditional attraction of theelectron backtotheplate,caused bythe imagecharge. Theothereflese{much harder tohandle, isthae‘therearesurfaceimperfections inthemecalsurface, whichchange theelectric fedlocally, andsince&appeas iatheexponent this Fogrmakealargedifference. Incidenally, weseethattheexponent maybe thickness isgiven by v “eg (59) 86 Quantum Physics ve+ct Letom{zivolteeV Fig.5-10, Energy diagram forunneling berween womeals separated byvacuum, Tunneling berween metals isposible onlywhen there ate campy staies ontheight. Such empty sates arecreated when eVis applied colowes thePer level onthesight. “Thesame effect appears ifwebring ewometal plates close together. Figure 5-10shows thesiuation bothwithout apotential diference, andwitha potential difference. Without thepotential diference,cunneling isnotpossible becausechelevelsonbothsidesofthe barterareGilled.Theeffectofevenaweak cleetic fieldiscochange theshape ofthebustier alle(Fig.5-10)—an effect,thatwecanneglect—and tolowertheFermiseaononesieofthebarter.This,ineffect, brings some empty levels incotrespondence withthefilledonesonthe ther side ofthebusier, andnow cunaeling canproceed, with transmission coaficient [T/tee2VRWA« 630Suchafactoractsa8resistance.Unforunately thisexpressionisverysensitivetothegapseparation 4,andsinceforaworkfunction oftheorderofelectron volts, thesepatation hastobeoftheorder ofangstroms, ithasnotproved possiblecomakemealplatessufficientlylatandparallel.Theformulaasbeen Applied cotheincerptetation ofcurrents flowing between twoplates withan‘oxidebetween them(Ni-NiO-Pb), wherethegapcanbemadeassmalas50A, andiesqualitatively correct. ‘Anincetesting eect occurs when themetal onthesight isin&supec-conducting state.Acharacteristic ofsuch«staeisthatabovetheFetmilevel One-Dimensional Potencials 89 q Terai 4 Nowe( ‘:, 7 seomcondcn Big.5-11. Energy diagram fortunneling fiommetal0superconductor, La concast tothemeal-meral cunncling shown inFig.510,notunneling isallowed intotheenergygap,Thisaffeecsthecustent-voliage characteristic as ; oo there isagapintheleveldensity, thatis,therearenoallowed statesbetwee anenergyEr~andEy+Awith&oftheorderof10->eVcomparedwiththeFermienesgyBroforder10cV.Theselevelsdonotdisappear,butatesqueezedupanddown,sothattheleveldensityjustbelowandjustabovethepupisety 7haage.IftheelectricGeldissmallenough,thatis,a€<Aye,thecewillbeno =tunneling, sincethereisnoplaceforcheelectrons togo.TheGualiative featuresofthecurtent-voltage relation andtheenergetics areshowninFig.5-11,Thesefeatures aceingoodagreement withexpetinent, (b)Tunneling isalsoimporcant innuclear physics. Nucleiareverycom-Plicated objects, bucundercertain circumstances itisappropriate toviewthem 4sindependent particles occupying levelsin«potential well.Withthispictureinmind,thedecayof«nucleus iatoan-patcle (ttlenucleus wichZ=2)and |daughter nucleusmaybedescribed astheewonelingof anaprile though abamiercausedbytheCouleeore!beeentheughndthepare Thea-partice isnotviewed'as beinginaboundstace:if itwete,thenucleus ‘ouldnoedecay,Rather,thea-paticle istakencohavepostive energy, anditsdecay isonlyinhibited bytheexistence ofthebartics. | {16youfdiditiul imagine why4tepaiion wouldkeeptwoobjects fomsezreing, thikoftheinva proces, «cata. leislerthtthehater eneeaKeep sheprice ove 90 Quancu Physics 0) iat q Fig.5-12. Potent baieoradec. Iwe waite it}t=e* 657) then where Rischenucleate! and isthe cuing pot, detemined bythe‘Vanishingoftheintegrand(Fig.5-12).Zsthechargeofdhe daughter nucleus, TaD,Cozere)ithechargeofthepileBeingemived,Tenrgcanbe done exactly eofn aye “(eye (Reyf+G-4) -vi[es(3)-G-*)1oo) [Atlow energies (elatvetotheheightofthe Coulomb busier atr=R,webave 55> Rand then (amZrZee*d\"* Rye ox (™)[5-G) | oo Gc, eyears ofeaster ais caeforhesudofaleeNow a eet Pipagedtbaton, teeed byseating ectensaoe aelcinsocartaheoooulbeexpecoecacy ene ante : One-Dimensional Porentals 91 . withb=ZZae*/E. Ufwewriteforthea-particle energyE=m0?/2,where»is its final velocity, then a Gxte aeazzs() 81) 4 ‘Thetimetakenforana-particle togetoutofthenucleus maybeesti- FEES, muted asfollows: theprobability ofgesting chrough thebatter onasingle ‘encounter is«2.Thus thenumber ofencounters needed togetthrough is Bn~&.Thetimebetween encounters isoftheorderof2R/e,where Risagainffitheouclearradius,andvistheavelocityinsidetheaucleus.Thusthelifetimeis 4 2R ratte 6-82) FThevelocityoftheainsidethenucleusisaratherfuzzyconcept,andchewhole FApicueisveryclassical, sochahefactorinfonofthe«cannot rellybepre-BME dicted without «much moreadequate theory. Ourconsiderations dogiveusan PRE orderofmagnitudefri,Fora1MeVa-particle, i4 foe_[ae [2s FiBEAlso,forRwetakemn RX1SX10AMom 6-63) "BBR andforA=216weget,forthefactorinfront,2.6X10-,Wecanalsorewrite BGin the form oxi 60) WegVE(Mev) 10thatonepredicts, forlowenergy a's,thestraight-line plot ilogisconse—1.73 os q cad VE(MeV) ‘withtheconstantinfrontoftheorderofmagnitude 27-28when+ismeasured inYrinaofsecondig3.13showsthagoodtotheiftinedaaof iBEE:©largenumberofaemfersisobtainedwiththeformula | 1 EA logs =G4—G4 i cand VE FIRE whereG,=1.61andC,=28.9+1.624,Thuscheverysimpleconsiderations | etetemeteones fu te s Hy go tongtarn) orswe aeee”) ftrorcsmhZan9 09 Sree ante YWfeonv0Berz-2 firmx2o {foxval sn 1MenHy :of oum ETS ; apm Blewwf 4ciao BEety /ad aeGene fv a |e Ha* oyis2 arA os28 x 5 y BjaFm253fofU200 fe|88soian/ofm g|syahe | em200,/8A0278, py|9 onyf eS86252 ar -deus we #0| / Jjr akereane {fen ecmines wih7toe62 fe{Tes th 2are 2 ffm Lfror 4fens "10 8 20 25x Fig.5:13. Plooflogy1/+versus Gy—G2i/V/E withG=1.6anda dy varying C=289+1.62" (FromB.K,Hyde1.PrianandG. Salboep, ToeNeclor Prope ofteFeary Elen, Vol.1,Precsice-Hall, Tne.(964), reprinted bypermission.) oz One-Dimensional Potenials 93 ; With more energetic aemission, heGfactor depends onR,andwith ©.R= red\"t, onefinds thatrisaconstant, thati,chatthenotion of2Coulom’ bare aking overcheroleofthepotential beyond thenuclear radius hassome "validity. Again, simple quliative considerations explain thedata a‘Thefactthattheprobabilityofareaction(c.g,capcure)betweennuclei Fisattenuated bythefactor any 6-65) Pimples thatatiowenergies and/or forhighZ's,such reactionsaerare.Thtis whyallattempes tomake thermonuclear reactors concentrateontheburningof Inydrogen (actualy heavy hydrogen—deuterium). UP HR Hel+ (4.27 Mev) : APSE SAH +p (403 Mev) \ekg APH Het (176 Mev) | sincereactions involving higherZelementswouldrequtemuchhigherenergies, [5Ranehighest maeendeemuchiheengi, |problems, Forthesamereason,neutronsareusedinnuclearreactorstofission |F<theheavyelements.Protons,atthelowenergiesavailable,wouldnotbeableto| 775 getnearenough tochenuclei toreact withthem. ‘ag E,One-Dimensional ModelofMolecule ie Someaspectsofwhatgivesrtecomoleculesareexhibitedbytheexample AE. ofaparticleinadoublepocentialwell(Fig.5-14).Thealgebraicworkisgreatly Ge,simplified ifweconsider asquare wellinthe limitofgreatdepthwiththewideh 4 ‘going tozerosuchthatVaremains aconstant. Inthatcasewegetadelta. 4% fonction well,which isveryeasytohandle, Justcoshowthis,consider fist2 F single atuactive porencil well ; N Cnt v1)=—™0 666) “Theequation tobesolved is,when E<0, PdeFA)suis)=*ia)ate 6-67)+ where f=2mELA ‘Thesolution everywhere, except atx=0,mustsatisfy theequation du/dx* —su=0,andifitistovanishatx—++©,wemusthave sam x0 =e x<o 6-66) 94 Quantum Physics L , ao Pree Fig.5-14. Double one-dimensional potential well.Theshape ofthewave Fanction fora bound sate iskerched in. “The coeficienss infront arechesame (and here chosen tobeuny—we can normalize afterwards) because ofthecontinuity ofthewave function. The derivative ofthewave function isnolonget continuous. Asargued before (Eq.5-13)wehaveaede» @),.-@_--i oo “Thelastrelation gives theeigenvalue condition > that is » “3 . (5-70) “Thedouble square wellwilbereplaced by Comite)Vix)=—Bex—0)+e+a on Because thepotential issymmettic under theinerchange x—>—,weexpect thatcere willbesolutions ofdefinite party, andwewilfrstconsider cheeven sclutions One-Dimensional Porentials 95 1,For the even solution wewrite a) =e x>a : =Acohex a>x>—0 =e x<-a (72) j. tndcontinuity ofthe wave function gives ots Acothee 6-73) [Because ofthesymmetry, itissufficient toapply thediscontinuity conditionforthederivativeatx=a.Nothingnewwillcomeoftheapplication atx=—2 Weget 2 mae —Ashea =—*em 74) ‘andtheeigenvalue condition is 4 » ahaa = ay (75) iguce 5-15shows thisgraphically. Thereisonlyoneintersection pointofthe ‘euretanhywith(N/3)—1.Icisobvious thatwhen y=2,cherightsideiszer0, E-whereas tanhy>0,Thus theincersecton point occu fory<A.Ontheotherhnand,sinceanhy<1,wemusthaveQ\/7)<2attheintersection point,thatis, d ox (5-76) TEwecompare thiswich(5.70), weseethatthe eneegy forchedouble wellis ager negative narber, thatis,cheenergy forthe double poceatal istower. Note thatthisisnotbecause somehow chestrength ofapairofpotentialsislarger thancharofasinglepotential,asmightbethecaseifonecomparedanelecron, 3\~ Fig.5-15.Solutionoftheeigenvalueconditionanhyarn 96 Quancum Physics boundcotwoprotons withanelectron boundtooneproton. Thelagerbindingistherebecause, asFig.5-16indicate, itiseasiervoaccommodate asharply ‘dropping exponential toasymmetric Function (herecoshx)with discon~tinuityinslopeasgiven,chanicicoaccommodate iroanequilly sharply‘koppiogexponentialontheohersideofthepotential.Intherealwot,asingle tlectton bound totwoprotons separated. by«smalldistance willhavealower“energythan1singleprotonplus4hydrogenstomfaraway,eventhoughinthefrstcisethereisamoreeffective repulsion berween cheprorons. Again itische ‘wayinwhichthewavefunction canaccommodate itselftothegeometrical situation thatisthedominane effect 2.Theoddsolution willhave theform a) x>e =Ashe a> «>a ane kn om ‘Again,because ofthesntispmmenry, itissuficient toapplytheconditions a¢ie=asay,Continuity ofthewavefunctiongivesAsich va= (78) andthe‘discontinuity equation reads » meet =cAcoshas =~Nem (5-79) (| { Fig.5-16. Bound statewavefunctions forsingle and double dea function attractive potentials F One-DizsecsionalPotentials97 Geo a 2srt tare .4 B.‘Fig.5-17. Solution oftheeigenvalue condition anhy=(4/7)-1). 2‘Combiningthetwoyieldstheeigenvalueconditioné Fs cothng=> 6-80) "5Figure5.17shows4potoftherecproclofthisequation,haeis,anhyageinse “sQ/y~1)~.Therewillonlybeanintersection iftheslopeoftheformer nthe fAorigin islargerthanthatofthesecond, thatisif 4 dot (an TE Aty=0/2thecerm(X/y~1)ivaleady at1,40thatthe iatessection hadto“HE occur fory<0/2,thati, x »3 «2 (6-82) Thus theoddsolutio’,ifthereisabound‘State,islessstronglyboundthanthevensolucion. Thewavefunction, whichhastogotough te fede ispbecween hewell,andthuscanonlyatcommerite tolesapyfalling exponential. Depending onthesizeof, theremayormaynotexistapindate1cusnowconsersuperposition ofthegroundsateau),withenergy E,andtheexcitedstateu(x),with‘energyE,(eand0standforevenandodd) His)=nla)+mle) (83) With«chosen50a0make" dss]? asamaasposible,thci,wih 98 Quantum Physics the“electton” localized, asfaraspossible, ontheright side. After atime f,the weave function will be Has) =le)«HM +cx) MH RM Eads) +eEO fe) (5-84) thatis,chephase relationship between theewoparswillchange. Inparticula, aera vim such char HBB wy (55) the“electron willbelocalized onthelefesideinexactly thesme waythatit ‘easlocalized ontherightat¢=0,Thustheeianoscillatory behavior, which maybedescribed bytheelectron going backandforthbetween thetwopo- tentials, with frequency EWEw= dae 686) ‘Weshall leave ittothereader coconvince himself thatthepetiod associated withthefrequency inei,forlarge),approximately equal tothe“eunneling time" across thebarr separating thetwowells, asmight bedetermined from themateria) presented iSections CandD.Thisisamodel fortheammonia, molecule, There ateways ofmeasuring such afrequency with high precision, tndthuswehaveatourdisposal averyaccurate “clock.”* F.The Kronig-Penney Model Metals generally haveacrystallin stuctute, thatis,theionsarearranged in ‘awaythatexhibits aspatial periodicity. Thisperiodicity hasaneffect onthe motion ofthefreeelectrons inthemetal,andthiseffectisexhibitedinthe simplemodel that wewll now discus. ‘Theperiodicity willbebuileintothepotential, forwhich werequire that Vix+2)=Vix). (5-87) Since thekinetic energy term —(K2/2m)(d?/dst) isunaltered bychechange x+x +4,thewhole Hamiltonian isinvariant under displacements bya.Fotthe ‘aseofzeropotential, when thesolucion corresponding toagiven energy B= BR/2mis He) = 6-88) ‘Fordncastonofcheammoniamolec,weR.P_eyoman, RBLeighton,and 1M,Sands,TbeRowmanLacesonPiya,VolIl,AddisonWeel,Reading,Mas,1963. 4 One-Dimensional Porentials 99 thedisplacement yields Het a)=irtel =aey(n) 29) thatis,theoriginal solution multiplied byaphase factor, sothat Worei=WGI (650) f;.The observables will therefore bethesameatxasatx+a,thatis,wecannotcell whetherweareatxoratx+4.Inourexampleweshallalsoinsistchaty(#)and Wx+a)differonlyby2phasefactor,whichneednot,however,beofthe form ¢**, Tosimplify thealgebea, wewilltakeaseries ofrepulsive delta-function potentials, Br s Vix)=So Bee—ma) Gon) f)Away fromthepoints x=na,thesolution willbethacofthefeeparticle equation, thatis,some linear combination ofsin&xandcoskx(wedealwith ‘ealfunctions forsimplicity). Letusassume thacintheregion Redefined by(e—1)aSx<ne,wehave W(x) =AysinA(x—ma)+Bycosk(x—na) (5-92) 4 andintheregionRey,definedbynaSx<(w+1)awehave FH)=dasesinMe—(0+1)a]+Bycosdle=(+1)4)(593) EContinuity ofthewave function implies that(x=ma) Aayasinha+BuyscO8ka=By 694) tndthediscontinuity condition (5-13) here reads g fay605ha+bissinbe~bs=*By 695) ©Alide manipulation yietds Aa=Aycosha+(gcoska—sinha)By Bur=(gsiake+coska)By+Aysinha (5-96) where g=Whe . ‘Therequirement thatchewavefunctions (5.92)and(5.93) berelated by (Ran) =#*HR) 97) issatis it ; Aas =6Ay f Bays=0By (5-98) 100 Quantum Physics ‘when tisisinserted into(5.96), wefindaconsistency condition thatreads (e*—cosha)le*—gsinha—c0shi)=sinagcosba—sinbe) thar is, 8 =69(2 cos ka+gsinha)+1=0 Muliplicaion by€gives cos@=cosha+gsinba(599) wetakeperiodicboundseyconditionsforour"ryts0that VRaiw) =HERD (6-100) then icfollows fam (5.98) that¢¥#=3,chais, o=hm m=o,41,42, (5-101)x Wedenore@by4,whereqsthewavenuenberofanelectoninboxoflength Ne,withperiodic boundary conditions andwithout anypotential, chatis, ‘without anyionspresent. Thus(5-98) should berewritten iatheform inhe cosga=cosa+30 _(102) “Thisisaveryinteresting result,because cheleftsideisalways bounded by1sthati,therearerestrictions onthepossible ranges oftheenergy E=4/20thatdependontheparametersofout‘eysal.”Figure5-18showsaplotofthe fanction cos#-+ sinx/2¥asafunctionofx=a.Thehorizontallinetepee-sents theboundsoacosgs,andtheregionsofx,forwhichthecurveliesoutside thestip,areforbidden regions. Thusthereareallowed energybandsseparaed bytegionsthatateforbidden.Norethattheone:ofaforbiddenbandcorespondstothe condition hae me w=£1,22,43... (6-103) “This,however, isjustthecondition forBraggreflection withnormal incidence.‘TheKronigPenneymodelhassomerelevancecothetheoryofmetal,insulators, andsemiconductors ifwetakeintoaccount thefact(tobestudied tare)thatenergy levelsoccupied byelectrons cannot accept moreelectrons.‘Thosametalmayhaveanenergy bandpaccally filled.Ifanexernal fedis applied, theelections areaccelerated, andiftherearemomentum statesaail-abletothem,cheelectionswilloccupythemomentumstacesundertheinfluence Ofahecleceie field.Insulators havecompletely filledbands, andanelecsic feldcannot accelerate electrons, sincetherearenoneighboring empry sates. Tchecece Beldisstrong enough, theelectrons can“jump” actoss afor-HfcheelecsricBetheintoanemptyallowedenergyband,Thiscorresptggds One-Dimensional Potentials 101 cone ae Fig. 5-18. Plotofcosx+(4/2)(sinx/x)a8«fonctionofx:Thehorizontallines representhebounds-£1.Thecegionsofxforwhichthecurvelinesoutsidethe atti are forbidden tothebreakdown ofaninsulator.Thesemiconductor isaninsulatorwithaverytatrow forbidden gap.There, small changes ofconditions, forexample, arsein ,_femperature, canproduce che"jump" andtheinsulator becomes aconductor. G.The Harmonic Oscillator [Asouslastexample weconsider theharmonic oscilatoe (Fig.3.19). Ia contrast totheexamples dealtwithuatilnow,thedifferents) equation that reedscobesolved isnosotrivial, andoneteason fordiscussing thisproblem is itolearnsomething abour thetechnique forsolving suchequations 102 Quancum Physics ve (" [a= Pp ————n=0 a] _ | _ Fig.5-19.Harmonicoscilasoreigenfunctions, andprobsbilry densitiesfortheloves foureigenvalues, Notetheevenness andoddness properties oftheeigenfunctions “TheclassicalHamiltonian isoftheform aaEat (2109) sothatcheeigenvalue equation is eats) )oe dt+Pertu(x) =Balx) (5-105) Weintroduce thefrequency oftheoscillator w=Vile (5-106) tite Ht (107) andchange variables to yezs (5-108) ra One-Dimensional Powis 10810falygeethesimplerformoftheequation ns=7t+k-y)u=0 (5-1Bunn 0) DAllquantities thatappear aredimensionless Foranysigenaive «7°» othecminvolving eisnegligible, andwe amusetherefore equ tht#0)asymprosaly suly theee ; AO—sui=0 Gato) Wemulkiply by24h/dy, which allows ustoremit chsinthefom d(sy a e\e) ~7y @)=0 (5-111),a\a) 7% orequimleay, d|(du\?~5)~Met|=—2ym™ (5-212) 5lG)-> ]--» Gu) PPsingsesdelepehve0gidoftegu Bon,Weassumethatchscanbedoneandthencheckththesoupesome.wedroptheghtsie,wefadthee dn : Sm (tre! ‘whereCis&constantofintegration. Sincebothayy)andduc/dymustvanishat Fntinicy, wemuse have C=6Thus Pa te 9by=e ¢) Fwhore solution, acepable tinny is : sy)=ert (114) ‘Wecannowcheckthat2a?=2-Hisindeednegligible computed with 4 ‘ 4yay=4Germ —aper ey(uct) o(Fev) Ayes forlugeyiwenowincovuce«newfonctionM4),suchthat 6)=AG)ev (5-115) theneefeel equation ieasly eentoakethefrm ee en re 104 Quantum Physics ‘Thismaynotseemlikemuchofasimplification, butwehaveaccounted forthebehavior atinfinity, andwecannowlookatthebebavior neary=0,Letus atcempe apower Series expansion hy=Eyer (17) ‘When chisiinsereed incotheequation, wefindthatthecoeficiets ofy* satisfy therecursion relasion (e+Doe+2)dag=Om—e+1)he 18)“Thus,givenananday,cheevenandoddseriescanbegeneatedseparately, Thattheydonotmixis&consequence oftheinvariance oftheHamiltonian underreflections. Forarbitrary e,wefindthatforlargem(say#>N) nya2oa 6-119) “This means thatthe solution isapproximately AG)=(4polynomial in9) +[reise wegne | ad NeNWF? TNNDOTSD ~ where, forsimplicicy, wehaveonlytaken theevensolution. Theseriesmaybe ‘wren inthe form n(nOme4GIne] {or(7)!late=m*evar*awarewhichisoftheformofapolynomial +aconstantX*e#,Whenthisisinserced.Jato(5-115), wegetasolution thatdoesnotvanish atinfinity. Anacceptable solution canbefound ifthe recusion felation terminates, that is,if e=2NH1 (2120) Forthatparticular value oftherecursion relations yield peNON=2).(N=2+MIN-2+ Gy, aun(22) on (2) aod ye(NEANN1).(N=2+BN ED Gg, ans=(2) ay ary ‘Thus che results are 1.These arediscrete, equally spaced eigenvalues. (5-120) translates into B= hale +9) (6123) One-Dimensional Potentials 105 4Formthatlooksfamiliar, sincetherelation between energy andfeequency isthe saine asthatdiscovered byPlanck forthe radiation fieldmodes. This n0acct dent, sincexdecomposition oftheelecomagnetic feldintonormal moses is essentially adecomposition incoharmonic oscillators thataredecoupled, F 2.Thepolynomials A(y)ate,except fornormalization constants, the Hermite polynomials H,()), whose properties maybefound inmanytextbooks Wearenotreally interested inthese dewil, andwewilsolve theharmonic ‘oscillator problem again, sothatwedonocpursue chesemates, Ii,howevet,‘worthpointingoutthatthereasonfotheimporanceofthehainonieoscillatorinquantum mechanics, asinclassical mechanics, ischatanysmallperturbation‘ofasystemftomicsequiibeium statewilgiverisetosmalloscillations, whichaeukimately decomposable intonormal mods, thatis,independent osallars. 3.As(5123)shows,eventhelowestscathassomeenergy,thezo:pint nergy.espresenceisapurelyquancam mechanical efect,andcanbeinerpacted inteamsoftheuncenaintyprincipe,Isthezero-pointenergythatisesponsible forthefactthathelium doesnot"freeze" atextremely lowtemperatures, but femains liquid down totemperatures oftheorder of10"?degrees Kelvin, at Fmotnal pressures, Thefrequency cislarger forlighter atoms, which iwhythe effect isnotseenfornitrogen, say.Italsodepends ondetailed features ofthe Ffinveratomic forces, whichiswhylguidhydrogen Joesfee P Problems 1.Consider anarbitrary potential localized onafinitepartofthex-axis, ‘Thesolutions oftheSchrBdinger equation totheeftandtothe rightofthe f,potential region aregiven by respectively. Show that ifwewite C= SiA+ SD B= SuA +SD ()-@ 96 B) Nba Sad, 106 Quantum Physics tharthefollowing relations hold Isal®+(Sil*=1 [Sal* +(Sel? =1 SySh +Sas =0 “Thisisequivalent tothestatement thatthemairix (SuSis s(s:) isunitary (int, Usefluxconservation andthepossibly thatAandDarearbiteary ‘complex aumbers.) 2.Calculatetheelementsofthescatteringmatrix,Si,Su,Su,andSeefor thepoceatial Vix) =0 x<nme =Vy -acxce .=o xca ‘andshowthatthegeneral conditions proved inProblem 1areindeed satisfied. 3.Theelements Su:...S:sarefunctions of&.Showthat Su(—#) =Sh) Sa(-A) =S2Cb) Sis(—A) =SCA) that i,that chematrix hastheproperty S-#) =8) +4.Considertheoddsolutiontochepotentialwell(e.g.,Eq.5-39),which canbeusedasamodelfor three-dimensional pocencalwellwithzeroangular‘momentum. Iftherangeofchepotentialisgivenobe14X10-"*cmandchebinding energy ofasystemis—2.2MeV,andifthemasstobeusedis0.8X 10-™pm,findthedepthofthepotentialinMeV. [Hinss, (1)First, conver distances andmasses intounitsofsome mass, sothat therangeisd(f/ue) andthebinding energy isofcheform«(yc"). Aconvenient tnassmight betheonegiven. (2)Thebinding energy isverysmall, sochatisalmost zero, Ifiewere 270, condition (5-11) would yield 7».Expand about thisvalue] 5.Without actualy solving theSchrddinger equation, setup thesolutions neDimensional Potentials 107 ‘0thatonlythematching ofeigenfunctions andtheiderivatives remain tobe dove forthe following situations: ‘withthefollowing conditions (a)fluxKt/m would beincident fromthelefeif thepotenils were absent; ake E<Vs : vary ‘withfluxofmagnitude fub/mincidentfromtherightifchepotentialwereabsent, Reve 6.Showthattheconditions forabound stare(5-33) maybeobtained by ‘equiing thevanishingofthedenominators in(5-25)at&=ix.Canyougivean argument forwhythissnotanacident? 7.Consider thescattering mattx forthe potenti » VQ)=aie->) Show thc ichastefxm te ka — ike —* Dany _ike Fike 3° ke—% FProve cateisunary andthai wilyieldeecondition forbound states when 108 Quantum Physic thecementsofcatmatrixbecomeinfinite.(Thiswillonlyoceutfor<0: 8,Calculate ainEq583, which willlocalize theparticle sf asposible fnthesight sideoftheorigin. 9.Workoutindetailthewavefunctionsforthethreelowesteigenfunc- tionsoftheharmonicoscar. 10,Consider theharmonic oscillator potential perturbed byasmall cubic tem, sothat (a)=nat(=-Le) Ifaislarge (compared tothecharacteristic dimension (5/ms), estimate howJongiteesparticleinthegroundstateto“leakout”totheregiononthefatright. Note thar wth cisperurbation alone, there inolowest energy sate, since folarge enough =thepocentsl becomes arial deep 11Consider thepotential shown below vee]‘Wo-. ith st v9)=ED eoome Estimace the liferime ofaparticleofenergyEinthispotential.(Theoutside potential represents acentrifugal barrier inathree-dimensional world.) Express your result interms ofthedimensioniess ratio //AR, where E=W'k*/2m. Take In. 12.ConsidertheKronigPenneypotentialwith haar (4)Make adetailed plotof bsax cone +SH a One-Dimensional Potentials 109 Basa function ofx=ka. (b)Showthacforbidden energybandssarejustaboveke=wr. (©)Show thattheallowed energy bands getnatrower as}increases q (a)Plottheenergy i%4*/2m asafunction of¢. 13.Consider themodel ofamolecule definedby(5.71).Showthatwhen a hi E and ee References F>TheKronig-Peaney model isalodiscussed indetail in E.Merzbacher, Quantum Mechanics (2ndedition), JohnWiley andSon,NewYork, 1970 Fora moredeiled discussion of“bundtheory” sec Kittel,tnradctont SliStatePic(thedition),JobnWileyandSons,: Tne, New York, 1971, Chapter 9. phForamorecompletediscussionofbarterpenetration, usingtheWKBapproxi‘mation,seeanyofthemoreadvanced textbooks listedattheendofthebook |chapter6 TheGeneral Structure of Wave Mechanics tnChapter 5theeneigy sigefaactons, chitis,thescutions ofthe aston o a Hus(x)=Bus(x) 1) wih Peeya2(hay() HmoTHe)a(té +Vx) (62) rteobtined foranumberof physilyinerening Haritontns. TheHam Conn opetorH wasempha, becuse isthtoperator thatdecriesthetimedevelopment ofasystean,Theinitialsateof«systemcanbedescribedbyaywavefunction Ye)whichisonlyeonseninedbytesogucemeathae favo He)<@ (63) E_thatis,chacitbesquare integrable. The¥(x)can,without lossofgenerality, be tliplied byaconstant, soaes nomad favs vex)= (6-4) ‘ThetimedependentSengereqetion he 7aMm)=WKxs) (65) describes tetimedevelopment ofthewavefonction, andgiven theenegyfigeafoncions urs)thspblemanbeolsedena:Whanae 112° Quantum Physics thesolution is@general theorem chatstates: anarbitrary function (x)canbe ‘npended inscomplete serofeigenfunctions ofH,tat, Hx) =ZXComets) (66) Ifwechoosetheeigenfunctions ofHtobenormalizedandifwetakeintoAccount tattheeigenfonctions comesponding codiferent values ofEare ‘orthogonal, 0that feb)neo)de=bee co) arenule proved inAppendix Bshen [aeszto vo)=3cofsiteare) =ZLCobew =o (6-8) that, theexpansion coefcients aedetermined. Nowtheiedependence for cach energy sigenfonction i seule) =onl) oe 69) ‘ascaneasily beseenwhen theabove issubstituted into(6-5), andhence Hoa) =EDCee als) (10) 1kshould benoted, aswehaveleaned from cinsideting 4large number of cuamples, chattheencgy eigenvalues maytakeondiscrete values and/or con- Cinuous wlues. Wespeak ofthe spam ofeigenvalcs being, discrete and/or ‘continuous. Thus (6-6) really reads He)=Ecanna)+fdBc06) can comesponding tothetwopossibiles, and(6-7)reads fhala)ma,(2)de=Ban 1) forthediscrete values, and fsis)unl)de=MEBD) 3) forthe continuovs one. Tiss northeonly possible choice. Aswesain our‘Slcnontofproblemswidhpotentialwelofbri,thesolutionsoftheenergy a ‘TheGeneral Structure ofWaveMechanics 113, igenvalue equation canbemadeupoffunctions chatfarfromthepotential are pp,momentum eigenfunctions. Thereis2relation berween theenergyandthe Emomentum (E=p/2e awayfromthepotential) anditturnsouttobepossible Ecnotmalizethesolutionssothacherighsieof(6-13)isplacedby8(p—p’) ‘of,inthreedimensions by4(p~p’). : Wealsopostulated aninterpretation fortheexpansion coefficients: |Ce|* fis theprobability tharanenergy measurement ofthestatedescribed by$x) B:yields theparticular eigenvalue F.Anyparticular measurement canoalyyield [>ancigenvalue, burincontrast roclassical physics, wecannot predice which oneE.itwillbe:weonlyhavetheprobabilicy thatitwillbeaparticularvalueE.InGeer quantum mechanics, asinclassical theory, ameasuremeat mustberepto-Biducible tohaveanymeaning, Thusifanobserver, uponmeking «singlemeas-Gy,urement onasystemfindschattheenergyis,say,F,thenasubsequent energyGjmeasurement forthatsystem mustagain yieldBj.Heace, afterthefistmeasure. ment, thestateofchesystemisdescribed byanewwavefunction, namely theBBE cigeafunction ws,(x); onlythenwillarepeated measurement yieldE,wich probability 1.Theexpression "ameasurement projectsastateincoaneigenstate BLofcheobservable” issometimesused, q‘Theexpansiontheoremmaybeviewedasageneralization oftheexpansion fofavectorAintermsoforthonormal unitvectorsinanN-dimensional vector ce Space 4 A=ath+oats+... tant 1s) Theunitvectors t,satisfy * tats =be 615) andaretheanalogsofs(x),Thecoefficients ayaregivenby 4 asl 616) :band theyaretheanalogsoftheCy.Weshalloftenusethelanguage ofvector Espaces intaking about quantum mechanics. Thus weshall often refertothe f_Conficieats Grasthe“projections” ofY(s)“along” w(x), andthequantity . Gofate vede (67) willoften bealled scaler product. Wewil,following Diet, introduce acon- fvenient notation forthescalar product 3 fi8°) HO)de=ly) (61) ‘Thesimilarity between theacceptable wavefunctions, andehecollection FofallNedimensional vectorsisactuallyquitedeep,JustxsthesumofanytwoFvectors yields avector q A+B=G (6-19) t: . 114 Quantum Physic ovtheproduct ofavector with number agunaver, wildhesumof faytwosquare integrable functions againbeasquare integrable function, as ‘wiltheproduce ofasquare integrable function withanarbitrary (complex)umber,Inbodsaxes,providedwedefinethenotionofasalarproduce, (A|B) =AB (6-20) iy)=[xovx) (621)imeoer,wehavealineatvectorspace,Theonlydiferencisthinquantuminechanics, dhevector spce isiaiite dimensiooa, Infact,sincein(6-21) iis‘hecontawusabelxthatplascherolethattheindexplasin AB= Dah (6-22) ‘weseethatthespace iscontinuously infinite, ‘Thisdoesmean thataproper Trachematel totment ofsuchvector space ismuch moee complied, sincequestionsofconvergence ofintegralslike(6-21)havetobefaced,andincon-fisto#inedimensional space, proving completeness much more dificle Tamathematical parlance, thesquateintegable functions formaHilbert space, andcheencigy Cigenfunctions form 8complies ofbaieo.Tavectorspaces,betheyfinitedimensogal otmotegenes,anoperatoriseind vobesomething thatuansforms avector intoanother vector, ointhis ‘ase,2square integrable function intoanother square integrable function. Wefecralyineesed inlnareratorthahavethepeopertythat Hats +Bhi)=alls +Blths 2) “Thesimple example discussed inChapter4showed!datheexpecationvaleof H,defined by (oy=[946ae om ‘wasreal.Thisiscobeexpected foraphysically measurable quantity, andit[rnenlizes othesatementthattheexpectationvalue,forally(s),ofaoperatorepeseating anobservable quantity, hastobetel,Wecalled operators that Ind thispropery, herman. Foran ubioary linear operator A,wehave are=[ore1)de 625) and cage=fcaveWe)de (628) j ‘TheGenelSuuctoreofWaveMechanics 115 |. Theopertor At(pronounced A-dagge) isdefined bytherelation 3 [sv 009de=foreaveae (627) Poet ceheingaFr eae einve Vv d i ayiaUE (4)'--z $)-—Simitcly, thehermixian conjugate oftheoperator (&-«) AB iseasly shown tobe 4 (nate‘); ae Forahermitin opertor 23 Cee=fuse $00ae =fre14ae =Wy ; =fre96aae (628) Andsincethisisreforall y(a),wesaychat wen 2) . “TheDizac notation forsalar product chatinvolve operators is § fore)avea=aie) (630) ; Thus! ‘ lain =[Lavin)* 6)de [ste dinion ofie(627 oatinte eneexgeceson aeofAisytocheckbywing#0)=04Roa)wsbegasaneayaneiN | imps sherp betas ncoeode toi 6), 116 Quantum Pays =froAlOlx)dee =Wile) (x) “Teress ftoutdive towatd generality isasend seen, cheftthat Hisaottheoalyoperator ofinterest, Other physical observables, suchasthetmomentumope'oturparty,positon,and3onaerepresentedbyhermanoperators. Wesalle thecers AyB,C... foroper, andsince weae nlydealing withoperators tharepretenscbterable, theyaeallhemi, tha is Ana bow tnd soon “Aeran operators haveeigenfunctions, tha thee exis xofvectorshathavetheproperyaheoperating oatemreproducesthem,Teepe foe«proporloaliy const, teegenalae Asg(x) =ag(x) (633) “Thespecrum ofeigenvalues, 1sforthe Hamiltonian, maybediscrete and/or contmuous. Thespectrum ofmomentum eigenvalue wasfound eobecon tinuous; thatoftheparity eigenvalues, +1wasdiscrete, Asfortheenergy cigen- functions, those coesponding todiferent valves of«ateorthogonal, andthe ters maybeehonen tobe nove 0thatwehave furtela)de=Baa’) (6-34) easinou ew aotation (ia\tar) =Blaya’) (6-35) Here O(aye") maybeaKroenecker delta dueifthe eigenvalues arediscrete, ora Diracdeltafunction&(¢—a’)iftheyarecontinuous. Itfollowsfrom(6-33) and (6-34) that amfu)Ante (636) chatis, |a= (w[Alm) (637) ‘Ths theeigenvaluesofhermanoperatormustbea.Sincetheresusofan individal easement oftheober decribed byAmast beoneofthe tigeovaluc, isnt be3, " ‘TheGeneral Structure ofWave Mechanics 117 JustasforcheHamiltonian, wefoundinChapter4thattheeigenfunctions ‘ofother hermitian operators alsoformacemplate st,30thattheexpensiontheorem HW)=Dcals) (638) where G=futongtedemGul) 639) holds,Theinterpretation ofCyisaguinthatofaprobabilityamplitude,catis,|Col?Hischeprobability offindingtheeigenvalue inmaking1measurementofAona systemdescribedbyYs).Againafer2measurement, reproducibility requires “hatchesystem befound incheeigenstate (+) Inboththeproblems discussed inChapter4,theparticleinthebox,and theficeparticle, wefound thartheeigenfunctions weresimultaneous eigen- |fimctions ofHandanother operatot, parity inthefrstcase,momentum inthe second, andwesawthatinbothcasestheadditional operators commuted withLetusnowexaminethegeneralconditions underwhichthishappens.Theeigenfunctions u,,corresponding totheeigenvalue «oftheoperatorA, Aad) =aul) (6-40) willbesimultancous eigenfunctions ofanother operstor B,when Baa(s) =buss) 41) ‘This, however, implies that aangyyBs)=Abas)=ble)=abuts) aod BAwg(x) =Baug(x) =aBas(s) =abia(s) tha is,that (AB~BA)w(x) =0 (62) Ifthisweretoholdforjustoneaitwould notbeveryinteresting, butifithholdsforthecomplete sete,thenitmeans thatforallsquare integrable func:tionsHs)=SCaml, DX(AB —BA)mlx)=(AB BAD Corals) =(AB~BA)(x)=0643) Ethatis,cheoperators commute [4.8] =0 44) 318 Quancum Physics Conversely, ifwehavetwoheemitian operators AandBthatcommute, sothat (6-44) holds, then ABu,(x) =BAug(x) =Bats) (6-45) tha i, AlBins)} =alBes)] (646) “Ths hefonction Bus) isalsoaneigenfunction ofAwitheigenvalue 4,1f thereisonlyoneeigenfunction of.Acotresponding totheeigenvalue 4,then thisimplies thatBug(x) must beproportional to(x), thatis, Buua(x) =bua(x) (6-47) “Then u(x) isasimultaneous eigenfunction ofAandB.Thissituation, inwhich theeigenfunctions ofAatenotdegenerate istheonethatwesiwfortheparticleinthebox.If,ontheother hand, there aretwoeigenfunctions ofAcorrespond ingtotheeigenvalue a,thuis,wehaveatwofold degeneracy AusD(2) =aa) Ani?(3)=asx) (648) 2scoaton lostated inthefepanicle ample, chenwecnonlyassethacBu?(x)andBx(2"(x)mustbelinearcombinations of1,"(x)andws°(x): BuSP(s) =Bans) +barns?) BuSP(x) =Bus(x)+bas) (6-49) Ieis evident, however, thatwecantakelinearcombinations oftheseequationsco ‘obtain equations ofthe type Beis) =bush) BeiP(s) =bss) (6-50) Forexample, Bows? +ul?) =(ar+Nn)wi?+(ba+Nb)al? =balae! a?) provided wechoose Xsuchthat bat Me i futMn * “ThisisaquadraticequationandtherewillbetwovaluesofA,corresponding (0theevoeigenalesbeismorepropetodenotesimultaneousigen functions ofAandBin(6-50)byuii(=)and4{9(x).Sincethesecorrespond to = TheGene Stace ofWaveMechanis 119 {different eigenvalues oftheopentor B,theywilbeorthogonal toeachother. FS)Inpractic,forewofolddegeneracy, thedegenerateeigenfunctions of4,iftheyGareakentobeorthogonal coeachother(geand«frthefeeparticle $e ase), willautomatially beeigenfunctions ofB. 4 Evenafterfinding eigenfunctions ofAandthenmaking linear combina-Eonsthatareigenfunctions ofacommuting operatB,theremaysilbesoneBSdegeneracy, thatis,chereareseven eigenfunctions ofAandBsimultaneously,Bee withthesame «andb,Thismeans thatthere must beathirdoperatorCthat EG)commutes wichbothAandB,andthefunctionscinbeeecombined tobe HEAsimultaneous eigenfunctions of4,B,andCwhoseeigenvalues distinguish the Boe.degencaare eigenfunctions ofAandB.Thiswillgoonuntilthere isnomore Fe egenctagy. Theserofmurcally commuting operators, B,C... Mofwhichfoursetoffunctionsisasetofcommoneigenfunctions iscalled&completetof b commuting errabe. Wehave 4 14,B= (4,=...=14,M)=0 j (8,d=(B, Dj=...= (8,m=0 s)) and soon 4 Aa al) =ae a8) q Bias ol) =bah of) Ms al) =mia ft) (652) ‘Thestatedesctibed byHase..n(s) hasdefinite values oftheobservables A,B, BG... M.Thisisthe laigest possible aroune ofinformation thatwecanhave abour «system allatonce.Thereason isthatifweconsider another operator thatisnotsome function oftheoperators 4,B, ...,M(since these commute, such4function isunambiguously defined), theaameasutement ofitwllnot JBive2sharp valueforthestate#at..n(t). Ingeneral, ifewooperators donot ‘omanute, chenatypeofuncertainty relation connects theprecision withwhich [the ewoabservables canbedetermined : Todemonstrate this,wemustfrstagreeonadefinition ofuncertainty. ‘Aatual defiition is ; (aay =(a) —(ay 63) socalled chediersin. Ithastheadvantage thatitdoesaoevanish evenif(A)=0,anditvanishesifcheexpectation valucistakeniaaneigenstateofA E Notethatwemayalsowegethisas re(aay =(a=(apry os) E since 3 (4=244A)+(AY)=Ca")~(Ayiay 120 Quansum Physics “This shows thac(4,4) deals with themagnitude offluctuations about themean, Ikisscrtighsforward toshow (seeAppendix B)that (aayiany —\AABIY 20 (635) “Thus forxandp,forwhich [xp] =if,itfollows thae ie (axap)? >4 (656) Notice thatinthederivation nousewasmade ofwave properties, x-space ot ‘pspace fonctions, otparticle-wave duality. Ourresult depends ently onthe ‘operator properties oftheobservables AandB. ~ ‘Letusnowtutntotheimportant question oftheclassical limitofquantum theory. Todothiswemust fiststudy thetime development ofexpectation values ofoperators, Ingeneral, cheexpectation value ofanoperator changes withcime, Itmaychange withtimebecause cheoperator asanexplicit time Aependence, forexample, theoperatct x+p/m, anditalsochanges withtime because theexpectation value istaken withrespect 10awave function that )itselfchangeswithtime.Ifwewrite (Ah=foeoAM)de (6-57) thea 4va), 5) 24qinfva)Sveade aves)+faAMES)de tas) aed +fiV(xs)Aesde aa 1 --C4),+f(twees)Med) +fea (tHs) aA) iye =(28) Efe mateoae -;fVA)AMDde The Genel Srcrte ofWave Mechanics 121 that is, 4 aa\ i : Gian= (4)+Ena (658) Inthedesivtion wemade useofthefctthatHis hermitian operator, We observe datifAhasnoexplicit time dependence, chen chechange ofthe expectation value foranytates 4 i : Faun Ful. 59) JUftheoperatorcommutes withH,chenitsexpectation valucisalwaysconstant,thats, wemaysaychatsheobereableis conan ofthe melo. cheHamikonianisoneofthe complete setofcommuting observables, then alltheothers are coostats ofthe motion. Tetwsconsider successively A xandA~p.Wefisthave 40 4 4 =FMaly : =H((2 ve) t([:=+Vix)x Now xcommutes with anyfanction ofx, Weal =o (660) fo chatweonly have tocalculate Uo) =Hol +oe > 4, ea E Thus weobttia a = (2) (6a) Next wehave Me tea@=k({Z+veve]) =-|vey 663) since pandpevidenly commute. Toevaluate thelastcommutnor, wenoe that 122 Quantum Physis pve40)~veopeed=*4weyyoat-*veo2yoo & () ny_FMyy ew tort ee lave)=222 665) and thus 4. faves) ‘Wemaycombine (662) and(6-66) 0obtain a (dv(x)aon=—(22) (oon ‘This looks veryouch lketheequation ofition ofacssical pont pastice inapotential V(x) Pea _dea)ET dea wos) ‘Theonly thing thatkeeps usfom making theidentification a= @) (60) int w\ od(frre 670) Under circumstances where theabowe iaegutly becomes aoapproximate equality, themotion isessentially classical, aswasfestnoted byEhsenfest. This |requir thatthe portal beaslowly waving Function ofcsargument. Iwe wate 40) F(x)=—ae (71) then FQ)= GbMex) =©) aay a)=C0)+Ge~(a9)PGE)+EO mele+ a Wftheuncertainty (4x)?=((x—(r))*)issmall,andthehighertemsiatheRE) expansion canbeneglected, thenwehave TS APG)SePGR)+x=(83)PU) a=FD) (on) = ESIrisindeedtuethatevenforelectronsandothetsubatomicparticles,(6-72)can{Sy_bevalid,Formacroscopic fields(672)isagoodapproximation, andthsallowsHe.uscodescribeelectronofprotonorbitsinamacetlenitorbymeansofdasic]ES Htions ofmotion, mit|AB Problems EFonlyhermivanifandBcommute,thati,ifAB=BA,and(2)cheoperator© (Ase Bis heemician, : 2,Provethat4+A”and(A—A!)arehetmitian foranyoperator, asisaa 3,Provethatif#1is«hermitianoperator,thenthehermitianconjugate ‘operator ofei(defined cobeSSiH"/n!) isthe operator 4.Prove theSchwartz inequality j wivr(ols) >|#19))* Notethatchisisequivalent tocos*@<1forthreedimensional vectors HintConsider(J+N4l¥+M8)DOandcalculatethevalueofdthatmini-sizes theLs.) 5.ConsiderEq,6-58and6-39,Calculae(ply)foranarbitrary¢inrerms ‘of(|4,), andshowthatitispossible towrice I) =DLOle) (ual) Inasense chesumover«complete setof ; ZX[eddael isequivalent totheunitoperator, 7.IfAishermitian, show that(42) >0, 8.Consice thehermitian operator Hthathastheproperty hat Wet 124 Quinn Physics WharaetheigenoftheopentorH?Wheaetecgeaaluesif Hisatreoicted tobog hemian? 9.Amoperator issaidtobeunitary ifichastheproperty chat w= uu=1 ‘Show thatif(J|¥)=1chen(Up|UP)=1. 10,Show that if ishemi hen sway 11,Show thatifthe {|form anorthonormal complete st,with (o|me) =dan then the set |v)=U|ee) withUunitaryisalsoorthonormal. (Themeaningoftheaboveisaunitary ‘operator acting onasetof“basis” states yields another setof“basis” states.)12,UsethedefinitionofdxadApgivenia(638)coshowha apae~ hn fora pice inninit bointheste charcerined bythequantum sun- 13.Show thatifthehermivian conjugate operator Atisdefined by(6-27), then fatavcar vor=[acerave | (Hing,Seefootnote1,p.115.) 14.Usethecommutation rettions between themomentum pandthe | position toobain theequrtions describing thetime dependence f(r) and {p)given theHonins 5 =Bk pmte .a) Honda(uhxt+ox+0) J Pg tates (b)H—+dats e Solve thefirstsetofequations (Hamiltonian (a)). i ‘TheGener! Sructue ofWave Mechanics 125 References ‘Thegeneral struceue ofwavemechanics isdiscussed inallbooksonquantum gmechanics, See,forexample, among themoreintroductory books, D.Bohm, Qeantum Theory, Prentice Hall,Ine.1951 |,RH.Dickeand).P.Wie,Inada QuantumMechanic,Addison-Wesley Publishing Co.,Inc, 1960. J.PowellandB.Crasemann, Quantum Mechanic, Addison-Wesley PublishingCo, Ine, 1961 E,Metcbacher, QuantumMechania, JohnWileyandSons,ln.(1970),isamong = themore advanced textbooks |chapter 7 Operator Methods inQuantum Mechanics ‘hedicostion ofthe gene suet ofwavemechanics placed equa Fy wcighe onseopeatos thtepsen hecowie sndseheaaJF Rncons. Albough thenerweeatone potdeatbed wsnals enewthonoemalbastfaitvecesianNinel ceseewah fwould craisly downgrade deminimptance theyhedoahenge [Seedtopaytheleading tleinowdscunion ofhesyed eatin |.Gap 5.Ins chap wewilhow, wangswan ee Paes4congoveryfrroarbioghegeriuetooneag ea|ted) tacthedesipion ofeigenfontions otsbesreon Seea ha Imreabu. Theinteisiportant betas aGe aveoaed[-functions thatdepend onxofonp.Weshallseclaterthatthereexistobservables dhaanno bestochted with Capeeinayditemayedfedane ee abseact notion ofwget mastbedeveloped They tomate itbees tomewhat chart inthecouse ofthesluten aftarering UeRessewelro‘The ont hstheform HeEsimate oa) wnaxandpetepeas.Wednovichatperepentedby(di) Theonlyvesigeoftheexistepesenuton datwesoa ea thesarenec efthefasamene! fon een vk tea)=% 72) “ree ef polenta ec sb, ve en qoo isopentane a ie ci 128 Quantum Physics CCssicaly theHamironian could befactored intovan ~wo(JB (VP) tdCee |RE butbecause pandxdonotcommute, wehave [Fea(VPe+oe) “(y2!ami)V28+!am =i eeomt2Ea2(px—99) =H Yo 3) Le asnow itrodce the notation - aAnNat im fro; Be pofe Be .Baa om 4 Sincexandparehermicanoperators,heubeingofthesecondopemtoewith« daggerisappropriate. Tetwooperstorsdootcommute,Wemaycomputefas Inas oe ee de tataVFeFale[feeVF] =h (7-3) ‘andrewritetheHamiltonian intermsofthe newoperators, H= YotoA (7-6) “The simplicity oftheHamikonian isreflected inthe simplicy ofthe commutation relations ofAandAtwithH,Wehave {H,A)=|eA'A,A) =of4’.A)A =-feA (7-7) andsimisiy [H,A") =(wA'A,A') =#A'[A,A"] , =fwAl (7-8) +eshamakrepeatedweofhetlefrcommunebenAppendBs (A+ TAG HIB, Clad 48, =AB.O+ 1A.eiofcouseeettatheoeropenaedae Operator Methods inQuantum Mechanics 129 Incideataly, iisauseful technical trickindesiving commutation relations involving hermitian edjoine operators torecall that :(4B=(AB—BA)=Bat—A'Bt=[B44](79) PInparticular (HAN =(A) ==[8,4 , =(fwd)! (7-10) from which (7-8) follows, Lecusnowwritedown theeigenvalue equation, which reads Hug=Eay (an) 1 Inthepast,whenever wewrote down suchanequation, theimplication wasthatHcontainedsomedifferential operatorsliked/deandthatuywasafunction (ofx.Thatwasappropriate whenoutoperttorswerespecificallytiedcothespicedefined byallsquare integrable functions ofx,butinwhatwearedoing now,[wearenotbeingveryspecific aboutwhatouroperators operate on.Weshallassume thattheyaredefined insomeabstaact vectorspace,andventethatsabsuractvectorspacecothespaceoffunctionsofxlater.Totranslatethisabstraction intothelanguage thatweusetodesctibe theequations, weshallnotFespeakofeigenfunctions butofeigensate,andwhatwecalledwavefunctionsFtwavepackets,weshallnowcallaeeects.Thustheeigenfunction tat..n(2)ofthemaximal commuting setofobservables canbereplaced bytheeigenvectoroteigenstateofchsmaximalcommuting set,43,.n;thelabels,,.=mgivethevaluesoftheeigenvalues oftheobservables ,B, ...,M,andthisdescrip.tion, without thex,doesexplicitly show themaximum information content. Lecusnowtake (7-7) andhave itactonua. HAuy —Atlag =~huAuy f With thehelpof(7-11) thisbecomes HAug =(E~ ha)Aap (712) ‘Thisequation satesthatifupisaneigenstate ofHwithcigenvalue F,thenAa isalsoaneigenstace ofHbutwitheigenvalue E—hu,thatis,withenergyloweted byoneunitof A he 73) 7Wemay therefore write Aug~AB)un. (714) ‘Theconstant c(B)inecessary, sinceevenifupisnormalized to1,Aueneednot 130, Quancum Physics be,Inouremphasis onseparttion fromx-depeadence, thenormalization condi- tion thee wasalways weitten as vieacs)de= Jsnow, using thenotation defined in(6-18), written as Gurls) =1 (055) ‘Weshall always aormalize alleigenstates co1,unless they belong, cothecon- tinwam, iawhich case hae)fe-8) Ce) Vocaip—p) 16) Iwenowapply (7-7) 0thestaveun, weSind, inexacy thesame way, tbatAuge, of,equivalently, A'uy gives astate ofenergy E—24Thus byrepeatedapplication oftheoperatorAtoanyaewecangeneratestatesoflowerandlower energy. Aisappropriately called alowering operator. There islimittohowmanytimesiccanbeapplied,sinceitisaconsequence of(7-1)thatH!mustalways have positive expectation valve. Foranarbitrary wave function wiry)=[reDM)defovwor (oh)be -|WILA)de ="f[aeny/del? de>0 oan ‘which werewnte inovrcoordinate-deemphasizing way as Wir lv)=GLb) =Wiph>0 (718) Simiily, since isalso ahersitian operator Wletly) =Gly) =davly) 20 as) andallscalar products ofvectors with themselves yield thesquare oftheirlength,hitis,apostivenumbet.Thusourloweringproceduremustendsome: Pwhere,andthereiagrandsate,whichwewillnowdenoteby4»,beyondwhich {lowering ends. This must mean chat Aum =0 (7-20) @ ‘Operator Methods inQuantum Mechanics. 131 BTheenergyofthegroundstateis 4 Huy=(AA +Yh)to=Vous ony Letusapply (7-8) totheground state - Hates ~Ah =fe — 4 HAlg =(faa+Mis) Abus (9-22) ‘Theenergy hasbeen mised byoneuniof andA apy described a raining operator. Wewillchange outnotation altl, namely, labelthestateby Fe checumber ofenexgy units«=faithasovercheground sateenergy Hc Tas wewie q Aly =oy (7-23) ‘Note that(7-12) implies that Ey Any=cg (7-24) Wey sothatA!andAmoveupanddownthesame"ladder." Allthestatesmaybe graeme bycopened application of°com. One Consequence isthatthe caagy spears gen by E=(n+})ho nw 01,2, (725) Fe have succded inobtaining theenergy specrum without solving any Aileen equation, Wehaveao genea!epesennton ofthecgemecto, Lay ; “=yal) * 9) Fhe wehvepotinthecone oamaliation constant Wit hehelp ofthis $ presentation, wecanprove theorthogonality ofeigenstates corresponding to ifn neice, Whats involved ianewan ofancxpsion othe rm a (ueA™(AT)*| wo) Andthisisdonebycommuting 4trough theAt thesgh, where, he “Forhealgstsioy ented end,weSib iyhewyinwichhii derived.With(A}*me=citewehave|én|*alin)=|eg|?=GA")wel(Ata)= “3(walAAan). Now wen (13) tdeie hon 2) =A+ CA), When tater ewe al ona tesco es onetn atsavalenlyf=MBWe.antowoofgene,conte 132 Quancum Physics end, they aton giving «vanishing resl. Using (7-3) wesee,forexample, that BUA =AAA BYLAY) =BALAN +AA AIA +h)At=3hA(ANP +A(ANY A ‘Thelastterm, sandwiched beeween theusgives zerobecause Ay=0,andthe firstrermcanbemanipulated inthesamewaytofinallygiveGi?A'.Now (io|A'sty)=(Auto|ue)=0 (7-2) sothatwehave proved (asl) =0.This procedure, when fllowed inthe ageoenl cnt, allows ustoprove that (ili)=0men (728) “Thestatement chatanatbicay state vectot canbeexpanded ineigenstates of nowsends inthe coondiote independent way voEGe (729) and since (qt) =Bom,Weave Ge id (730) Wedigress bey fiom chemain chest ofthischapter copoint outthat therising andlowering operators maysao beused toadvantage insolving the tarmonic oscar equation. {aspace, (7-20) reads (JZtifa) aa=o oa) Usingchexrepresentation oftheoperatorpp=(H/ild/ds)thisis (mx+h4)ds)=0 (72) This isasimple diferential equation, whose solution is w(x) =Comer (7-33) “The constant Cisdetermined bytherequinement that a(x) benormalized ro unity teof”dere , -o(#)"d 4 ‘Operarot Methods inQuantum Mechanics 133, that is, 3 ce(=)" (7-34) ‘Wemayalsoobtain theexcited sates bywotking outindesl im : sls)=7 (ANml)vai -aal2)"VS- VE oo Vatte)ANoVimsds)© 35) ‘Thisis,infact,averycompact wayofwriting outchegenetal solution ofthe diflerenia! equation, Wehavesucceeded inmakingthepointthatonecansolvefoetheeigen valuesoftheharmonic oscillator usingoperatormethods alone.Forthisproblem allthatisneeded tospecify thecigenstates istheenergy, thatis,theintegetB= 01,2... appearingin B=(nt Bho aadthusthecomplete setofcommuting observables consists ofHslone «Thus thelabel#ontheeigenstate w,describes itswhole content. Wewould therefore bequitewilling togiveuptheprivileged coleofthe eigenfunction inxapace, ‘4(), excep foronepoint: us(x) doesprovide swithmoreinforaation inthat itgivesustheprobability density [via|(x)|*] offinding theparticle atx.Does thisadditional content single outhe*-space wave function afterall? Letas reall theroleofthewavefunction inmomentuin space4())thatappears in ‘Chapter 3forexample. AstheFourier transform ofthex-space function it ‘ighthavehudsomeclaimtoprivileged roe,bulate,in(4-59), forexample, ye explained chat$(p)was“merely” anexpansion coefficient ofanatbieuryW(x)incigenstaces ofchemomentum operator, andthatiswhyitsabsolute square yielded theprobability offinding«momencum pforthatsate.Similay, thefacethas|¥(x)/*yieldscheprobabileydensityoffadingxforthepostion ofthesystem couldbeinterpreted bychestatement thatUx)istheexpansioncoeffcient ofanarbitrary abstract stateineigenstates ofthepostion operatorep.Wewrite theeigenvalue equation abscacty a8 Kobe=Xe (7-36) keepingxas.subscrpctosessthattilabeloftheeigenstate,justasmisthe labelforaeThespectrum ofxzp,&henmitian operator, iscontinuous, 30that 47payisconned ineeabelm.Sateswitwnenaeposcve ptysates, Andshovewith oddhavenepeive pay. Tisfellows fomteaethesae ecses‘and Aweof 134 Quantum Physics theexpansiontheorem,instedofakingfulike(7-25),alyads o-[ecw 3 Since thecgensttes defined in(736 orm aethonotal et, (albe) =ae—x) (738) wwe deve Clx) =el¥) (7-39) snthisquanitheprobabiryampleforSinding«paceatmorespecifically, chemeasurement ofthe observable xwillyieldtheeigenvalue»with probability |C{s)|¥. Allwehavetodoischange thenocation, rewriting (737) a8 vefaves (40) toshow charthewave function inspe hasnopivieged oe, andwewet |only a84mater ofconvenience, Thebasic principles delwith opemeos aad ther eigenvectors andeigenen tnabst sace, andthe reste ate of ripreentation. The latter is,ofcourse, crucial inobtaining numbers, which is star physics all about ‘That iswhywewilnotlytomach ssn onthe formal scructure ofthetheory, andcontinue using wave functions. Later wewill Ive todal with opentors thahave nocsi analog, sucht theisinie ‘pinofelectrons andother particles, ndthere wewilvec oufreedom 80 toeother epreeatatons, "Weconlode thiscaper bydiscusing thetine development ofasysteminourrepresentation independent, way.Thetime-dependent. Scvedinget 0~yy) ony {snowanoperator equation insaabstract space. 4)isavector, anditpointsinsdceciontatdependontime,Theequationcncaybesaved.Thesla,toa i, vs)=eA (0) (7-42) where ¥(0) isthevector attime#=0andtheoperator~M*"*isdefinedby in ital as) or at “The solution (7-42) allows ustodescribe thechange with time oftheexpects tinale ofome opentorAthatdoesataveanyexpiedependence: (Ae =WO 44) Operator Methods inQuantum Mechanics 135 =e Yf0)|Ae™™*4H(0)) =(0) A Yo) =WO AO H0)) q =O. ~ (7-44) We used (ety apn gen (v4) along theway, and defined Aya dt480 (7-46) What(7-44)saysisthactheexpectationvalueofatimeindependentoperatorA - ‘onastatethatvaties withtimeas(7-42) maybewritten astheexpectation valueof«time-varying opertorAQ)[givenby(7-46)]inthetime-independent state0). Thisisveryusefl imtheFormal discussion ofquantum mechanics, since it isconvenient costupbasis oforthonormal eigenvectors intheabstrace vectorspaceonceforall,andnotworryabouthowthebasisvectorschangewithtime.‘When wedothis,weareworking intheHeienberg picture, whercas keeping A without timedependence means thatweareworking intheSchridinger picture Theresult isthe same, wharever picture weuse:thisisanalogous cotheoptionofdescribing2rotatingbodyrelativetoafixedsetofaxcs,ofofdescribingthebody attestinarotating coordinate syscem. Thechoice isoneofconvenience Ifwedowork intheHeisenberg picture, theastatevectors afefixed, andwe neednotrefertothem. Howanobservable vaties withtimeisdetermined by (7-46), which yields 3 4 GHgun4ituFmgyitr a)=FEgt4iEgnayyeit A=" ye An =5NAW -|a =|HAO) -) 4formremarkably like(6-59).Thatequationwasanequationforexpectationvalues, butsinceitsfoumwasindependent ofthestateiawhich theexpectation value wastaken, ithadtoelect operator properties, and(7-7) shows that cexplicicl. For the harmonic oscilatoe H= AA +tho 136 Quannim Physie and since Hi constant ofthemotion, wehave H=oA'@) AO) +Hho (74) Wecanalsoshow, using (7-46), chat (a0, A100) =8 (74) Hence (7-7) and(7-8) silhave thesame form, andweget 4 Ai) =iad 44)=inst) (730)ra=if 50) ‘Thus thetime dependenceofA(t)andAT()isobtainedbysolving(7-50),with che resale that As) =e™ Alo) A= #40) crs) Using therelation (7-4) itiseasytoshow that 0)=p(0)cosat~max(0)sinat x09=x0)coset+Osinot (752) expressingtheopertor() andpl)inersoftheoperatorsx0)and0) Problems 1.Use thecommutation elation (7-5) and the definition ofthestate given in(7-26) toprove thar Ain=Vik ns | (Hins,Useinduction, thatis,showthatifthisrelation istrucfor#itistrueforsrt sand esablsh itducly for =3 2,Usetheabove relation toshow thatif(4) isanypolynomial inA’, thea MA4=LA, Operator Methods inQuantum Mechanics 137 [Nore thatrepresenting Aintheform a anne |isconsistent withthecommutation felaton (7-5)ndisquiteanalogous tothe presentation haaries 3.Calculate theform of(va|+|uq), andshow thatitvanishes unless amet f(Hintisslficenttocalculate(in|Alta)since(ealA!itn)=(Atlin)=4 Wn|Alse)*. Usecheresults ofProblem 1.) 4.Use theresults ofProblem 2toshow that MUAY) sy=F(A’ +Mh)a (Hint, Expand theexponential in«series, andusethefactthat feesa=EAP) ; rn) a 4FO) =Ff) towork outthisproblem.) 5.Usetheresults ofProblem 4toestablish theoperator relation OAFAN) eM=f(AT +A) Notethatanopemtor relation mustholdwhenitatsonanarbicary sae,Letsoabitrarystatebeoftheformg(t).Thuswhatmustbeprovedisthat 4FAN) OMgC!) ty=f(T+i)6A!) ap ‘Thiscamalsobeproved fomthegeneral relation OAM AMaat]+ELA, 14.Al)+ 6.Usetheaboverelationtoprovethat GAN 8gkDaonan Theprocedute isthefolowing. Let ptt 908Re) Differentiation withrespect todyields ‘ ae (aAbat)OAHoyedpy)4peME (24+bat) AtBy)+oeSE 138 Quantum Physics chats, (A+ bat)M4FO) =a4on)400aan Use Problem 30show that aF (t a (GAT—abhi)FOX) sotha Fin)=204etna 17.Usetheprocedure ofproblem 5toshow chat 4!(A)eMwmf(A~Mi) Show fom this that, ptt Gh gagah ‘sing themethod outlined inProblem 6. 8.Use theabove resule toshow hat fw LANA fA) Ag Fim) Noe that«=(1/-/2ma)(A+ 4") Usethisexpression tocalculate wole!**| 49) 9,Show thus theresult obtained above isthe same astheone obtained fom idels)ul) 10.Usethegeneral operator equation ofmotion (7-47) t0solve forthe time dependence oftheoperator x()given that Heca+mgt) 1,Coasider theHamiltonian describing «one-dimeasional oscilla in anexternal electric eld. POs amare —etal) ‘Calculate theequation ofmotioa forthe operators p(t)andxi)using Eq.7-47 and the commutation elation a Ueto= Operator Methods inQuantam Mechanics 139 Showthattheequation ofmotion isjusttheclassical equation ofmotion. Solve for{)and(6)interms ofp(0)andx(0). Show that bx] #0 forn # ‘Thisshows thatoperators thatcommute atthesametimeneednotcommute at different times. 12.UseEq,7-35t0calculate theeigenfunctions forn=1,2,3.Note.Be surecokeeptrickoftheordering ofandd/dsintheexpansion ofthebinomial 4 References ‘Thematerial discussed inthischapter isalsoereated inalmost allofthebooksinthereferencelistattheendofthebook.Thestudentisencouraged tolookUpsomeofthem,sinceitisalways usefulcoscethesumebasicmacetial presented from different points ofview. |chapter8 N-Particle Systems Out dicusion ofasinglepailiseaslygenealivedtoanN-panice systan.TheNparsatedescribedbya'wavefuncion Yon,seee imonmalzed such that f[aes denxx,Xm)Pm (8.1) j;Theinterpretation of[¥enay... xy)|? isageneralization oftheiaterpretssionof|¥4x)|¥, chais,iyieldstheprobability density forfinding particle 1a¢ fsosparideZac,pileNatxy.TheGanedevlopment efechmacefuncrion isgivenbythesolution ofthedifferential equation aiheevn, ani)=Hin, tNA) (8-2) where theHamiltonian sain constructed inconespondence withthe casifom Hm Psves cae) (63) (22 a ldCoe Ee) ‘hewholeformalism ofquncum mechanics developed beforeiseasilygenene,withtheprovisotatopenersdescribingvaglepailshricommte whentheyrefertodiferent pics, forcoe A [px= 7fw (8-5) 142 Quantum Physics Ifcherearenoexternalfields,suchasthecommongravitationalfieldofthe ‘earth,orexternallyimposedelectricormagneticfelds,chentheporenilenergy {anonlydepend ontherelative separation ofthe particles, thais, Vi= V(x ~~ x, XN —XW) (8-6) “Thismustbethecase,becauseintheabsenceofanyexternalagencythatsome-howdetermines an“origin,” thedisplacement ofthe whole system should nocchangeanyphysicalpropertiesofchesystem.Inotherwords,theformofthepotential (9-6)is«consequence oftheinvariance ofallphysically significantquaativiesundertheeansformationmoute en ‘Averyimporcint special caseof(86)isthe caseofrwo-bory forces, inwhich ase ve DM (6-8) “Thesummation isoverall indices ‘andj,subject cothecondition i>j10avoid double counting, andthecounting of#=j.Actually,inthedescriptionof tlecttons inanatom, wewilbedealing withthecommon Coulomb potential, 25wellastheelectron-<lectton repulsion, andthere thenucleus provides an‘tiga.Thepotentialinthatcaseisachrdimensional generalization of Lwed+LMowx) 9) ‘When there arenoextemal forces, then inclassical mechanics thetotal ‘momentum isconserved. This follows from theequations ofmotion Px; amm Voy =ass acss 9) (BAO) 4consequence ofwhichischar ‘ dn me a Zh us =- 5 View ae... en=Xy) t BG Day Ven )(ay =0 “Thereason forthevanishing ofthetightsideofthe above equation isthatforeveryargumentinV,thereareequalandoppositecontributions thatcomefrom¥0/Oxc acting onit.Thus be p=Dae 1) is«constantofthemoton. NePanice Sycems 143 tnquantum mechanics thesimeconclusion holds. Weshalldemonstate itbyusing theinvariance oftheHamiltonian under thetaasformation (8.7) “Theinvariance implies thacboth Huslxs, x2 19) =Buel, x, +xw) (813) and Hite, +4,2+ay... +)=Eup(xs+4,0244,aw+a)(8-14) Bold. Letustakewinfinitesimal, sothatteamsofO(t)canbeneglected.Then 2 :wher+4, ew+a)alm. xe)“a6,w(x,+n) 2 q taanwhen, XW)+ a Hscosa)te ale se) andhence, subeeacing (9-13) from (8-14) cary (2tS2)eatssay)=a8(32)esteem) ar}=@(E2)eons... =(%2)Hashim) (8-19) ewe now define path Seadn (e19 Weseethat wehave demonstrated chat (HP—PH)we(x:, xn)=0 (8-17) Sincetheenergy eigenstates foeN-paricles presumably formacomplete setofFstates,inthesensechaanyfunctionofmyx«24eanbeexpandedinttsofallthea... x)theaboveequationcanbetranslatedinto (HPI¥en,..-,xn)=0 (ts) forallYs,n),thais,intotheoperatorrelation [HP] =0'(8-19) ‘This,however,impliesthasP,thetormomentumofthesystem, iconsant of ‘hematon. Thisi2verydeepconsequence ofwhatirealy xsateanene shoe 144 Quaneurn Physics thenatureofspace,Thestatementchatheteisnoorigin,thatis,chatchelawsofphysics areinvariant under displacement byafixeddistance, ladstoacon- Servation law.Inparticle physics therearenopotentials ofcheformchatwe consider here;nevertheless theinvariance principle, asstated above, stilleadsto 2conserved tor momenta. ‘Oumain incest willbeinthesuo-partle sem, which wediscuss next. oeewononinteracting particles wehave thesimple Hamilronian ayeBemYom (29) ‘Wemighe expect thatsincethetwoparticles aretotaly unconelate, thepeoba-bilityoffindingoneatx:andtheotheratxistheproductoftwoindependentprobabilities Plas 2)=Pos) Pls (2 “Thusweexpectthatthesolutionof ee Re . (-seni 2m2)sey,3)=Eels) (8-22) should beseparable into alas,29)=dC) 6609) 623) ‘Subscicuing tisinto(8-22) anddividing byals,=)weget =W/amyPo)/dat),—(8/2)toale/de?) TCfm\Polsdides)4PhaEoeWe) |e(ead én) * als) on “Thetwoterms intheequation depend ondiffrent variables, andthatiwhywe secboth ofthem equal tothe constants FandFssespectvely E=hth BPH) pyeamdat Pte Bolo)ae ae Pate (625) ‘Theewoequations areeasily solved, andweget afi) =cde (626) with pm2m gymals : weOEgs= 27) NePanide Stems 145 Lecusnowrewrite thesolution using thecoordinates mnt mis xoRt me a thari,theseparation berween theparticles, andthecenter ofasscooeiate we wtite musi+mse fins+hats=afer—3)+9BT eeae a) wefind that amhthek stabs—my onem =A sotha thesolution hasthefoem alae)=COREpie 29) [whereK=y+feithewavenumbercoresponding tothetotalmomentum, [and4isthewaveaumber contesponding totherelative momentum, Thefisfactorrepresentschemovionofthecenterofmuss,andthesecondfacotitheFiner wavefunction. Theenergy maybewttten as we wera -ee (142 30) Ferrey ME(ata) 9 Thefirstfactoristheenergyofthetwo-perticle system,withmasssy+aysmoving feelywthherotamomentum thesecond teistheitera enegyweintroduce theredid mas, defined by tetyh @31)aint thentheteamisB42/2u, whichiseectvely aone-particle energy namely, chatofafeeparticle withmass and momenculn WhentheHamiltonian in(8-20)isaleedbytheaddin of«potential thatdepends onx1—xyonly thenwehave Bo woa (-23- Fesdep)Hts)+Vinsa)oan)=Bate)(032) Using thecoordinates x=etmeecathe "say 146 Quanrum Physics soda wexete ae x- Be (34) altdealgebra shows thattheequation takestheform eo ee= BO yy)ala) = BaleX) Ifwesite she,X)=09) (836) wwefindthattheequation for$(2)is BPD)|iy yde+V(x)G(x)=abl) (8-37) thatis,aone-particle Schrédinger equation withreduced mass, andenergy p-Se (638) . 2hans +) aChapter 9wewillobtain theseparation inasomewhat moresophisticatedway.Wenowturntotheproblemofidewicalparties.“There iscompelling evidence thatelectrons areindistinguishable. Ifeissecrenotso,thenchespectrumofantom,ty,helium,wouldvaryfomexpetitmenttoexperiment, depending on“batkind” ofelectrons werecontzined ia jt.Nosuchvariation haseverbeenobserved. Similarly, nucleat spectarealways thesame, adicating thatprocons areindstinguishable, asareneutrons Similar : evidence fiom‘highenergy physics experiments indicates verystrongly that i ‘cher perticies, forexample, pimesons, ateasoindistiguishable. Thisis@ | purely quantua-mechanicalpfoperey: inclassical mechanics itispossible to I follow theotbits ofllparticles (inprinciple) sochatcheyarenever relly indistinguishable. ‘Weshall leam thatclecrons arecharuceried byanintemal quantum umber, called thepi,andthuschitstates mustinclade intheirdescription thespinlabel,Thishisfurther effectontheconsequences ofindistingisha- bilgy, which wediscuss next. "AHamiltonian forindstingushable percicles must becompletly sym-meticinthecoordinates oftheparticles,Foraewo-partcle system,ichereis0dependence onthespinlabels, cheHamiltonian is n= PEvee) (639) NPariie Stems 147 with Posen) =Vbeses) (6-40) Wewrite thissymmetry symbolically as 042) =HQ.) (en andicunderstood thaiftheHamiltonian doesdepend onthespinsoftheParticles,chenthespinsaecobeinchudedinthelabeling"1,"'"2."Awavefunctonfoeaa.parce system, withllcheparticlesicentical,willbedenotedby M4, 2... N),andthisstandsforthemoreexplicitvi,e1:4003-24, oy) wheretheoisdescibe thespinsates Foratwo-particle system theenergy eigenvalue equation teads H(,2) exf1.2) =Big,2) (o2) |Since thelabeling doesnormaccer, wemaywritecis4s H(2,) wx(21) =Baxt2a) (o43) (Ontheother hand, using (8-41) wealsohave H(,2) wal.) =Bue(21) (44) TFwenowfollowtheformalapproach thatweusedinourdiscussion ofparity,vewillintroduce anexchenge operator Ps,which, actingonasateinttchangesallcoordinates(spaceandsp)ofparis1and2.ThedefinitionofPssoees that Padi) ~v2.) (645) Eq,8-44 may bewriten asfollows HPs we,2) =Eg2.1) =BP (1,2) =PasEae(.2) =PusHtue(.2) (646) ndthis,asusual, implies theoperator telaton ‘pad =0 (47) ‘ThusPivtke pay, is2constant ofthemotion. Also,ikeparity Pad?vlt.2) =v4.2) (6-49) sothatcheeigenvalues ofPyare 1.Thecigensates arethesymmettc andantisymmetric combinations 02) =Fea) +42.0) 148 Quantum Physics 1 e42) =Jalvtia)~420} 9) “ThefactthatPyisaconstant ofthemotion implies thatasate thatissym metic aan inital te willalways besymmmetac, andananisymmeric state willalways beantisymmetric. Teisanimportant lawofmatere thatthe spmmeny ofantsymmetty under theinterchange ofowopurtcles isacharacteristic oftheparticles, andnotsomethingthascanbeartingedinthepreparationoftheinitialstate.Thelaw,which wasdiscovered byPauli, tates that 1Systems consisting of‘identical pariles ofhal-odd-integeal spin(ie,spin1/2,3/2,...)atedescribedbyantisymmetric wavefunctions. Suchparticles acecalled fermions, andaresaidtoobeyFermi-Dita statistic.2,Systemsconsistingofidenticalparticlesofintegral spim(spin 0,1,2, .-) | aredescribed bysymmertic wave functions, Such particles arecalled bosons, fandatesaid coobey Bose-Finstein stasis.“ThelawextendstoN-particlesates.ForasystemofNidencialfetmions, thewavefunctionisantsymmeaic undertheinterchange ofanypairofpatticles. Forexample, athrce-particle wavefunction, properly antisyrnmetrized, has the fom 1 yous)=4 - v.23)=FglK.23) —8)+¥23.0) ~43.21) +¥G.t2) —v4.29) (850) whereas thethree identical boton wave Fonction hastheform 1b23) =Se 4K )HCL)=Tela) +¥aAL3)+¥232) +32,1) +3.1.2) +100,3,2)) (8-51), etusnowconsider @veryintceting special caseiawhich Nfermions donotinteract with each other, butdointerac with acommon potential. In that case we (32) where . n=E+ve (53) “Theeigenseaes ofthe one-particle potential acedenoted bysn(s)where Henle) =Buin) sa) NePanice Sysems 49 solution of Hug(, 2, IN) =Ewe(, 2, »N) (8-55) x(1,2, NN)=wax)alsa)... wegen) (8-56) whee Ex+ Ext +Ev=E (8-57) In(8-56) wesuppressed theo;labels thatgowiththex.Ourtasknowisto sacisymmetrze (8-56). Ifthereareonlytwoparticles, weevidently have 4 1 902) =Solanla) sale) —mle) une) (858 (1,2)=lam) se)—an)am) , With tte pasties, eheform is 4 1 (323)=tne) me) ens) ~tn) wn) ns 10.23)=Febans)anoanes)—tne)wm)a) t+mew) myles) ey(s) —wn,(x)we,(%s) (9) d “+eG) a) ans) —a) ma) wn) (59) For Nparticles, theanswerisadeterminant, theso-called Slaterdeerminant: Ly “)=| anlen) ene)... ene 0.2N)=|ned(23)...wsGew)| |e sols)...ani)| |sre)ab).aeto)|Ggggy (Cleat theinterchange oftwoparticles involves theinterchange oftwocolurans inchedecezminan, andthischanges thesign.Ifwoelectons actin thesume energy eigenstate, frexample, E,=Ey,aadiftheyateinthesaespinsae,thatis,thespinlabelsareche'same 0;=0,thenthedetetminant vanishes when 1 x-thavis,theelectronscannotbeatthesamephice-Thus chezequizement ofantitymmety incoduces aneffective intescion between twoecnioor ‘qualitatively weseechattwopaticewin thesnes tndcosayawayfomchother,sincethejointwavefonction vanishes whenthetsepetion. goestorio, Ths even noninteracting parle behavewiftheeweesepahane 1ThewavefunctionforNidenticalbosonsis‘tocallysymmetric, endthegeneral formseben! byexpunge lerminat m(60)andann hag 150° Quantum Physics imeracion becween chem, Wewillsethasacomplete setofcommutingbserablesforelectronsincludesanadditionalwo-valuedobserabeastociatedwrththespin.Thusastateofgivenenergy, angslar momnencom, parity, andso ‘,cambeoccupied by«woelectrons (ofoppesite spinvariable), butby90‘motethantwoelectrons. ThisisarestrictedversionofthePauliexclusionprinciple.“The statement “no twoelectons canbeinthesime quantum sate” sutikes onebyisglobal nature, Suppose wehave2hydrogen atom inthe {ound stateoneathandanother hydeogen atom iaitsground stateonthe tnoon. Does thismein thathewoelectrons must beinopposite spin staes?‘Toanswerthis,wenotetha¢specification ofthestateofthetwoelecrons requites notjuste staerment thatthe electrons havespin“op” orspin“down ndchaecheyaintheground sates ofthierespective atoms, butitalso requires«specification oftheenergyofcheatomsHowwelldoweknowthese? Suppose weconsidera boxofwidth L,andsuppose theatoms aelocalized in 0<x<L/Aand 3L/A <x<Lrespectively. Then themomentum ofthe atoms canbedetermined with anaccuricy thatisresrced bytheuncertainty Pinciple. Theposible values oftheenegy aegiven by ent OME (0) from which wededuce thatpossible valves ofthe momentum aze -™aaa Measurements ofthemomenta ofatoms arerestricted bytheuncertainty. relation nh ah ; aw eo andhence therenespcs canonlybedetermined withanaccuricy aBM Bent ’ap~Ae~oe (663) “This, however, islarger chan ten Bn—Ea ME (8-64) afac,foratoms separated by1meter, say,moving withvelocity 10cm/sec fn~10!,50thaethere is00posslty hatin 2ractoscopc scuation theretillbeconicewithclassicalinition.fnef,ifchetwoatomsaelabeledATidB,thequestion iswhether chte is«difference berween using thewave function Ya(x1) #x(x2) and als) Yolo) —Yale) veld va ea jo (8-70) Een[vant(S) 152 Quantum Physics ‘Thos theenergy perparticle Efe --* 71N72ne 7) {grows with K*,Equivalently, foragiven energy, theaumber ofbosons filling thewellisproportional toE,while thenumber offermions filing thewellisproportional toE",Thehighestleveltobefilledinthefermioncaseistheoneforwhich m=N/2, anditsenergy is Aten? = @n) ‘Thesubscripe Fhasbeen putinbecause thisenergy iscalled theFermi energy. ‘Wemay writeitintermsofthedensityoffermions,whichintheone-dimensional problem isN/b =p,as Wet Brae 673) Weshall return tothesignificance ofthese remarks inChapter 9, “Theexchision principle plays anextremely important roleinthestructure‘ofaroms.Theenotmoustichnessinthevatietyofchemicalpropertiesofthe various elements isdirectly trceable tothefactchatonly alimited number of slectrons canoccupy agiven energy eigenstate Problems H 1.Whatisthereducedmassofanelectron-peocon system?Howdocsit i difer from thereduced mass ofanelectron-deuteron system? What isthe reduced mass ofasystemoftwoidenticalparticles? 2,Prove that theexchange operator Preis hermitian. 3.Consider rwo noninteracting electrons inaninfinite potential well‘Wheeischegroundstatewavefunctioniftherwelecttonsareinthesamespinstate? 4.Consider wo elections inthestme spin state, interacting, with « potential Vian al)= “te Iam Se = 0 dhsewhere ‘Whac isthelowest energy ofthetwo-tlectron state, assuming that thetotal momentum ofthe ewo electrons is2er0? NePartcle Systeme 153 [itins.Separatecheequationinamannerleadingto(8-37)andthenapplythePauli principle 5.Consider «woidentical paricles, eachofspin0interacting withpo- tentialenergy : Visa) =Klar —x)=G2+aT" where xand—x,aetheequilibrium postions ofthe particles. ‘Whatisthespectrum ofthetwo-particle system?Whtisthespectrum of thesystem when theidentica) particles have spin1/2? j 6.Consider twoidentical patcies described bytheenergy operator a Hm Hip) +Hp) where 99)=FoSmuts Separate outchecenter ofmassmotion, andobcain theenergy spectrum forcis |system. Show thaticagrees withthatobtained bysolving Hl) =Bae) with 4 Hee) =sales) inte) Discuss thedegeneracy ofthe energy spectrum. References Seeanyofthereferences listedatthe endofChapter 6andalso D.S.Saxon, Hementary Quantum Mechanic, Holden-Day, Inc,1968.5D,Park,Intradustion ttheQuantumTheory,McGraw-Hill Co,1964, |chapter9 TheSchrédinger Equation in Three Dimensions ‘TheHamioianfor«singlepricemovingindeedimensonalspace vas BaP EPSys) (91) wichwewienthefoe aBan on Tetheedimeninal moseatun phastherepesnaion pete 3) Forewopaces inte dimension, thegeal fu ofc Hamilton it pt pt Hean+Om+Views) (9-4) Irtheprea depends ontheepantion bewecn thepti lon, thai if Veries) =VOir =|) 9) shentheHamiltonian isinvicta dhedplacemet ofthe whol sytem, b ontancntarand aueaw inOhpra, isiapler ites ‘evan ofthe tual momeoram ands separon fale tw fells weil achieve tesept byfndng Ions hse ealtnrees ans Fonction ofthecommuting opened tndPretet pena cnet tigi eqeton ens Ppfire) =Pfr) 9 (0-6) 156 Quantum Physics hac is, 4cw.+99find=Pfleved on Hfwe write Sent)=ri—rsa+Br) (9-8) chenwithR=anitpe,(mad edtEMo» ak,? a owFj(a+8)FaveR) =PMR) (99) thats, thevaiable x=11—ryisaconstant parameter asfarathisequation is ‘concerned, Thus thesolution ofthis equation is eR)=ofa)one (20) 4 Wewillnowchoose «and#10simply theenergy eigenvalue equation, which . WT_Bas Bae—pedaeyoOa -am am,VEMDa]Ar)& o(9.11) Sine ~ Win W ove Wi= 9, +BME (12) ‘hisequation takes theform Rees 2 oP -E[-Ho)+Pye) | esaie)——2 ps Pe -E[v0)—PRP —Oe >] +Vilel)ae)=Beesu(r) (9-13) afertheexponential fucor hasbeendivided out,following thedifetentiationwithrespectroR.Thisequationsimplifiesifthecrosstermsateeliminatedwiththe choice a= B=ym (9-14) Tethen reads os 1)ate)= - )ae 4 -Pratswe)+VUr})ale)=(0Rwrs)(r)(9-15) ‘TheSchrdlinger Equation inThree Dimensions 157 here wehaveintroduced thereduced mass4by j ae 16) bThisisreally@one-particle Schridinger equation withenergy B=By~ 17) q TO Fes+ms) ” ‘Thostheenergy thatenters intotheeffective one-pattice equation isthe rolenergy,lessthekineticenergyofthetwo-particle system,whosecenterofmassmoves withmomentum Pandwhose rocalmass ismy+ms ‘Thequantity 7isnotspecified bytheabove equation. If,however, we require tharthevariable Rbecanonically conjugate totheotalmomentum P, thais,ifwerequite chat 4 & Wek=> 18) [and s0.0n, then wescethat é A Wu+Prosar+Baal=5(a+a)= (a9) if implies chat ateun (9.20) chats, 4 1 Tote o2n ‘Thereason forcarrying outwhatisafterallaverysimple separation ofvariables [in thisseemingly complicated wayisthatthisprocedute willeveasanexample‘ofhowtoproceedinthefurtherseparations oftheone-particle SchrOdingerequation. Such «separation ispossible when thepotential depends ontheseparation becweentheparticles,|r|alone.With|r|=r,cheHamileonian aeFive (922) isinvariantunderrotations;Vi)iscertainlyafunctionofthe distance from the‘otiginaloneanddoesnotdependontheangularvacabeschatlocatethedzec-‘ionofthevectorrp?isalsoasalarquaatiy,thelengehofthevectorp,andthusindependent oftheorientation ofp.Equivaleatly j*=—APV"isineeoaatunderotations. Thesceptical readercamcheckthisexplicitly bycoasideing thespecial ase of«rotationthroughanangle@aboutthesans,with 158 eso Physics : w= xcos0=ysin oy=xsin8+ycosO (9-23) itiseasy tose that Pa WEE REL ERE A and (2)+Q)-(rs-sno3)+(0b2+coo) : ~(a)+() | since theHamiltonian hasaninveiancepropery,weexpectaconservation\aw, aswesawinthecaseofparity andinvariance under displacements. To ‘Buhay theopercors thecommute withHietwsconsider Aninfitesimal fotation about thezis. Keeping rms oforder @onlysothat vex yaytee 20) werequire that Huglx —Oy,y+Ox,2)=Eun(x —by,y+Ox,2) (9-25) weexpand thisofrstonder in@andsubact fromit Hayley) =Basley.2) 026 we obaia H(:>-y&)adlega)=E(->-y&)melee) (0-27) Sincethetightsideofhismaybewrittenas (<2=Z)mae andsince denl) frm acomplete seweidhatwith weK(eSrg)sem (9-28) che commutation sation {AL] =0 (9-29) ‘TheSchrddingerEquationinThreeDimensions 159 jholds. Listhez-component ofcheoperator . L=rxp (930) Whichistheangularmomentum. Hadwetakenrotationsabousthex-andy-axes,wwewould huve found, iaaddition, that ‘ (4.1) =0 (HL) =0 os) ‘Thusthethreecomponents oftheangular momentum operators commute with theHamiltonian, thatis,theangular momentum isaconstant ofthemotion. ‘Thispacallels theclassical result chatcentral forces imply conservation ofthe angular momentum ‘Wemight betempted colookforsimultaneous eigenfunctions ofH,Le, Lmand Lebutthese donotforacomplete setofcommuting variables, For / eample q (LesLal=[9p—=f2Pe— apd =Dit2B]—Lepr=Pal~Lyfe.xp)+[2pmxp =ylenel pet ate & =|Oh x) . sib, 32) Similerly { UyLa]=dhe {aLa)=ify 33) ‘Thusonlyonecomponent ofL.maybechosen withHtoformthecommuting setofobservables. We can doalittlebetter,however,since(9-32)and(9-33) imply chatL?commutes withallceecomponents ofL: (lyWA=[LyL$ La?+La UyLol+(leLod =Lay Lal+[lagLa)La Dall La)+(eya)Ly =Lgl +illegLy ~ihgle —ikl, =o (9-34) and50on.Wethuschoose asourcomplete setofcommuting observables the ‘operators H,L,(@purely coaventional choice) and1?Wecould alsohave included patty, sincetheHamilconian ismanifestly invariant under x»—x, +yor—yand 2+~z,but,asweshallseelater,specification ofL?determines thepity. 160, Quanrum Physics JnChapcer 10wewilldetermine theeigenvalues andeigenfunctions ofLyandL?;herewemercynotethatthetusegreatlysimplifiesthesolutionofthe‘Schrddinger equation. Thisfollows fromarelation derived below. Le=Xp) =[leXpal+IlrXpl+IleXPh? (,2-:2)(,2 -22 -*(be-=3)6 a”3) a ayfa_ 2ea)CR-*e) -#(2Pale_2) ay 7Ox) 7?de fae(4 BM) ep(M4 “*[:&*ee(oe+x) +o(S+2)-hok : ae*ap)7andy dye \ me iPy? 22] dade OeBy Oe (933) | salle algebra shows. Similarly i ve (e252 422), 4,24 22ote (2x rete) (ratyte) | a ce oa a a |aoa(eaeetegetybySpr an7Oe+7ae+7aay>aoe 2, 24,2tueQirettyZerg) 036) “Thesum ofthetwoyields 7 (422) p(242422 aresei(ZrZrZ)ee(gry +22) (9-37) ‘Wetherefore gettheidemticy Lt (ep) =Ppt ee 638) Since weaedealing withopeciors, keeping trackoftheorder oftheterms i crucial, Icfllows fom theidentity that re1[+ewater] (2) P09). Lisp t(,2V opt2 -£[3(¢a)(-2)+.S-u) ao +VP) we(e)=Exe(r)(9-40) Ifweworkinspherical coordinates (Fig. 0-1),whichisthenatural chingtodo, welt) =YAO¢) Rely) (9-41) * where, 162 Quamum Physic LYA(6.6) =2¥3(0,9) (9-42) istheeigen equation forL,thntheoeationsepanes ito(9-42) anda iy talequston, Ourprocedure iely0dire hantheconvena)duaioy ofwuaes Tedoes,however, sesscheoeofthesymmetryefetcntnmng thecompletecommutingstofoperatorandwitheiselpthefepartion canbeefed,Wehaveconceal onchereductionofthethredimensionaleergyeigen coun inapes cordnae, sincecena potential, forwhich .Prey nebyftthemostinteresting ces.Oneeetsituation thatisof incre tosis theese when thepotenal iofthe fem Veg) =Wile) +ViG) +Vale “The equation B(®, ee) ;E(B BtBonen +1+vi+Hodove =Euslxy2) (9-43) i iseasysen0besoledby | sek) =nal)#46)wae om | wheretheFunctions ontheightaesolutions of | ee . | [-2B+ree]nc=on | By vy|raG)=ra) £4 v4)|r)=00 ea[-£ftne}oncr=ener oo) and Eeutate [Apuiculay interesting example isthe ue-dimensional geneiatn | ce prea holewthinte wallsIfthettcedimensionl boxiscabial | inshape, withsideL,then Via) =© «<0 mo o<eck =o Lex (9-46) andson. Ths, aie fom «normalising factor, thegener sluin is -“TheSchdingeeEquationaTheeDimensions 163 sels)=sinsin2gin 6598)=sinsgig om) and : BnBeottntmd 0-48) ants OF mt Notethaehereisquiteaotofdegeneracy incheproblem: thereareas manysolutionsforagivenFastheeatesetsofintegers{nna}thatsatisfy(9.48),Thedegencacyiusallyassociatedwittheexistenceofmatualy commuting operators, andthisexample isnoexception. Herethese opertors ateHy,Heand Hydefined by He=FE+V4) y=FEvay 0-48) = +ve so that . Hetyte=Ht 630) Icsincresting toaskforthe ground state energy ofNaonintencting ‘deolfermions, forexample, clectons, intheboxofvolume L’.Forexchtripleofintegess,(11,1),21,1),(1.21). twoelectronscanbeaccom.modated. Iseasiercoaskchequestion indiferent way:Howmanywiplesof integers (msoga} arethere suchthatEgiven by(0-48) islessthantheFemi«nergyEy?Eachtripletformsalaticepointinathreedimensional space,andifthereareverymanyofthem,theni4verygoodapproximation tosaythattheymustlieinside asphere ofradius Rgiven according to(9-8) by fbntbog=RenME ntthad=eeME os) anucheirnumber isgiven bythevolume ofthe octan ofthe sphere forwhich allthemare postive. Ths thenumberof lance points is j 1degy14a(2p!ees (Re) os sndhence thenumberof electrons withenergy lesthancheFetmienergy Epistwice hat, tha is ce ya[ite 164 Quancum Physics ‘Thenumber ofelectrons isproportional rothevolume oftheboxL*,which is tobeexpected. Interms ofthedensity ofelectrons, KR oak Da os) we have tet (30 ‘Tocalculatethetaalenergy,thenumberoflaticepointsmaybewrittenas 2fon (056) stinice ‘Thefactor 1/8comes from ourrestriction topositive integers in(9-48); inthe above integration thisrestriction isremoved andmust becompensated forby thefactor infront. Ateachatice poict theenergy isgiven by ReemPew 50) | 30thatthetotalenergyis { Bet BeEf tain \ Reape-*.t de | = ps (9-58)*20mL* © SinceRisrelatedtothenumberofelectrons by nar) te (959) a3 wwefinally gee Pi (any , Ifwe writ thisinterms ofm=N/L? weget it (3) te=EE (3) a . aa=SE(BY 6) “Thefactthattheground stateofamany-clectron system inapotentialconsistsofalargenumberoffilledlevelshasmanytamifications. Typicalvalues fof Byaof ondeof10eV.Ths, aondinayempertar eyewce sontsbethmaly exces mowofthemed orb ed eee {iCscdyoceoiel Themplenion of seteeieeewelldescribedasacrystallatticeofionswithoneortwofreeelectronsperatom,alythewsconcur wthemeee eeaes eanee emmetal,onlytheelectzonsneatthetopofthe“Fetmisea”canbeaccelerated, |Kherthose dc deeper enue 8Sete ca eee F.Coicmed hvelongean ieptt, Clogs oonet ata |treorges elonfyce bce eo ee Tal ee ‘These matters arediscussed mote fullyinbooks onsolidseatephysics. Problems 1.Consider apalemovingin+clndclly syameeic pata Vip),whetep*=s*+3%,Whatischecomplesetofcommuting observables thayouwoell wo ect ieooe rae 2.Ueyourcondos femPre toeps theSlinger a‘equation incylindrical coordinates. 2Gin tattheumberdesoffieccoincopperis3>10" corcae i)heFem eng cece ae ae Stcoon moving mehnee ae esee a 4.Aus comms ofNeons andZpons mhNZ=A AEtheradiusofthe nucleus isgiven byR=ruA™%,withry=1.1fm(1fm= 10-*em),andiftheneutronandprotonmassesarebothveryneatly1.6X10-* Powreeyesions forthe ene enagy oftepn eee es a suming tthepoten solReece ecec ane Fermi energies ifN=126and Z=822 3.Conse nensiniagroundse,wthmassJens» vayng from10"to10!gmcm”*. Calculate theFermi energy asafunction ofp.Note dca me porte aon gusbcos nae een Noe barecaengy tlecmen ce y etie 6."Te meanelton energy gente deca asivenby lars . @)=afm 166QuaiPhysics fornonrelativistic electrons, and 2farmigre+meen=mel @)==—__, ———2fen8. moregenctally Calculate thegeneral expression forehemeanenergy asfonc- . tionof&=pr/ma.Usethistocalculatethepressure,definedbythethermo- dynamic formal OE) Peaim) ; ithenonrelativistic formula andintheulrarelativstic domata, where £>> 1 References , Foragood discussion ofangular momentum imthecontext usedhere, sg J.L.Powell andB.Casemann, Quantum Mechanic, Addison-Wesley, Inc., 1961. “Thisbook,aswellaseveryintroductory book,worksouttheseparation ofthe 4 thew-limensionalSchrddingetequation. | H q|chapter10 Angular Momentum (urskinthichapteristoindtheeigenvacs andtheeigenfunctions ofbeoperators LandLtSncetheangusomentan hastedence B,wemay wate ceelgeamlucequronsinthe tne Lin =ah L*¥im=A+1)Yim (0-1) where and1+1)ateelmubets. Thepeculitway ofwing cheeigen‘aleofwillproveiconvenience er.Theretesealwaycanoe‘Theconvention! wayistowateouttheepemten Lisphere oieWebve ¥ x= rsin Boos ¢ y= rin On sareat (102) sotae Fde=Sit8C089de76086cos6d—Fin9singdp4=sinOsi6dr+c05856d+rainDeondbd=costar—rsino (03) ‘These canbesolved cogive dr=sin8088+sinsinby+cose t=} esteorgde estandy~sins) tisin cos. =et oa+bay) (10-4) ter 168 Quaotum Poysic With thehelpofthisequation wecanobtain 2% 2, Wd, wd“aeOeOrtaeOF*Oedp ait @ sine2. . =sin90s9-5+1coscosoeSO 2 a1 @cose 2 'pnHintingSacoos +oe i22_sind \ a (10) andchus wefally obtain a a 2 fo4.2 -,2).82 10-6bey(oyre)as Gos “Theother twocomponents oftheangular momentum aremore compactiy I cexprested,if weinzoduce Le- Lei, 07) Thea kc) a aJ atbeplea) a a a a !-A[aa(Zsig)- mamral i 643cogpeee248%2 ' =strcost(sinveee2+hearst2aAe2) 24(cos92—8092) =firsin84(ood 22) 08) a atenhet(22+iro) G09) (One canthen construc theL*opeacor byobserving that LyL. =(Le+thy(Le —thy) =LE+byt=fhe, Ly) (10-10) soar Me Lo4Leilbal =LL LP~he (oat) Angular Momentum 169 In chelastlineweused(9-32). Wethusgetsecond orderdiferential operator involving @and¢,andcheeremains thetaskofsolving thediferental equationsthar(10-1) represent. Thisisdiscussed inmanytextbooks oaquantum me. chanics orclassical electrodynamics. Wewillproceed algebraically butdigressforamoment codiscuss theeigenvalue equation LAYig=RY (10-12) ‘andsomeapplications. Theequation,using(10-6),reads q aq SYin(0$)=imYin, m 5p Yano) (04) (10:13) 30thatthesolution isoftheformYia(0,$) =Orm(6) 4a($) where j A) q PO) ©ino.(9) 1014on @ (ora) ©The solution tothis,normalized suchthat f4/9)?=1 (10415) : 6(@)=ane (0:16)me ae 1Teissomerimes argued thatsince«rotation through 360",chatis,«mansforma- tion$+@+2,leaves thesystem invarnat, itinecesaty thas erm (10.17) $0thacmianinteger.Tisisnocquizcortec,sincethequantitiesthatentericophysialobserabies areofthetypef.”dps*/e) Atal),withwavefone: tions¥4@)oftheform , . = aad 0-1 vo)ZsVa (10-18) 1werequire thatthesearbitrary wavepackets donotchange (except foran‘overall phase factor) underthetransformation @+@+2,thenweueledto theconclusion tatthemostgeneral allowed valueofmatem=+imegerwhetecisaconstant.IsonlyifweviewtheoperatorL,tspartofthetraloe(Catala) thatwecansaysomething abouttheconstant ¢.Weshallarguebelowthattheeigenvalues atedistributed symmetically aboutcr0,0thats =0or +©=1/2,andforthe opercors considered inthischapecr, weshallnescice oonselvesto¢=0,thatis,thecondition thatmisanimager, 170 Quancom Physics “Theeigenvalue equation forL,sppeats inanother context. Consider aclassicalrotator,rottinginthe=plane.IfthemomcatofinertiaisJ,thenthe energy is Bee (10.19) {| tnd thus theHamiltonian is Le a-= (10-20) “Theeigenvalues oftheHamiltonian arenowimmediacely seentobe Bert2,= 0-2) andtheeigenfunctions are¢+™*, There is«degeneracy, since Hcommutes with 7L,,and theewo eigenfunctions foragiven Eqcorrespond tothetwosenses of,rotation.IfwehaveNparticlesrigidlyfixedonacircle,withequalangles2x/Nberweenneighboring particles,andiftheparticlesareidentical,thenthesolucionoftheenergy eigenvalue equation Hoa(9) =Bex(@) (1022) willagain bee4,Thephysical system isunaltered under @rotation of2/N radians (oranincegea!multipleoftheangle),andthesoluconsshouldretecethis. ‘Thesame kind ofarguments thacforced mcobeaninteger nowimply chat demNX(aninteger)" Theenergyistherefore i _Nm? a (10.23) Jetusnowretutn toourequations (10:1), endtyt0obsin thecigea- values inamanger reminiscent ofouttreatment oftheharmonic oscillator in (Chapter 7.Theeigenfunctions ofthehermitian opertors LyandL?willbe ‘orthogonal, iftheeigeavalues arediffernt, andwithproper normalization, we will wie (eel Yin)=Sidon (10.24) Since (Vial(Let++2)Yin)=LaYin|La¥in) +(LaYin|Ea¥in) +oh? 20 (10-25) "Thesendmigh:lokbackcotheDicke-Wicke Gedarkenenpeiment discussesinchap. ‘Angula: Momentum 171 icfollows chae +20 (10-26) j “Theoperators Lyintroduced in(10-7) ateveryuseful inwhatfollows, and weshallsethattheyplaytheroleofraising andlowering operators. First, wea: 5 ready sawchat F Vann +Lett, 027) [Tn thesame wayweseechat Debit Let at, (10-28) Tefollows fromtheabove, a5wellasdtecty fom(9-32) chat Uy,LA}=2h, (10-29) E ~Theremaining commutation relations are (Le,La)=We+iy,Le]=~illly =BL, =~ft, (10:30) and [Lb =aL (20:31) rom thefacethat[L4.L] =0,italso follows chat (LJ =0 WL) =0 (1032) [This implies chat LLY =LUVmn=MEBLY (1033) thatis,LaYuaaralsoeigenfunctions ofLtwiththeeigenvalue characteieed by1Ontheother hand, LabsYid=(gLBs)Yom=WLVin+BLyYin =H +1) LYin (1034) 50thatL,¥im isalsoaneigenfunction ofLoybutwithm-value increased byunity. Similarly wecanshow that 3 LLYq=Bm—1)Ling (10:35) $0thatL_Yia ianeigenfunction ofLewithm-valueloweredbyunity.Thuswe «allLynising andlowering operators, respectively. Wemaywite . 5 aim ©Cl) Yona, (1036) 172° Quan Physics Iefollows fear thehermiiy ofLeandLythat Lita (Let ib =Lewily=1y (1037) Hence, «consequence of (oaYig|LaYee) 20 (0038) istae !(YaaLglaVia)20 (1039) andthetefte (10-27) and(1028) imply hat (Vigl(Ut=LfAL)Yiu)20 (10.40) hati, M4 DE mbm M4) 2mm (041) SinceAU+1)20,wecanake/>0withoutlostofgenetlty?Then(1041) shows that -Igmsi (10482) Ifthee is minimum value ofm(= m.) then forthe conespondingcigensate jLYim=0 (1043) ‘Wemaychen caleaate mbyusing (10.27) andapplying it Yin: weget | Hb a) =mR me uot) Simi, ifchee isamaximura value ofm(=m) then Yin =0 (1045) | andanapplication of(10:28)tothemaximum eigenstate gives | AL =mg +mht (1046) | Hence aol met (1047) Since themaximum valu isto bereached fiom theminimum value byuni steps(epentedapplicacion ofL,),wefind(Fig.103)(a)eathereare(21+ 1) sees, ei,2+1isaninteger, and(D)thamcntakeonthe wales peer en) “Thepossibility that1shalo inceeal, that, =1/2,3/2, willbe dis: 2we eset nd tnt|&=,wewouldmelene=—1~1andseplce she cl ih de ee oveLNotingwoldangeceU1)=104 Anguls Momeneum 173 ———-" $0 uaa «li re <a ; Fig. 10-1. Specoumn oftheoper Le ®t agivenvacot cused inChapter 14where wediscus pn,Inthischapter wereste utselvesEco inegeal vale of ‘Wemayalsocalculate checoefiints a(n) defined in(10-36) Wehave Cath)|*Vmas]Vins)=(La¥inlLain) :=in| LelaYin} : =(inl ~13 fi) Yin) BUG+1)=mw0)soths,withaconvenientchoiceofphase,wegee Cb) =HUNG+2)—mtay (10.48) Thisivasfaras operator methods cantakeus.Weshallnowusecheexplicefotm ofthe operators L,andL.toobtain convenient exptesions fortheeigenfune- tions interms ofthespherical angles @and6.Thisdevelopment willpallthacofBq(7-31)10(7.35).Wewaeasaltensuggested Vials) =Ort) (2049) The condition (1045) reads (2 5ico? ys hen(3+ie)ono)¢ pve (2 vonatn(24eot6)Ou)=9(1050) 174 Qutam Physics ‘Thesolution tothisequation iseasily found tobe 20) =(ino! (os) ‘Theappropriate multiplicative constant willbeobtained laterfrom thenormali: _ationt Comin, Antbituy Sate isObtained bythelowering procedure Yinl@,d) =C(L!™ (sin8)! (10-52) ‘Consider first )(—2+score?) (sinay Lvidee)=nee(—24ace)ain use(2—peor)ind)! =net(-2—react)in Since onecashow tha franatbitry Function f(@) @ Lda yyG+teoe®)10=ay4i[sia5)0](1053) wehave obtained! Yus=CT(-$)iGinsiinn) 4059) “Thenextwepisthesame,exceptthatFsreplacedbyP—1andtheoperationin(10-53) acts ontheform obeained in(10-54). Thus i aoe (~S)[0gail) "| vaneAE (4) ein1(-4)cin Yuan0San (--)[S00aml) oO t a se eT =coOe SLEtino] (1055) Tncerms ofthevariable x=cos@,—1/(sin @)(d/d¥) =d/du, and (10-54), (40.55), respectively, cead »md oy YonLohao vee encrfe a “4 Yur=CGgeUh04(1056) “Thegeneralfonis vm(ZY 057) = Tico ae) Mm 7 Angular Momentim 175 Theeigenfunctions are<obenormalized, Sincewearedealing. withspherical angles whose angeofintegration is0<< 2x,0<0<rooo Fig.9.1)andwhere theintegra overthesurface ofthesphere (r=constant) is j far-[afsnowwemustimpose alta) t=[afatc(Z)" 0-of (10-58) [The integration istedious. Wecontent ourselves withwitngdowntheappco-5priately normalized Yrq(0,6) withthephases thatateconventionally established. d = rafZEE=IYon)cine Yen(O6)=(—1)[aeTbmi)PGeoseere (1059) with Yipee=(-0*Yin (10-60) Theassociated Legendie polynomials aegiven by (>my =ate fad Peta)=ayn (4)a=)!(10.61) with thevalve fornegative obzained from Qu)=(=1)9LH pm 3 Pm) =(1)FS Pm) (1062) Tewill beenough, forourpurposes, colisafewoftheeigenfunctions: 1 Yoo5Jez BS Yueee sine [3 Yu=Vecos0 Yarn2evesate ay :Yas=afBetsincone 176 Quantum Physics ] I>oscet Yap=vieGcos—1) (1063) ‘WiththeknowledgethatL?,actingonaneigenfunction, asin(9-40),isto bereplaced by/(/+1)f®,wecannowwrite theradial differential equation that determines theenergy cigenvalies andeigenfunctions. The equation, which svewilldisciss ingreat deal foe «variety ofpocencial, is eo dad 14 K+y-alra ta SJenin +VP) Reim(s} =ERein(r) (10-64) ‘Wenote thcthere isaodependence onmintheequation, Thus, fora given J there willalways be«(2/+1)-fold degeneracy, since allthepossible m-values. willave chestme energy. : Problems i 1.Amoleculeconsistsoftwoidenticalatoms,eachofwhich,inits |around sate, his spin 0.The molecsle has, among itspossible excitations, rotational exctions. Ifonly rotations about thez-axis areconsidered, sothat H=14/21 andthe separation between theatoms isconsieved fixed, whit is therotational spectrum? Iftheatoms have spin 1/2andthey areboth inche |samespinstate,whatischespectrum? i : 2,Express thespherical harmonics listed in(10-63) interms ofx=1sin9cs$7=rindsin,and=7cos8 3.The Legendre polynomials Pu) =P,a) canbedefined inteams of theexpression (1061). Usethisdefnition toshow that P(e) satisfies theequa- (1=#4)Pre) —20Py(a) +M+ 1)Px) =0 4.Show that theLegendre polynomials Pi(u) satisfy therecurrence seations AP =al+(U2) Pe UF Pigs =U4 1)oP—(1 PY G+)Pigs=QL+1)wPi+Ppa=05,Use(1061)toshowchat SEPy) =ae teyc Angular Momentum 177 6Usetheprocedure outlined inthischapter todiscuss rotations infourdimensions. Thegeneralization ofLisnowthesetofoperatorsthatmaybe vwiteen as Lay=~ieds —x) Gj=1,234). totroduce UsJuJs) =(Leaandss and (KKK) =(Laulawlsd) (@)Findthecommutation relations ofallsixoperators among themselves. ()Show thatcheoperators J=G+K);JO =0K) cachobeyangular momentum commutation relations andhattheycommuse vitheachother.Usethefinalresultodetermine themaximal setofmacualycommuting observables, andthusthequantum numbets thatwould beusedto label aneigenfunction 7.Consider anelection inanarbitrary potential V(r)andastateof ‘ngular momentum 1.Showthactheprobability offinding itatthe poiatrisonly @function of|r) (Hint. Notethatthesolutions forthe(20++1)m-values atedegenenste, and tutifnospecial alignment isprepared, allm-values ateequally probable, Usethe formula . ‘ pea)ZX[¥inlO9)|? =a 8.Aparticleiospherically symmetric poentia isinastatedescribed by thewave packet Wlxy2) =Chey++20)oo" ‘Whatistheprobability hacmeasurement ofthesquare oftheangular mo-‘mentumyields0?Whatistheprobably thaticyields6f°Ifthevalueof6found tobe2,whatatetherelative probabilities form=2,1,0,—1,--2? 9.Consider thefollowing modelofaperfectlysmoothcylinder.Ieisa fingofequallyspaced,identical particles, withmassM/Nsothatthemass theringisMandissmomentofinertiaisMRE,withRtheradiusofthering,Gateuate thepossible valvesoftheangular momentum. Calculate theeaetgycigenvalues, Whatistheenergy difference becween theground stateofsereangular momentum, andthefstrotational state?Showthatthisapproachesinfinity asN—2.Contrast chswithchecompare enetgy foranicked”cylinder, which lacksthesymmetry undertherotation through 24/'Ntadians,‘Thisexample implies thatiisimpossible tosecaperfectly smooth cylinder ig 178 Quancam Physics rotation, which isconsistent with thefactthatfor«perfectly smooth cylinder such #rotation would beunobservable 10,ExpeessL?incexmsof0/28and2/06.Writedownthediferencial‘equation obeyed byOrndefined inEq.1049. (Hint, Usethevariable x=cos6).Show that(sin8)isthe solution oftheequa sion for = m. References ‘This isstandard material found inanyofthebooks listed onpage 501. For2 deeper look into theconsequences ofinvatiance under rocation seeespecially K.Goxtiied Quantum Mechanic, Vol.1,W.A.Benjamin, Inc, 1966. 4]1M.E.Rose,ElementaryTheoryofAngularMomentum,JohaWileyandSons,Inc, |1987. | |chapter 11 |TheRadial Equation ‘Theradial Schtédiager equation (10-64) maybewritcen as oad % de i(+2£)nmin-+[rw+o‘Jomo 2al +AERa)=0(UN) here weave placed theabeEbyinthe subst ofthe eigenfunction Rus(7). Wewillexamine thesolutions tothisequation foravatiety ofpo. teal eticed bythe condition thatheygo mae ata oe Ifeexcep forthe important apc aseofteColo conte Wead alsoassume thatthe potentials arenotassingular as1/attheorigin, sochat LimFV(r) =0 (11-2) F Itissometimes convenient tointroduce thefunction taim(t) =Ratt) (1-3) Since ; #2 4)aul) 1 )(Z+24)a alt) ana) iflows eae : Saal)5de fears ; |Thislooks verymuch like«one-dimensional equation, excep eat (a)thepotential His altered bytheaddin ofarepulsive cenctifugelbare M+ ie .Vi>Vin+a (11-6) v0 180 Quancurs Physics wy N \ \ S avon? Ne tee Xe]vot) Big. 11-1. Effective potenctl acting ioradial equation for#=#R() when the realpotential isasquarewel (b)thedefinitionoftata(r)andchefinitenessofthewavefunctionatthe| ‘origin require that ! sain(0) =0 a7) whichmakesitmoteliketheone-dimensional problem forwhichV=-+~in | thelefchaod region (Fig.11.1) |First weconsider theradial equation nearcheotigin, dropping allsub- | scripts forconvenience. Asr—>0,theleading terms inourequation are x M+)fe WEDwo (118) because thepotential does notcontribute forsmall enough rwhen (11-2) is satisfied. Ifwemake the Ansitz a(~ i) wefindthattheequation willbestsied, provided that ae Y= M+ 1)=0 (1-10) thatis,:=+1ors=—LThesolutionthatsatisfiesthecondition(0)=0,thatis,thesolution thatbehaves liker*¥iscalled cheregular soltion; thesolu- tionchatbehaves ike isthe eregular slain. TheRadial Equation 181 q Forlargerwecandropthepotential cerms, andtheequation becomes ou oot ax (nay ‘Thesquate integsability condition implies that r=[ariel ~[p24faclrancs Yin(0)|? q -fParlRata)* (2) that is, fAhia(DIP=1 (13) sothaethewavefanction should vanish atinfinity. IFE<0,4that 2Eat (1) theasymptotic solution i Wn)~eer (11-15) Jf>0,wehavesolutions thaaeonlynonnalizable inabox(ceediscussion inChapter 4).With mE A eee a6) thesolution willbealinearcombination of¢*and«-*™,theproper combinationbeingdetermined bytherequirement thatthe asymptotic solution tioncon. Ftinuously tothesolution chatisreguat atheengin. Wenowconsider seme examples A.The Free Particle Inthisexample¥()=0,buethereissillacenuifugalbarierpeesent, “Theradial equation (11-1) takes heform #24 May tee)« ; Ifweintroduce chevatable »=hr,weget eR 2ak K+) j 2B MtRe ptt eteeo ans) 182 Quancum Physic “Thisequationcanaczslybesolvedintexmsofsimplefunctions.Thesolutionstreknown asplerial Bes! foncuons, Theregular sation is @(sia i=cor(+ ZY(#4) (aay) po) Ne and theiegula oneis uta=-cor( ZY(22) (1120) po) No “The fist few fonctions aeliste below joa)=22 mfg)=—SP > ° fine cose iycose sine 1)=SSPSEiy=SPA wo(3-2) 3 \ino)=(2-2)sinp-cos : aa 7 w= —(2—*) corp—2sin : ' nin—(2.-4)ese~ sine vay “The combinations thatwillbe ofincest forlage parechespherical Hankel funcsions 12) =7h) +inl) az) AP0)=1lt vas) ‘Again thefrstfewspherical Hankel functions ase B= = B wr=-“(v44) i=(4% —2) a2)aere (OFspeci ire ate (a)theBehavior neartheorigin: forp&J,eras ovehae ye10” 55 aS aia) TheRadial Eqution 183 and n(y)=e—V9522=Y) (1126 cB For 92J,wehavetheasymptotic expeesions 1 be i)~ +sa(»~) uz) and k lp)=—>cos(--) (128) > 2 Eso that HG)=-4porn (129) ° ‘Thesolution thatiregular atcheeign is Rilr) =ler) (11-30) Tesasymptotic form isusing (11.27) : (~—Lpraiin =penny RO)=ek fein x31) Wedescribecisassunofanincoming’andan“outgoing”sphericalwave‘Thenomenclarure isattived atinthefollowing way.Thegencalization ofthe one-dimensional fx fs i=Live win—yeHe) 2 J=GMO wie—woHe] (aus2) Weshllsethatitisonythe" inthe sail dection thatsofinterest fe lagerThustheradial ux,integated overall ange, 4 wee’ fanle 25 Fora solution ofcheform =c=vag) ren) vith fartoair=s (ass) 184 Quancum Physics weget 8ce[2"(4we—S)—compteconjugate forZicr[™(+#-“)oksinge] aMEL O36) » # The +signs descibe outgoing/incoming Sux. Te factor 1/r* thatemerges from ourcalculation isactully necessary forfloxconservation, since chefx going though tesphesia sutice ateas +s fre.=Gocepeten of isn) Foroursolotion (11-31, theincoming fxis,aside ftom 1/e, Bh)ijaen|*-Biee |--23 (1138) | and ths isequal inmagnitude totheoutgoing fue. The oetux istherefore 2et0, aicshould be,since there atenosources offut. Ingene, fuxconservation demands thatanysoltion—and thisincludes solution forwhich V(x 0—whose form forrvtyurge must [bytheargu- ‘mentsfollowing (11.16) be \ymBt —gay)feb Ri)~~5h sit dem a1-39) requires |Sa)|? =1 (11-40) 18otherwise theoutgoing flaxwould diffe from theincoming one. Afunction ‘hose absolute square isunity conalways bewriten inthe form Si) =eu re) ‘Tetealfunction8,4)icalleddhephashiftcausethesiaunconitheasymptotic tegion (11-39) may berewecten a8 sin[br~fx/2+(8) nayen=a= nap Aside from thephase factor afoot, thisifr fom thefeepatie solution {AP whose asymptotic formis[sin(Ar~f/2)\/r, onlybythe silinphase, 34d). Wenote thatthe Ax inthe tdiection involves Kl? 1 b=2(v2Bo—comptescongue) ~E69 TheRadial Equation 185 andalagedistances, suchafu,whenmultiplied bythe ateafactorA,til vanishes as1/rrelative tothedominanc termintheradial flux.Thisisthe justifcation forignoring allbuttheadil faxatlagedistances. B.The Square Well, Bound States Consider thepotential VO=-Ve rca 7 =0 n> (11-43) “Then theradial equation hastheoem PR 2dR M4) mwPt eTRT EA DRa0 rca PR 2dk Mb) OEry i aoegeR=0 >a (11-44) Welookforbound statesolutions, forwhich B<0,Wewre 2yy ag q ReWot Ea ee . ‘ a(145) Thesolution forr<a,whichmustberegular attheorigi, i RO)=Ayer) ans) Thesolution forr>amustvanishasr—»a,Thesecond oftheequations(14-48) isjusttheequation forthe sphetial Besselfunction, exceptatFereplaced byia.Thesolucion thatbehaves lke«now becomes theexponentallyfling one, cha, wehave 1 RG)=Bh§?(iar) (11-47) forr>a.Thecwosolutions mustmatchat¢=«andsomustthedecnativesThis leads tothecondition ‘dip)de [se(de| :[FO] - [Mae au [7).. PO)Lim us) ‘hisi«verycomplicated tanacendental equation invalving/; Ve,andEFor+=oiesimples greatlyfoneweshefunction u(t)=rR).Wheeeninobeuined bymatching 4sinerandBe"atr=ThedeneoeBieon 186 Quantum Paysics H H i- + H ve Fig. 11-2. The shape ofthewave fonction a(7) =r() foranatractive square well when there exists one bound state =0). | ‘exerciseforthereader;theshapeoftheadialwavefunction«()fortefistandsecond bound statesisexhibited inFigs.1.2and11.3. iLetusreturn toEg.11-48 forthe caseofverydeep poteatial forwhicha®LTathatcasetheleftsideoftheequationsimplifies,sincewearejustifiedinusing theasymprocic form ofjs). Computation shows thar(11-48) takes the form ie —Sevea(u— =apni) — Therighthand sidedoes notcontain Vo,andif|,<Vothelargeness ofna er | |i Fig. 11-5. The shape ofthewave function 4(¢) =rR(7) foranatcracive square wel when there exist two bosnd states (= 0).Only thewave function forthe Second bound sae issketched inthis igure. ‘TheRadaEguaion 187 implies catthecorangent mustbedosetoaro,Thuswehaveapproximately ke ox usye .Hm(ty (150) Since for|B) <Vowehave E+xa(1+38) (sy where 7 w=Ue (asa) (11-50 eads By eb ale ;wr (0053) “Thusthelevelschatarefefromthebottomofthewellarapposite equallyspaced, forall! <&ts,withthespacing hae Seat (154) ‘Arelated problem isthe inne boxintree dimensions. Here VOQ=0 ra == pa 139) Iathisete, writing [ UE aEay 136) thesolution that regular atr=js 2. R= Ail) (as withthe eigenvalues dered bythecondition chatthesolution vanish at P= aythats, by jike) =0 158)“TherooforafewvaluesofFarstedbelowTN eI Boe Imo 1 23 3457669988936 628773 910 rose ose Se 188 Quantum Physics Ifthefirs00tfora given [islabeled w=1,thesecond root»=2,ands000,‘andifweusetheacceptedspectroscopic notationforthe/values,* S:l=o Prbm Dil=2 Fil=3 Gilad then the order inwhich the levels occur is 18;1P;1D;25;1F;2P;1G;2D; 1H;38, Suppose weconsider «model ofthenucieus tharconsis ofprotoas and neutrons ioide such aninfinite box. Since neutrons andprotons atespin3par| tikes, thati,fermions, nomote than twoacutrons andewoprotonscanOccup. 4given sate. Ifweconcentrate onprotons, weobserve thatinthe 15stateonly Weprotons canappeat IthenextlevelwehaveJ=1,50thaehere acethree states, andhence xprotons willfli. Forthe1Dlevel, with fivepossible imvalues (since!=2),tmproconsaterequitedtofillthis“shell”Thuslevels willbefilled when thenumberofprotonsis2,8(=2+6),18(=2+6+10), : 20(=18+2),34 (=20+14),40,$8,68,90,92,106... ,andsimi for theneutrons, Astudy ofrealnuclei shows chatforthe“magic” aumber of protons andneutrons, 2,®,20,28,50, #2,126, ...,these nuclei exhibie special haractetstics chat can beassociated with Sled levels, chat is,closed shells. The dlfference berween thereal “magic” numbers, andthose obtained inourprimi- tivemodel comes aboue because thee isanaddtional potential that depends on thespinandthatshifts thelevels about somewhat, thusreoedering thenumbersTheshellmodelofthenucleus,whenproperlyconstructed, explainsmanyofthepropertiesofnuclei.Whatisaotobviousiswhynucleishouldbehavelikeacollection ofparticles inabox. C.The Square Well, Continuum Solutions With E>owe write ME,ar, (11-59) "The histori originofthisnotationwathedeciptionofspeclinesa8Shap, Principal,Difse,---sandthersbsequeseHectcation, fedoesnottebesese,but Stuck,Thenoutionbeidfsfiomthatwes!insomicphysics,wheretheconventionalvation asthe Fale cote inde, Sota theorder would bewen inef,1S,2P,SD,25,4,3P,36,AD,6H,38, TheRadial Equation 189 ; ‘Thesolution forr>¢willbeacombination oftheregular andimegular solu- tions ofthe feefeldequation 4Rule)=Bille)+Cale) (11-60) while thesolcion forr<aasthe theregula solution, ei, Rie) =Aj) re) where w=BELVO (1-6)= ‘asbefore. 1dt Thematching of) Ea rmagives are) [ae+nal [Oe] =g|Blidde+Gaa aves [HN LG)+Cri)Ja, fromwhichthetioC/Bcanbecalculated,Tisaiocabeelatedcothephaseshifthaappeared in(11-42). Wedothisbylooking attheasymptotic (ge9)form of(11-60) ae) mY ¢ le00~2[sin(te£)—Sen(«-*)] caso hich sto becompared with (11-42), reweinen 8, Jpn be ie)adio=ffon(tr)const+con(«-4)sea We seethatthe relation Le ;rr) (11-65) hols, Theactualcompusation ofC/2fom(1-6isediousexcepforI=0. Asforthe bound sateproblem, theutea)=rR()simples thecaleadonaretly.OneonlyneedstomutchsinwtoBinbr-+Cosfratr=ate }obenanexpression fotandy,Theresisforthiscaseaeacheratialy dowsinFigs.1.4and1.5.Theyshowchatanatuactive poeatal enters thewavefunction, whilearepulsive potential tentstopush exe,Wecal fetunothesemaces inChapt 24,mhen wedisces culison thecey +Beforeconcludingthischapcr,wefocusonanimportcanonshacau beobtained bysolving thefrepate equation intueways, Onecolenncns 190 Quantum Physics feo | —~| [yn! Se/ ||nb)sationforv=0 Fig,1-4. Contowure solution a(?)=rite) foratracive potential (=0) obeained asasuperpositionofourseparatedsolutions(11-30)multipliedbythe sppropite spheri harmonic Yin(88) He)=EY Anji Yoo) (1-66) [Anothe solution ofthe ficeprice equtio, which reads (+ A)Yr) =0 (11-67) before theepaation intoangular andradi psi made, isthe plane wave une (0168) satesAbb aan 050 > Fig. 11-5. Concinwum solution w() =rt) forrepeive poeta (= 0) TheReta! Equation 191 Wemaycherefore findAigsuchchatYe)=#*in(11-66). Note thatthesphericalangles(4)arethecoordinates ofthevectorrtelativetosomeabi.tuaiychosenraxis(seFig.91).Ifwedefinechez-axisbytheditectionof& (unl sow anarbieay direction), then Ate dere (1-6) ‘Thasthelefsideof(11-66)hasnoasimuthalangle,¢,dependence, andthuson therightsideonlytermswithm=0canapeshence,makinguseofthe facr that rates)=222)" lane (170) whet theP(cos)aretheLegendte polynomials, wegettherelation ert ey Aijd)P(cos6) arn) Wemay usetherelation va y= be - F\which is»dtece consequence oftheothonormalty relation fortheYiqand (11-70), roobaain 3 Asie) =Hae(20+1)f©ape) (11-73) |The integral canbelooked up,ofwotked outbycomparing bothsidesinthe limicthatkr—>0.Inanycas,whatresults isthe expansion seo=&(at-+1)lin)Phcos8) 7 which wewilfindexceedingly useful indiscussions ofcollision theory Problems 1.Consider #=0bound states forananactive square well,Findthe «igenralue condition forabound sate.Whatisthedepthofthe poteatial foraseae thatisbarely bound? + Assume thatdhedeuteron (consisting of neutton and«proton, equaljinmassisaboundsitewith!~Oancthepotentialsquareinshapetndofcge 192 Quantum Physics ry=2.8X10 am,Given thatthe binding energy is—2.18 MeV, findthe depth ofchepotential. (Hint, Expand about thecaseofzero binding energy discussed inProblem 1) 3.Considerneutron-proton scattering,assumedcobeinteractingthrough 4sguarewellpotentialofrange26X10"cnandcepth20MeV.Calculatethephase shife as«functionofeneegyforverylowenergies,for!=0. 4,Calculate the =0phase sife frasquare wellpotential. Usethepro- <ceduce outlined following Eq.11-65 towork outboth eheattractive and the repulsive potential case. Discuss various limits, such asElarge and small, V> large andsoll 5.Show thatfor/=0scattering byasquare wellofarbitrary range and k depth Vo,cisalways possible cowre thephase shift asanexpansion Featy=—Fbrat/2+0(8% bean anexpression for#andreginermoftheparametersofthewal 6Considerapotentialofarbtearyshapechatvanishesforr>a,Letthei logarithmic derivative ofcheradial function inside thepotential 1aR(r) Road | bea slowly varying function oftheenergy. Consider !=0. (2)Ifthe potensial hasabound state with energy, Bp,what isthe valueof WEn)? (b)IffA) isindependent ofE,what isthephase shift asafunction of energy? (©IfAB) =fla) +(E—Ba)fi,how does fienter intothephase shite Ics simpler towork out(b)and(€)above interms of&cot&(b),insteadofthe phase shift, andtat isapreferable way topresent your results 7.Giveageneralargument forwhy&,(4)shouldbeanoddfunctionof&.‘Check chatthisissoforthesquare well[using (11-65), forexample]. Show thet SK=#) =58) where Sie) isdefined ia(11-41, 8.Calculate thefunction $,(4) for«potential Vj=e ree Voj=0 >a ‘Consider the!=0phaseshift.Whatisitforhaverysmall?Whatsitforkevery large? Note hacthispotential is«model foranimpenetable sphere. ‘TheRadial Equation 193 9.Usethesolution (11-63)together withthevaluesofchespherical Besselfunctions neartheorigin given in(11-25) and(11-26) toshow that tan8)+043 +0,Howmpidly doesitapproach zetoforagiven2 10.Consider theJ=0radial equation forthepotential VO)=Voleterrite—9tmnt} (knownastheMorse potential). Findtheenergy eigenvalues bysimplifying thedifferential equation. Dothisbydefining «newvaciable x=Ge" withC chosen tosimplify cheequation asmuchaspossible, andthencreating the ‘equation inchemanner thatthesimple harmonic oscillator problem wastreated inChapeer 5, Plotthepotential. Showthatforadeep,widepotential, thelow-lyingound states approximate those ofaharmonic oscillator, andexplain whythis isso. References ‘Thegeneral properties ofsecond-order differential equations inthecontext of quantum mechanics arediscussed in J.-LPowell andB.Crasemann, Quantum Mechanics, Addison-Wesley, Inc,1961 |A-comprehensive discussion ofsuchequations mayalsobefound in P.M. Morse andH.Reshbach, Methods ofTheoretical Physics, McGraw-Hill Book Co, Inc., 1953 |chapter12 TheHydrogen Atom “Thehydrogen ator isthesimplest atom, since icontains onlyoneele. stop, Thus theSchrodinger equation Becomes 2one-paicle equation afer the Center ofmassmotion iseparated out.Weshalldalwithhydrogenlike stom, thaci,atoms containing Oneelectron only, butallowing for2nucleus more complicated thanasingle roron. Thepete theni ro--2 (ay and theradial Schrdinger equation is a24 Ea Ze +1))#424)p, mfp,Me=o(r (Harz gerele GMP} eno29) We will coocentate onthebound sates, that is,E-< 0solutions. Ie1con- Fvenient comake 4change ofvatables, _‘ueiyp= (uel), (23) The equation then reads &R2dRK+) (»i) PRL2mMWADE(ANeg Urry Bm wm Ry od were wehaveinrodced thedimensionless premier 8 ef yin ya(yr ; Bsi) a(n) (29) “Tesecond formmates iteasier cocompute withi,since a=1/137 andthenergyisexpesiedinunitsoftherestmas;theisformdosehowever,nae._clahathevelocity ofight doesnottally appear inthe equation sheet that cis suicly anutes equation. 195 196 Quancum Physic Weuycosolve (12-4 inwhats bynow afalar way, Fist, weextract theage pbebavion, Foclarge phe only terms tharemain intheequation are @R SB_trse (26 andthesolution, which behaves propely atinns, is R~ Asinout trement ofthe harmonic osiltor, wewate RG) = GW) (27 ‘substitute thisinto (12-4), andobtain theequation forGp). Alitte algebn, which wedonotreproduce, leads tothe equation 2c 2)6protmen] £6_(,_2)46 PtMAY)Gg ary e035 [oo wo Wenow write apower expansion forG(s). This takes theform Go)=DLS aot (29) ‘ThefactthatRG) andhence Gp), behaves ikep!atheotigia wasestablished atthebeginning ofChaptet 11forallpotentials satisfying (11-2). When (12-9) jissubstituted intothedifferential equation, wefindarelation between various coefficients 4,.The recursion relation isobtained from thedifferential equation abeyed by HG)=Sage (210) whichis Si(Pty ao(241)we Vy oe ascaneasilybeobtainedbysubsticating G(9)=o'H(p)into(12-8).Wethenhave = 12z=[ne-newtragt(*42 1)0-1~dart|=0 (12-12) that i, Eetheen +LF2)aan)+= 1Iwadi= 0 Since thisms vanish cca byetm, wegettherecursion selation TheHydrogenArom197 Cn ee oes “ay GtDae + (213) Forlarge mthisratio is aon 1 stat 02.4) and,asforthe harmonic escilator problem, wecanshowthatwedonotget solution R()chariswellbehaved atifiniy, unless theseries in(12:9) tem rates. Thismeans thafora given iforsome »=1wemust have Nem titi (245) etwsintroduce theprincipal quantum number ndefined by namtley (216) Then, ifollows from thefacethan, >0,that Lazite 2.misaninteger 3.the relation dew “ implies thar pat (27) 4result familia from theoldBohe mode. Novice thaitisthe reduced masthat appeats intheexpression; this,ofcourse, isaotpeculiar cothediferent equation approach, IntheoldBohe theory, roo,4proper teaument ofthe clesical orbits, subsequently coberested bythequantization ofangular ‘omentum condition, would haveintroduced thereduced massintheenergyformola. "Thepresence ofthe teduced mass, mM po (a2s) whete misthe electron mass, andMithemassofthenucleus, means chatthe frequencies BixBymph (a _ dllfer slightly fordiderene hydrogenlike atoms: Inpariculat, thedierencebermeenthehydrogenspecrumandchedeuteriumspectrum’-wheteMi,the Fig.12-1.Orbitsfor«potenrialthatdoesnothavetheexact1/rformdo.noc‘lose upon themselves andprecess asshown here. Theorbits remaig planar aslong 45thepotential isradial. nuclear mass, isveryclose tobeing ewice theproton mass—was responsible for thediscovery ofdeuterium byUrey andcollaborators in1952“TheenergydoesnotdependonJ,thati,for2givenmthecnctgiesofall thestares such that+ 1<mare degenerate, Wedidexpect a(22+1)-fold degeneracy oftheencrgystatesforagivenJ,sincetheradialequationdidnot depend onm;herewefindchatalthough theradial equation does depend on/, there isanadditional degeneracy. Such adegeneracy wasformetly called “acc- dental,” since there wasnoobvious reason forit.This, however, depends on ‘what onemeans by“obvious.” Itisalready known inclassical mechanics thatthepotential1/rhassomespecialfeatures:theorbitsconsistofellipseschat‘maintain their ofientation inspace, instead offorming. precessing orbits (Fig.12.1),Smallmodifications ofthe potential docauseaprecession.Suchmodifi- ‘ationsmaycomefromavaieyofsources,forexample,theperturbations duc toother planess, intheKeplet problem. Iaconsidering theplanetary orbit of ‘Mercury, itwasfound thasafter allowance wasmade fortheeffects ofother planets, 2precession oftheperihelion intheamount of42”petcentury re- mained ueaccouated for,andthiswasfinaly explained byEinstein's general theory ofrelacivity, which predicted justtheright amount of1/r*potential co beadded tothe Newtonian 1/r. Tnquantum mechanics, 100,there areperturbations, $0thatthe Jategen- 2Therese,ofcourse,ethershiftinspectlinesthatativefomrelativisticeffets and fom theexeace ofelecon sn. Thee wlbediscus’ ate. “TheHydeogenAtom199 cacy isnotreally what isobserved. Infrstapproximation, however, wehave, foragiven», chepossible yalues of=0,1,2,-.., (a—1),andforeach there isthe (21+ 1)degeneracy. Thus thetotal degeneracy is Letty = (2229). Stietyspeaking,cheeareewoposiblesatesfortheelectronbecauseofitsspin,Sothatthe teedegeneracy isreally 202 Lecusnow return tothediferental equation. Ifweset =minthe secusion relation (12-13) s0that ktltinn cam OeEF et (an wefind chat Feeyee a lS we CUSGENEF AtD Metaey a~@+1 var 2” WiththehelpoftiswecanobtainthepowerseriesexpansionfocH{p).Equiva-Jently, weobserve thactheequation forH(p)isthatfortheaumeiied Laguere polynomials: Ho)=L279, () (12-23) ‘Thepolynomials aretabulated andtheirvarious properties canbefound inthe mathematical iterature* Aerconversion backtotheradialcooedinate randafternormalization, thefstfewradial functions canbecomputed. These azelisted below. Weuse A f aot(220) me inthetabulation Ryi(?): “ ZV" (ze Rel)= ~2)2m oa )"0-%) :1Anexsemely usefulbookisM.Abrowiaaod1.A.Segun(es),Handhlof MatientcalPontos,NationalDosenofSanduPubcon, 200 Quancum Physics L(ZY"Zeann mo(aa) ae zy er22)_ q Role)=2ZY"|1—2522) art 2AV2(ZY"2, Br) arm moBL) EC-S): v2 2)"(ZYem “Thefollowingqualcaivefeaturesemergefromthesamplingofeigensotu- tions: (@)Thebehaviorof#forsmall,whichforcesthewavefunctiontostay sonalforarangeofradithaincreaseswithficonsequence ofthecenifugalrepulsive barier chatkeeps theelectrons from coming close tothe nucleus. (b)Therecursion relation shows thatHig) isapolynomial ofdegreey=m-—1~1,andthusichasa,radialnodes(zetos).Therewillbe—~/ “bumps” intheprobability density dstibusion PU) =PAR) (12.26) ‘When,foragiven»,hasitslargestvalue!=n—1,chenthereisonlyonebump.‘As(12.25) suggests, andascanbeseen from thesolution tothedifferential ‘equation, Rewnar(t)=potHie (2.27)HenceP(r)7228willpeakatavalueofrdetcemined by Tea(ent—mde °(12-28) that is,at =m :3 (0229) ‘hick isthe Bohr arom value forcircular orbits, Smaller values ofJgive proba bility distributions with more bumps. Onecanshow thattheycorrespond to Aliptical orbits inthelarge quantum umber limit.(©)Plotsofchedialprobability densityP(7)forfindingtheelectronata discancerfrom theorigin canbeconstructed with thehelp ofthewave func- ‘ions. Figure 12-2 shows chegeneral pattern. Wemust remember that thewave fanction alsohasanangular part, whose absolute squate isPi(cos9).Plots of theassociated Legend:e functions Pim(cos @)aegiven ia’Fig. 123. Asm increases, thepeobubilty density isseen toshife from thez-axis cowaed the Ais HX oss ~ ~y 0 ot ae os ad—ir) i.me oe cos oe mh “7 os 3 f om 4 oatff f oof f oxo ok \ a ae ers = i, | Fig,12.2, Therialwavefants a)=r) adtheelpebaiySet faction) fraacafr dn lsaisltabc maser) snthepesind aoe neaete txBenbythesoldneyanh pte ees teas 202 Quancum Physics 1. 4“ in units oe ert or} l\ aeoR? 03{\a2| \ owsofAd \ w Fig. 122, comimed ‘equatorial plane.When|m|=J,then|Pi!(cos#)|?=sin*@ascanbereadoff from Eq.10-55. This functionispeakedabout@=1/2.As/increases,thewidth ‘ofthepeak canbeshown todecrease like-!, andthus forlarge quancum faumbes wegetthecascal picture ofpasar cits, The finite width ofthe peak canbeunderstood from thefllowing considerations When |r|=&we fave Lit=Prand consequently La?+Ly?=1.Ths theangolar momentum‘etorcanneverbepretyontalonganais,lncidentally, thedegeneracy relly inno distinguished ras, Thos asate that aneigenstate ofLewithCigemvalewlbe“ented”inhex-dietion,Thewavefunctionwillnowbe /‘ |098 os] in +008 \ oar, oor / fA Jei\i \ Je a | or os aN 0.06 oas ray /lant ootJ \ j a+ 1 « Rin? i we $Y a i \o4 i \ V Fig.12-2. continued o™ 07 \om i\ oes {nea WN oo a Joos ~ ‘ Nn \ spo ab fh \\ ose i\f \ I \y ~ fw 4 ND a rn 12 r nee oor ee a ee i/o oor /. oa / NS ez aeea a rr” 1 ke “+peers neates ® Fig.12-2.cominued ‘ 206 Quam Pyses Onnw ' -ae ee | | 4.nmaninob-ero | eo.10 Fig. 12-3, Shapes ofthemace Legendte pyoomis asa fanction of Ihesgl ewe thesans andteeta lg, Jonred beeythesans 4lineat combination oftheYia(04), butbecause ofthe degeneracy, theenergy will bethe same asforthe z-oriented orbits. a)nen theweve fonctions, wecacalcu oo)=fFaroteato (2230) Some wal expectio vales ategiven below = fo 104 TheHydroges Aram 207 =oF set a vr)ze[on+1—0+)) ()-2rl an 1 z(>amen (asp Problems 1.Compute thewavelengths ofthe2P—+15wanstons ia(1)hydrogen, (2)deuterium(oudearmass=2Xprotonmass),(3)positonium(&bound Fstateofanclectronand2positron,whosemassisthesameasthatofanelectron. 2.Aneecton isin thegrovnd sate oftritium, forwhich thenucleus consists of«proton andtwoneutrons. Anuclear teacion instantaneously changes thenucleus toHe?,thai, woprotons andoneneutron, Calelate the probability thattheelectron remains intheground stateofHe®,What isthe probability thattheelectron isfree,withmomentum p? [Nove.Themomentum eigenfunction forafreeelectron isPPM agh)-22) 3.Therelativistic analogoftheSchrdinger equationforaspin0electron (Ghus norapplicable cotherealelectron isthe operator version of (EV) =fe+nit that i, Bwaywy(te (a-+4)ve-w+(F)‘ (4)Find theradial equation, (b)Findtheeigenvalue specrum bynotingthelosereitionship ofthe radialequation obtained in(a)withtheradialequation forchehydrogen stom,problem, 4.Using theexpression for{I/r).1 calculate theexpression for -(2) foraoatbitary hydrogen acomeigenstate (withZarbi). Showthatgenerallyforthispotential (T)=-1V) “This is«special example oftheVin her 208 Quantum Physic 5.Anelectron inthe Coulomb feld of«proton isinastate descibed by i the wave fonction Hara)+bal)—Yaa)+VIOvale] | (@)Whar istheexpectation value oftheenergy? ! ()What isthe expectation value ofL'? { (©)What istheexpectation value ofL? 6.Anelectron intheCoutomb fld of»proton isinastare described by the wae function ‘Whatitheprobabilitythatiwllbefoundinchegroundstateofthe hydrogen 2.An elecon isin then=2,11,m=0stateofthehydrogenstom, ‘What itswavefunction inmoments space? 8.The expectation value offle) inanystationary sate is«consant. Geulae 4ep)=+ (tee o=Fey)=5ep!) fora Hamiltonian H= pi/2m +Vir) sand show thie (B)=even Use this toestablish the result ofProblem 4, 9.Usethetechniques developed inthis chapter todiscuss thethree- dimensional harmonic oscillator problem, with HaPbimate an Note thatthe associated Laguetre polynomials alsoappear inthis problem, References ‘Averythoroughdiscussion ofthehydrogenlike eromsiscobefoundin B.U.Condon and G.H,Shorey, The Theory ofAunic Spcre, Cambridge University Press, Cambridge, 1959. “Theproblem isdiscussed inevery book onquantura mechanic. |chapter13 Interaction ofElectrons with Electromagnetic Field InChapter :2wedisused theintercon ofan clcron wththewai Coulomb fet uc to4pointchaugeTogomnivetastehemeccaean tnexenal eugncc oxetic Bed,weSunfatree hecata meee VvBry)=0 (3-1) 1Bees) VXE(rs) +OF ° (03-2) V-E(rs) =Sap(rJ) (13-3) ¥xMeg)~2ORLAAey (34) whee oe) aa) tthe charge andcent denss thatatethesures of theceeomagneve elsEs) aBis). Theconservation ofche paca 20D4wie)=0 (33) isautora sie ‘Wemaystaytheetsmoequationbyexesingtheelsintemsof ascalat potential (rf) andavector potential Aird) Bie)™VXAles) Aes)Bes)==EMvoeey (36) 210 Quantum Pipe ‘TheldsBandBdontdetermineyandAunig.Newpoems,givenby Alles) =Ales) —¥firs) es) #668)=a0)+2MED 3 steelyseenoythesameEanBfel.Thetansformation fomtheset(A,B)to(A46)isknownasagaugesransformation, andtheinvariance ofFandBallows wstochoose theabizary fancuon flr)inthemostconvenicnt way. ‘Thesource-dependent pairofequations (13-3) and(13-4) nowread 1 Vert)—-_2(WA)=4ep(r.s) (13-8) sod 1PAs) 15 ae, VRXA+Soeot©OeYOie) which mayberewricen a 1DAG) 18) fe a5, THAR)+etr(ra+7*)Mes)(13-9) Ufthecharge distribution isstatic, thatis,p(r)isindependent oftime,itiscon- ‘eentochoose thegauge suchthat VAs) =0 (13-10) Thischoiceoffis)isgiventhenameofCoulombgage.Inthatcasewehave —V¢r) =4xo(r) (3-11) thatis,wehavea vimeindepedent seepotat andthentheequation forAes) seas fe)4LAG_fe ~vn) +LPMD ey (asta) Whenthechargeditburonisnostati,eismoreconveniettochoose theso-alled Lares pange orwhich VAG)++2! 4 (sa) Thisleaves theequation forthe vector potential vnateed, butnowthescale quition alsoobeysawaveequation, Atechni pointworth ating is che«teetion ‘VX (¥XA) =—VA+VW-A) exertionofHeronswithHecuomagnctic Feld201 sedtoobtain (13:9) isonyvalidincaresiancootdinats. Ths, VA(e), aic ‘appears, must becalculated interms ofx,y,and2Theequationdescribingtheneractionof«pointelconofmasswith anelecuomagneic fl thecasial Loreate foeequation ate~[ea+x20) (3.4 Wenowassert thathisequation willbe obtained ifchecsscl Hamikonian foranelectron intheabsence offields PeHe2 (13-15) ischanged bymaking theakeration j Poptfats) (316) ‘andadding thepotential ep(r)(weshalldealwithstaticsalarpoceatals),fothae a L € id Heif+“aca|+ele) (3.7) ‘Weshal ave theproof ofthis statement asanexercise fortherider! The conesponding Scddinger equation withthestatic potential taken overtothe fighe side is 1 “)A(Ertia)ve=te+oveivien css) whE Yt p> The leftside is iby+fa)(Sovtas)ATT FANG By ik i e =-EyBaw ways fm( 2pEYOEwalta H ® ii @; =~ Py Hayy 2é Eeny4AvetoGAW (13-19) © Foraconseane uniform magnetic eld,B,wemaytake +Antex (320) "seeforme 4.26.tNeee taeichesorwigsincewemayAddsheginofnyfncon voAwithoutchangingBTchoihowerersepaca 212 Quantum Physics “This means thatthe cece components ofAare : A=$68, ~2B2B,—x8,x8,—3B) ! and consequently VX A=(BB, +4B,, By,B) =B ©Hence thesecond erm in(15-19) becomes ick it rxpey-- “~pexeuex¥ ue xw =ferxtwee (3.2)ape RG Que * and the thid term is e e en Sex By =fem—ey=SEtt99 13.22) a Slew ORY SEY (322) ifBisthedirection chatdefinesthet-axis.Thisisoftheformofatwo-dimen- sioval harmonic exilator potenti ‘Letuscomparethemagnitudes ofchecwoteams,Theratoisestimated with(L,)takenoforderand(e+38oforderawithatheBobrrads: (e/aue)a8VeBB((/20)RB4fee/a348fad ~—_ Seats X1705 X10-9) B ~9X10"gauss (1523) «©Thusinaromic systems,withthekindoffields availble ithelaboratory thats, B'S10°gauss, thequadratic termiscertainly negligible. Thetermlinear inB, compared withtheCoulomb potential energy canbeestimated inasimilar way (ire) KB Weg LB B Cjay~2eas8274e/acs SX10"gauss (324) sothatchelinear rmwillonlyslightly perturb cheatomic energy levels. The Guedratic temcanbecome veryimportant under twocondition: ifthemag etc fedisveryintense, iisbelieved thatfields aslarge asTO!guuss may Exist onthesurface ofeaston stars, and this would radically aker thestrctare IocerctionofFecwons with Hlcromagnetic Feld 213 ofators? Thequadratic term willalsobeimporant when weconsider the macroscopic motion ofenelectron inanexer ld Lecusfsconsider thelinear term alone, andpickthez-direction wocoincidewiththeeofB.ThentheHamiltonian withy=0iseredbythe addition of n=BL (13-25)ae Ifwedefine thefrequency, called theLarmor frequency, a o Bow 13.26)Que (13-26) anudealwidenergy eigenstates chataresimutancously eigenstates ofI?and La,thentheexccatetm(13-25), when acting onaneigenstate, yields 2number, meme, Hittain(t) =heozmiinin(®) (13-27) ‘here misthez-component oftheangular momentum eigenvalue, with —:< ‘mSI,Thus theexisting energy levels, withtheir(21+1)-fold degeneracy aresplicinco (21+ 1components haar ually spaced wthenergies given by Baaitstay (3-28) “Thesneofthespi Ba (e)eQue Duc\e/ac?)ae? -(=)(2) Que Vi\e/at, syd)qBo =hate) aTS B -(2, ” Sincethereateselectionrules(obediscussed lateraccording towhich onlytransitions inwhich them-value changes by2eootunity atallowed, turnsoutthachesingle linerepreeating 2tanstion with B=0splitinto srelines,a8canbeseenitFig.13-1.Thiseffec isthemarmal Zeeman ef. ‘Actually, wales theeleezon spinstateintheatomisoneinwhich thesens ~ +SeeR.Cohen,L.Lodenqusi, andM,Rederman, Phy.Rev,Letters,25,467(1970). 214 Quaneum Physics rr) ; am ape tea on neo eee |~~TTTTT ne, Alig _ met .1eoet seven “Si ge an = ae Fig.13-1. Normal Zeeman effect: ofthe15possible tanstions berween the d=2and/=1sates, splitbythemagnetic Geld, only9,cotesponding toam=immy=1,0,oceutintheformofthelines zero, theinteractions oftheelectton spinwiththemagnetic Seldchanges the pattem predicted above. Themote common aomalaus Zeman efect willbe Aiscussed when wehaveleaed about spin Tis ofsome interest todiscuss thesolution ofan electro inaconstant magnetic fielduncer conditions where theB?term isaotnegligible, andwhere theCoulomb potential canbeneglected. Under those conditions, withBagain chosen todefine thezdivection, theSchrBdinger equation ids x oy ee aE SELtSRGteeBE(1329) ‘wherewehaveused(13-19),(15-21)and(13-22).Thepresenceofthe"potential” G+ 99)suggests theuseofcylindrical coordinates fortheseparition ofthe variables, Waiting x= pease y= psing (03-30) InventionofHetonswthHecromayneiceld215 wefllow theprocedute ote’ athebeginning ofChapter 10ative 2ggg2se2 dx 8ap Oe 2 2 ee ‘ ByHa tO ay (13-31) and beace a a 12 Le waBy Byte se 22)antOFdptoOg (13-32) Iewenow wie Vr)=tmp) cmeete (13-33) Wefodthatthe diferent uation aid byunl) ee eeeeee ee Bt oS are (Bee) eno (13-34) Teweintroduce chevariable [eBx=Vvme? (03-35) wepyrite etn nhom Btek™ eT eto (13-36) were fue (pH dhe (,ee)an (1337) Iciaysenightorward 0deen that(the beavior of) tiny, determined from isw(x)~=", and(b)thebehavior ofu(x)neatx=0,determined from Fe ide ontde xd 8=O isw(x) ~x'*!. Wethus write wx)=xletGl) (13-38) 216 Quantum Physics i ‘anddetermine thedifferential equation obeyed byG(s). | ‘lite algebra leads co 2G|(zal+1yea -; £65(LAOae)E+Q—2-aim)G=0 (1349) ‘Thiscanbebroughtintothesameformas(12-11)ifwechangevariables10 yor (13-40) | ‘Theequation then takes theform a6(*+1)#A=2-2\m]AS4(eo 3) e 7 4 % ‘Wecannow proceed asin Chapter 12.Comparison wich (12-11) shows thatwe must have 4,-tbel.,, (13-42) 4 z 25aneigenvalue condicion, withme=0,1,2,3,«<<»Thisimplies thar— F#E/2y, theenergy with thekinetic energy ofthefreemotion inthe2-diection subtracted out,isgiven by RE wh en Canetit Il+m) (3-4) and 6G)=Lt6) (ase4a) (©ue discussionofthissolutionwillbeconfinedtotheclassicallimit,Todo this,we6streviewcheclassicaltheory.GiventheHamiltonian(13-17),without thescalar potential tenn, wehavet yaPtWoa 03-45) * land with A=—Jr XB, weobtain eXvaexpt ex(aexD =L—Ebe(e-B) —HB} (3-46) “The render who isot falas with mechanics 48formulated byHasion can convince hina thatthe equtions ds a1t/9py dps ~~ a0d 3008,ae Stuiilen soNewt's eqton frH=p2/ay+Vie.Theequationsaohaforde Shove complicated Hemroian (pF sA(e/e¥/2 ToteractionofHleeronswithBlecttomagactic Field217 with chehelp oftheidentity &X(bX6)=blare)~efa-b) (13-47) Wetake the=-component ofthisequation toobain aeX=LtSBE A) thar is, B 00=Ly+S ot (13-48) ‘The expression forthe force ontheelectron Fa-‘vxB (03-49) Yields therelation wt_eBire (13.50) forcircular motion, Thisrelation, together with(13-48), afteralittealgebra, yields, epst A330 vr Ey a (13-51) and ‘Wenow rerurn totheexpression fortheenergy, (13-43). Because ofthesmallnessoff,theenergycanonlybeofmacroscopic sineforreasonableB,ifCt+1+[re]+m)isverylarge,Wehavetwocases:(a)Ifm<0,thisimplies chacmisverylage. Nown-determines thedegree ofthepolynomialLIE!),thatis,chenumber ofthezerosinthefunction,’ andifthatisvery large, thefunction cannot belargeforsome small range ofywhere theclassical ‘orbit would belocated. (b)If>0,thecoefficient is(2m, 1+2m),andthiscanbelarge,withm+small,providedharmi$large.Theenergynowis <a p-e (13553)2 ne +SeeBg.12:25,thedevelopmen leadinguptiandchediscsonp.200 i 218 Quantum Physics iinagreement withtheclassical resule. Note that | iL,=him (13-54) ispositive, asexpected. |‘Wecanaloshowthatsheadsoftheorbit,asdeterminedbythepeaking ‘oftheradial probability distribution, corresponds totheclassical value, Lecus takem=0.InthatcaseL{*(y)isjustaconstant, andthesquare ofthewave function is,according to(13-38), PGs) =lot (1359) “Thisbas«maximum where Yr Pk aim|xtisi! —etl pa F=ein alt) tag | chatis,at |x=vinl (356) ‘which yields a \to=(hm) (359) ‘This problem is«bezutifilillustation ofthecorrespondence peincpe “There aceseveral interesting quaacum mechanical ects connected with theinceaction with «magnetic feldthatwenow turnto,TheSchbdinget ‘©equation (13-18) appears toviolute theprinciple ofgauge invariance, since itis A(rs) tharappears intheequation, andunder thetransformation AA Urs) (13-58) theHamionian ischanged according to WL ee eealattiayog (Fesiariny om Ieispossiblewosavegaugeinvariancebyusingthefactthat«changeofthewave functionbyaphase factor, which maydepend onx,husnophysic conse-quences. Thusifwerequirethac(13-58)mustbeaccompanied bythetrans.formation Ves) > Yrs) (13-60) thenthelefthandsideofeq,(13-18)becomes Lae tage hoataatalgPtiAtivs) (Gutiat ivy Interaction ofElecrons with Elecwomagaetic Field 219 1 Lee kata =(Geiat fei)[A(Eve+fanSey+oeay Ln fh e e ¥=La(Festadtuyssva)'y (3-61) “Thus withthe choice A= hef (13-62) that is,with thecansformation law Her) 9MOD Yes) (13-63) _gruge invariance isresored F aaGld-iee region, B=0,which implies that q vxXA=0 (3.69) tharis,Amaybewritenasagridientofafunction A= (3.6) Inafield-freeregion,wemaythereforedescribethemotionofanelectonintwoways: either wedoaotconsider thepresence offldatall,andwrite iyFs(Es)+veo|y=we (03-66) fortheenergy eigenfunction equation, cewewrite cheequation with thevector potential given by(13-65) Lh ey snd ake Ve etinar y (13-68) ‘Thefunctionfir)maybewrittenintermsofA(e4)bysolving(13-65): fea[denen (3) where thepuchofitegration istaken fromanarbitrary xedpoint, focexample, cheorigin, ofinfinity, tothe point r.Theintegral onlymakes sense fB~0,thatis,inafield-freregion,sicethediffereneeintheintegralslong«wodif,even paths, labeled |and2,is 220 Quanum Phys | hs'"/fon2 if Fig.132.Theimagef4¢e)alongpathandpthargent the sum, sinceheferences equal themagnetic faxenclosedbythedoeoop fde'-A(r's) ~[anes =geAlr’) =[ivxawaas= [pss %(370) hee wehave wsedSoke’ theorem, andwhee seh Auxofmagnetic eld through thesurface spaned bythe twopats (Fig 132), Thos ony =0 wlsheps factor in(3.68) beindependent ofthe coir ofpain eeTneIntegiSuchamindependence isequifweinsthatthewevefonctionbeSingleTeewopuhsincludefu,thenthewavefunctionsofelectronsevelng tloog thetwopath wllacquie diferent uses. Anlatesting consequence isha fanelectron moves eld tegion that notsimply connec, batsuounde a"hole" conning fox&thenupon compaing xeste theelectron acquires anadditional phase factor «*®/¢, Therequirement that theelecuon wave funtion besinglevalve, a0Btthephase Retr iunity, impli thtsero! fi guased chea, motte. (3-71) Such 4scuaton ass inthemotion ofeeeons in+superconducting tingsuounding «region containing fax. Tebasexpetinents, dove ia196 were bsed oathefolowing there: atng, made of superonductor +B. Dever andW, Paiburk, Py. Re La, 7,48(961): R.DAM and M. Naseer J\\ AA Hy)/)\\/\ rT jjTI\ qth[|cup)) \\ \{ yh \ \ .T<te atb= an (13-72)ro) > ne vow (13-73) | 222 Quantum Physics! a=weeee Qn.rane som Fig. 13-4. Schematic sketch ofexperiment measring shift ofelecom inte: ference pattern byconfined magnetic flux. ‘whereyhdenotesthepartofthewavefunctionthatdescribestheelectronfollow-ingpath1,andyathepareappropiatetopath2Inchepresenceofthesolenoidwe have PmYadtNhited 4.yyibaa = re +ya)neler (13-74) ‘The flux thus causesarelativechangeinphaseberweenyxandYr,aodthiswill change cheinterference pattern, This efet, frstpointed outbyAbaranow and Bohm, hasbeen observed expetimentally.* Problems 1,Show that with Pavea2B+Hin) theequations a, OH a” be 2Hds Ox" +2. 6,Chambers, Pl Ren, Lat, 5,3 (960. Ieverction ofEecons with Hecwomagneic Field 223 yieldahequtionsofmotion fs ae Ox" 2.Show tha theHamiltonian H=2[p+| 7 Yields dheLoven foce equation a 1oe«[Pen+yxae| {Nae Inyour aleuation use 4piggy=OAHOM|DAdeDA a= +e te ybaOr sincetheflds thatenter intotheequation ofmotion (andcheHamiltonian)‘mustbeevaluatedatthepositionoftheparticle.) 3.Gleuate dhewavelengths ofchedhe Zeemaa line inthe 3D+2P transition inhydrogen, when theateriin eldof1"gauss. 7 4.Consider anelectron confined toategion between twocylinders of |. radia andbrexpectvely (b> a).a)Sepante theScinget tion sn cylindrical coordinates (cf.Eq.13-32), andshow thattheequation canbeSolvedinternsofBeselfonctions,Whatuetheconditionforthedeci.tioofthe energy eigenvalues? (0)Discus thedegeacay ofthe entgy sign functions? Whi sitdueto?ForBese fancions, ceoteaferpaces 8—— 5.Inthisproblem wework outanexample showing howanenclosedmagneticfuchangestheangularmomentumef«priceia4pionousidethefxrue,Consider amagnetic Sdconfined inscjindscl Retony cn LattheBuxbe Intheregion» >athreisnomagaetic eldrlhence tevectorpoteisofthefarm ‘ A(of.z) =VA(p8,z) (a)Thechoice ofgauge ¥-A=0implies that va=o Showthasslutionofhiequation,sifyng(13-70), 1 : a-ha 224 Quantum Physics (8)Calculate theangular momentum about thesymmetry axis wexv=t=[ex(Le+fa)] incylindrical coordinates, andshowthatfortheaboveAitisgivenby tek yee an re (©)SolvetheeigenvalueproblemLy=Wy,andshowthatsingle-valued inesoftheeigenfunctions leads tofluxquantization. 6.Show thatforasystem described bytheHamiltonian p= AloAeeor m0 thefxj,which satises 2vvevinoJeet se isgiven by j= [yee—wy+2aoeygLTRWEZeAe ShowalsochattheHamiltnian equationsofmotionofProblem1implythat 4, tino where Lerx(»+£aca) 7.Consider theproblemof«chargedparticleinanexternalmagnesicfield B=(00,8)withthegaugesochosenthatA=(—yB,0,0).Whatarethecon- stantsofthemotion?Goasfatasyoucaninsolvingtheequationofmotion, andobtain theenetgy spectrum. Canyouexplain why thesame problem iathe ‘gauges A=(—yB/2, xB/2, 0), A=(—yB, 0,0),and A=(0,xB,0)canstill represent thesaine physical situation, even though thesolutions look sodif- ferent inal htce cases? 8ConsiderachargedparticleinamagneticfieldB=(0,0,8)andina ‘crossedelectric field E=(8,0,0). Which ofthethree gauges mentioned in problem 7would youuseforthispeoblem? Solve theeigenvalue problem Note.Thesolutionoftheequationfu 1de sBatge(-S)e-e Imteaction ofFlcwons with leromagnetc Feld 225 with ioegal, arekoowa asBesse! functions, forthe regular soltions we (ZY j Je)()&nwti tndNeumann function frtheregular solutions )=2je)tog2—2(2)822 Nala)=JJala)og3=()Liem A(z) @- lor eyYER ran (log7=0.572...) a=(214%7] “They have theasymprocc behavior zy mor 1 nso~(2)"an(2-4) [140(2)] Adeuiled discussion ofthie properties may befound inanybook onthe special functions ofmathematical physics. References “the various aspects ofelectron motion inamagaetic eld aevery iateesiogly dseused i RP. Feyaman, RB, Leighton, andM.Sands,TheFyomanLeeonPhysi, Vol. 5,Addison-Wesley nes 1963. a= |chapter14 Operators, Matrices, andSpin| Aprops daca ofatoms isnot pombe without considenion ofhespnoftheeonInspoftesuggestivenametispoppeteconten Basuocasa aang tn iltoonbecome ens eefeeal somewhat shtace mths Foruntcly wenesome paaanoe heho fuer deparare om descipcon donated tocooeines ea aee dctsted boththehumonic olan (Cape 1)sndthese meet cigenlue peoblen DYin=WU+1)Vw Lin =hain (aaybyopormethods,Forteharmonicoxiwefoundsts,df!by 1 aeg ’ j =Gaye (AIHe (04.2) foeics Hig=theo +$)tay, 14-3) sdecouldaoalee theaction heating ondlowetngopentors ons, Ane Voth ny sd . Ann=Vik (143) We also showed Menta) =San (14-6) 4seement thccanbemade toBoldfrtheeigenstates ofanyherman trenton (1hr) Hwee theslat poof Aree aypapel . (ttm|Hite)=(Wm!HH!te)=(+4)FeoBae 228 Quantum Physics WalAte) =ealAta)=VODRbas in|Atte)=(Hn|Alita)=Vthdng (4-7) where wehave introduced themote symmetric notacion {usloe)) =(aslolas) (148) These quantities may bearanged inarays called ratries. The conventional notation foramatrix Myhasthefirstindex labeling therow, andthesecond labeling thecoluma ofthearray. Thus ifweconvert thescalar product (4m|H! ty)into Hay wefind that a rno 3/2 o o . 0 0 (3/2 0 Hohe°°oO2 ” : : - 049) Silly 0 6 0 oO > a _ na ee A=vELgo vs 0 : (14:10) and o vio 0 o 0 v2 Oe A=Vif 0 0 0 V3 Pont aan) Weshallcallhearray(talFlin),whereBisanyoperator,andthewaeanycom pleseset,amatrix representation ofFinthebatisprovided bytheu,."This appellation‘needssomejustfciion. Theproductoftwomatrices,frexample,satisfies FG)=ZLPalGes (42) andweneed (0verify cisrelation forthe“matrix representations” ofche operators Pand G.Todothis, letusconsider thestate Guy, and, using com- pleteness, expand itintheform Guy=XLCon (14-13) Operators, Matrices, andSpin 229 ‘Thecoeficients Gyategiven by G=Glas) (14.14) Hence AlFG)as5 =wlFE Coxe) 7 =XGed Fin) =EXWilFlue(unlGay) (14-15) hich isthe sameas(14-12), provided wewrite us|Fling)=Fin (14.16) ‘andsoon,Ieiausefulmnemonic device towttetheunitoperator intheform r= lan)Gul (aaa) andinthatformiccanbeinserced beeween thetwooperators FandGinthe matrix element (u«|FG|4;) togive(14-15). Farther justification forthe matrix connection comes fromtherelation Wen|Flin)*=(Fig|te)=in|FT|atm) (14-18) which shows thaiftheoperator Fistepresented by«mattis, thenthehermitian ‘conjugate operator F*willberepresented bythehermician conjugate matrix,since chelaver isdefined by (aq =Fe (1419) Notethatinourdiscussion wemade noreference tothefactthatwe started outwitheigenstates oftheharmonic osilator Hamiltonian. Theony‘hingthatiscialaboutthem,isthattheydagonaice thematrixrepresenting H.Withanother complete set,Hwouldnoebediagonsl, andreading ofitscigen.‘ues,chatis,themattiselements whenitisdiagonal, wouldnotbeexsyTheYinweredefinedtobestatesthatdagonalizeL?andL,simultneously. ‘HEwestaywithafixtd/,chati,withstatesinwhichonlythem-value isvaiable,thea,withanabbreviated notation,thesecondoftherelations(141)reada nal len)=Batbur (1420) Furthermore (10-40) with(10-52) implies that . Goat|Laxtym)=RUG+1)~mleteypeBaym (14-21) 230 Quantum Physics “This leads tothematrix representations a) L=i{ 0 0 0 ° o = (14-22) o vio Ly=fil 0 0 v2 ° o o (14-23) and 0 0 0 L=il vi 0 0 0 v2 0 (14.24) for the/=1angularmomentumoperators.Therowsandcolumnsatelabeled withm=1,0,—1inorderleftcosightandtoptobottom.[eiseasytocheck ‘thatchematrices satisfy thecommutation relations. Forexample vi\/000 2o\/ovi a Watl=#(o ov2](v2 o0)-#(v2 ool(o ove 00 o\ ovo, ovio\o oy 200 ooo 10 oO =W\0 20}~flo20)=#8loo o}=2hL, (1425)00 002 00=1 Genctal seasons berween sates canalsobewriten inmatrix represenci- tion. Consider, forexample, arelation like y= 46 (14-26) Ifwetake chescalar product ofthiswith aaymember ofacompleteset4s, wwe have Gadd)=(esl4a) (1427)Furthermore, theinsertionoftheunitoperator,intheform(14-17)berweenAand¢yields daly)=EelAlen)(eal) (04-28) Hfwewrite (w_|@) asacoluma vector ay qld) fwdle)||as fale)of10fa] . (14-29) sndsilty (ul) Fb dal)||ae (ul)>(ws|¥) =|% (1430) thenthematrix representation of(14:26) is B=DAnon (2431) ‘Thusmatrices represent operators, andcoluma vectots tepresent states.Thescalarproduct(6|4q)=(va16)*iswrittenconventionally intheformof2row (im) (@i,a3,a}, ...) (14-32) $0thatthescalarproduct ($y), forexample, canbewritten as iH)=Elm)univ) =Las, (14:33) ‘Aneigenvalue equation isaspecial aseof(14.26). Iereads Momab (14534) and iereads LDAaya (1435) inmatrix form. Thisisequivalent co a er dn tn—a An \ An An An—a a}=o0 (14-36) andcherewillb2nontrivial solution ofthsequation onlyifthedeterminant ofthemactx vanished det|Aa—abeel=0 (0437) ‘Thisis«goodwayoffindingcigenvalucs (andeigenvectors) foroperatorsrepresented byfinitemaces, butforinfinitemaces thisissmfoeonely10 50simple "Tris indeedforunate carthereianalternative torepresenting operators 9 252 Quantum Physies byfunctions anddifferentials, sincenotalloperators canberepresented inthat ‘way. Thesimplest example isthatcomrespondiog tocheangular momentum T=4.Bq.(10-51) and(10.60)elluschat Yanai =Cavin00? (0438) and(10:54) allows ustocompute £089 vn LM*oo (1439) ‘This, however, isnotproportional toYi,-1/2andFurthermote iissingular at @=Oand x.Thus for/=3there areroubles, and wemust tum tomattixrepresentations; Insteadoftalkingebout/=4,weshalltalkaboutspin,S=J, reservingtheFeter/fortheorbitalangularmomentumassociatedwith¥Xp- “Thespinoperators aeS,,Sy,and5,,andtheyaredefined bytheir commutation felations [5.5] =mS. (14-40) and s0on. Wewishtorepresent them by2x2mattices. (14-20) yields a) Sah (14.41) 0 =, and(14-21) gives o1 0 0 Sh Ssh (04-42) mr No 0 10 ‘Wemay write thisrepresentation as S= ie (14.43) where on oF 1 0 od = «= (144) 10 io 0-7 atethePauli matrices ‘They satisfy thecommutation relations lespal =2iee (14.45) andsoon,astheymust,cosatisfy(14-40),andtheyalsosatisfy 10 afeataet= = (14-46) o4 Operators, Mates,andSpin233 and nner (447) hich arerelations pecstar tothespn§representations andJonotholdfoe the!=1mauices, forample Tueeigenstates ofSwilbeeepcseted by2ewocomponent cola ‘ecco, which wecllspinor. Tofindtheseeigenspnocs, wesole that is, (10(”(«<4o&/\, . ‘Thepluscigensclution has»=0,andtheminus eigensolution has w=0, 1 ° x w= (14-49) ° 1 forthecigenspinors costesponding tospinup[S,=+(1/2)K]andspindown[S=—(1/2)i], respectively. ‘Anatbicary spinor eanbeexpanded ichiscomplet set ow 1 ° wal’ ea (0450) o ° : ‘ndtheexpansion postulate yieldscheinerptetation tat|ay|*and [a-|2,whenpopetly normalize, sochat slay? +laltaa (14-51) yieldtheprobabilities chatameasurement ofS,onthestate(2)yields+(1/2) Rand ~(1/2) Arespectively 234 Quinta Physics Icisnotnecessary tokeep5,diagonal.Ifwelookfortheeigenstatesofthe ‘operator$,cos$+5,sin¢,wemustsolve eoeesono(”)-w( )as hac is, 0cosgising/# «=. cosétising 0 . ’ This implies har ve aw aime (14-53) Hence A= 1 (14-54), ‘Theeigenvectors corresponding toX=+1and\=—1are en he 5 1 1 respectively. Itisinterestingtoobservechaifwechange¢t0.4+2xthesolutions changesign.Thisischaracteristic ofoddhalf-integerspinwavefunctions(fermion ‘srates); although thisdoes notviolate quantum mechanics, since —1isjust@ phase factor, itdoesmean thatnoclassical macroscopic wave packet anbe constructed thatasoddhulFintegal angular momentum. Given anarbitrary statea,theexpectation value of$maybecalculated. We have @lSlay= LY lies riled car] of,equivaleatly, | soebad (ay, a) S| ‘Thus 0 1\/a (Se)=(ha) gh 1 ol\a perms, Macs andSpin 235 =et,2)-Wea_+aay) “ 0-i\/a (Sy)=Hila’, a2) io Na soeee iis . =We,2)-~Fa-ata 2 ia soea (8)=Wa, 2) =Wilegl? ~jal a 0436 [Note thatallofthesearereal,asexpected forhermitian operators Weshallseelatertacheap ofanelectonapesintheHamiooian fordehydrogen atom, forexample, coupled totheeabial eagle momenta, When anslecton islocalized acyst laice si,forctample oftenpomibletouethespinastheonlydegreoffredomthattheelecuemposses,‘Theeconwillhaveanintinsicmagoedcdipolemomentbyweeofisspin anddatmagnetic moment is | m=~Es (4sn where g,thegyromagntc rai, iveyclose ro2, a2 (14%...) a zoonsia a4: ta(i+e )2.002319: (14-58) td misheelectron mass Forsuch loazed elect, theHamitonian inthepresence ofaexternal magnetic eldBisusthepocetal energy ah=u /new p=hap 439) a.) TreSettingsencforshesey)=[2-0] eee vo= [20 "A“cc eeeon moving iacide wthangle women Lil fem4 cancatagp wns mapecte momen =tLe Seaheise ey ce Sede eon capte (1430) oxybyeo Re eine ce Sercnivide Dis: ease fomwhic hee £2fuses Heat £72comefomgustan datelyenic Totsonlaiod Sree eees50)idsovonnGousimirtodG UlesieckGoan 236° Quantum Physics HO Bg nyeMO. Bo ayy (14-60) 1€B istaken +0define thezexis, andifwewrite a0) a Wo= =e] (4-61) af) a chen theequation becomes a 10\/a fe-au( ( (14.62) wa}*\o ala “Thesolutions comespond todiferent frequencies, Wehave, for =ep8/4me, ()-(}).andfotw=—(egB/4n0), (”)=()‘Ths,ifheintialsacis . then theste ata late time will be wi)=()w=28 (14.64) be “ame Suppose thatac+=0thespinisaneigenstate ofS.with eigenvalue +(1/2) f, chati,it"points inthex-dtection,” This means thac 0 1\/4 ‘ wh =m v o/\s . ()-a(aatis,(" )= (en,ataterce of vy on ~ y= wee) a=wa, (:‘Ya(e ) “& =tecen( 7)3608Dat (0465) Operators, Matrices, andSpin 237 Similly o=i * Ligoin a (5)=Ma(°5)aa(e) apis ~thet aFie ie sinat (14-66) ‘Thuschespinpeecesses about the2axi, thedtection ofB,withfrequency . do=Ee 14s Taasolid thegyromagnetc factorofanelectoaisaffectedbythenature oftheforces acting iathesolid. Aknowledge ofgprovides veryuseful con- straints onwhat these forces could be,andiistherefor important tobeableto measure g.Thiseanbedone bythe paramagnetic resonance mathed, which wenow descibe. Consider anelceuoo, whoseonlydegreesoffiedomarethespinstates, undertheinfluence ofalarge magnetic fieldBppointing inthez-direction, and constant iatime, andasmall oscillating fieldB,coswt,pointing inthexdiuec- tion, The Schrodinger equation now reads oO)gp[BoBrcoswt\T ali) at -( (14.68) La|4\ecosw Be/LmH ot,with =Be 8Btee “doe (14-69) FOanal)+onconet (9Doocones) —od) (1470) Let AQ) = BO) =Be)om (a) “These satisfy theequations AO conotBO)Om 5 AO=cosusBY) 238° Quancom Physics =fod" BU) SB) tiesiP=cosetate =beaIO AG) (14-72) Iaobsaining these, wemade anapproximation. Wewrote 08asPtmpfddonter4famawe dear Sincewewillbeinerestedinvaluesofw=2uy,andsincebocharelarge,theterm thathasbeen dropped oscillates very rapidly, andwemayexpect catitscon- teibution averages toxeto, Amore detailed treatment suppots thisobservation. ‘Wemay eliminate B() =HAM rao ' By-2 (1473) andusethistoobtain asecond order differential equation forA() PAOa5,AO42guy ge~lta—«)TE+Sl)=0(474) Atial solution is AQ) =Ao) (14:75) Whenthisisinsertedinto(14-74),therootsoftheequation -x+tmwrt ao that is, ie—weVay—wt ryeMee VGolFt 14.76) determine “The most general solution is AQ)=AP +AOY a4) and hence B=—2AM EAA Cb) ‘This faly yields a= ALOT APY Wa >eh OA,OttAM) (0479) Operators, Maries, andSpin 239 Ifat¢=Othe electon spinpoints inthe positive -direction, then 4(0) =1and Ho) =0,travis, A+A=1 MA +XA =0 sothar dM Any de ae-y (14-80) ‘Theprobability thatatsomelatertimethespinpointinthenegativez-directionis|O(¢)|* wore 2M pe MA al?We BL -y Le q ot recien|! GaroF+al sc VETO coy)Gano +ot 2 ‘This quantity issmall, since ax<ca. When thefrequency ofthefeld Byis eaned” tomatch oe,thea theprobability becomes 1=coset apt (14.82) thatis,iapproaches unity.Sincetheenergyofthe "up"state isdifferent from thatofthe “down’ state, such anenergy difference, absorbed from theexcetnal Geld, signals theresonance frequency, sothatue,tndhence gcanbemeasured with great precision Problems 1,Ifthegroundstatevectorfortheharmonicoscillatorisgivenby 1 . ° w=| 0 240 Quantam Physics vse (142) and (14-10) tocalculate ay,as,#3,What isthegeneral pattern? Satisfyyourselfthat | in|ttn)=Boos X 2,Givenavector 1 2 -tft¥=Vel o calculate wich theharmonic oscillator operators (14:9), (14:10), (14-11) the quancties + @ ©)&), GD, 0). (©Usethistocaleulae ApAx. Note,TheexpressionforpandxinermsofAandA"aretobefoundin(7-4).]3,Calculacethetopleft4x4comerofchematicpresentation ofxfor the harmonic oscilatr. 4.Use(14-20) and(14-21) tocalculate chemats representation ofLn Ly,tndLyforangular momencusn 3/2,Check thatthecommutation relations {Ln t]= Ls ands0onatesatisfied 3.You aregiven theHamiltonian yep tary ts BeabetanOtay y Find theeigenvalues ofH(8)when theangular momentum ofthesystem is1; (0)when theangular momentum ofthesystem is2. (Note, Thematcix representations ofLy,Ly,Lsforangular momentum 2are obeinable from 200 0 0 01000 L=k{ ooo 0 0 000-1 0 000 0-2 Operators, Mavices, andSpin 241 020 00 oove oo Lekf ooo Veo} rawy 000 02 000 00 6,Calculace theeigenvalues ofthe matrix 8 4 6 ua(4 4 4 6 4 8 What aetheeigenvectors? 7.Consider anangular momentum 1system, represented bythestate 1 wee 4 ~ j vi\ 3 ‘Whacistheprobabilitythatmeasurement ofL,yieldsthevalue0? 8.Consider «system ofangular momencum 1:What atetheeigenfune- tions andeigenvalues oftheopetator Lily +[yL.? 9.Consider asystem ofspin1/2.What aretheeigenvalues andeigen: vectors oftheoperator S;+Sj?Suppose ameasurement ofthisoperator is ‘made, andthesystem isfound tobeinthestarecomespondiag tothelarger ‘igenvalue, What ischeprobably cheameasurement ofS,yields 4/2? 10.TheequationfortherateofchangeofanoperatorintheHeisenbergPicture isgiven byEq(7-47). Consider theoperators 5), ...What atethe ‘equations ofmotion ofthese operitors, iftheHamiltonian isgiven by = sy-B Ime andthecommutation relations ate{540}, §,(0] =iS), and50on,1B= (6.0.8), solve forS()interms ofS(0) 11,Aspin1/2objecisinaneigenstateofS.witheigenvalue++A/2at time¢=0.Acthattime itisplaced inamagneticeldB=(00,8)inwhichiis allowedtoprecessforatimeT.Atthatinstantchemagnetic fediveryrapidly rotatedinthedirection, soatitscomponents are(0,8,0). After another tine interval Tameasurement of5,icarted ou.What isthe peobebilty thatthevalve f/2 will befound? . | 12,Work outthebehavior of«spin 1particle inanextemal magnetic field, ChooseB=(0,0,8)andtaketheiaiialsarecobeaneigeasateof Som=5,sin@cos$+S,sin@sino+S,cos0 wih eigenvalues f,0,~Ainsuccession. Uns, Usethematt representations given by(14-22) co(14-24) References ‘Thematerial onspin isstandard, anddiscussions may befound inallofthe books listed atthe end ofthis volume. |chapter15 TheAddition of -Angular Momenta Supposewehavewoelections,whosspinsatedsctbedbytheopentors SsandSz.Eachofthesesetsofoperatorssatisfiesthestandardangularmomen-tun commu laos [Sis Sig]=iS and soon, [SamySta)=Sag (15-1) andsoon,buttewoSeofopemtors commute with eich ote, since the Gegres offcedom asec withdiflreat pails aeindepend, thatt [S:,SJ=0 (15-2) Lecus now def hetoa spa by S=S +3, (15-3)‘ThecommutationrelationsobeyedbythecomponensofSat [SesSs] =[Sie +SreySty+Soul =[SieSig} +[StesSe] =(Su +Su)=iS, (3-4) todtoon, Weaetherefore justified iacling Sthetotspi,WemaynowStremioeheigeluesandegenuncont atSaadOSSTQE wospin Systm ttaly hasfoursee wedenote thespinor of thefirstelectron byx{),sothar SxY =4G+1)LY Sia) =sti? as) 244 Quantum Physics andsimilanlyforthespinorxofthesecondelectron,thenthefoustatesare XPxP PE, AP, PP (156) The eigenvalues ofSforthefour stazes are SxDxP =ue+Se)xX? =Sin) xP+xDSax®) shat is, SxP? =Aaa? Sxx? =xP =0 S22 =HP? (15-7) ‘Thateatewostareswithm-value0.Onemightexpectthatonelineatcombinationofthemwillforman$=1state,voformaeipletwiththem=1andm= ~1states, andtheorthogonal combination wilfrm asinglet§=0state Tocheck thisexpectation, Jetusconseruct theloweting operator LS +S as-8) andapply thistochem= 1state. Thisshould giveusthem=0state that belongs totheS=1tripler, aside fromacoefficient infront. Indeed, using the fact that S29 =fx? (039) ‘whichcanbeestablished bynotingchat oo4 o =") /1 ° Kw 7a =f (15-10) 1 0 i of} \o 1 wwe gee SxPAP =SAP) XP+xSP =HP +I? 2B? +OP =va asa) ‘Thelinear combination hasbeen normalized, andthecompensating factor in front, -/2h, agrees withwhat onewould expect from (10-36) and(10-48) with=m=1,IfwenowapplyS_tothislinearcombination, andnotethat 5% =0 (5:12) TheAddition ofAngular Momena 248 we get O72 422? hme sweEES PMgone 40 =VIO? (15-13) asweshould, foranangular momentum state $=1 The remaining state, constructed tobeorthogonal to(15-11) andproperly normalized, i 1 Feat? —2A asa andbecause ithas noparners, weconjecture thatitisan$=0sate, knonderto checkthis, wecompute S*forthetwostates 1 Xe=7PAE xP) (as-is) We have $= Gi+S)=S84S8428-8 =SP+SP+SastSS+SS (a6) First ofall, SiMe=FePSA! &PSY) =wx 5.7) andsimialy ;SPX =IX (3.18) Nex, wecalculate 2SuSmXy =2h)(— Hi)Xy=—PPX, (15-19) a SS+SS)Xam7arMSAD+SeaPSex? FSuxDS AP Sx Sox?) hich, withthe help of(15:9) and(15-12) yields (SiS +SiSoy) Xm LHX, (15-20) 246 Quancum Physis Thus 2 SX, =PG41-.40X= BX, ° =WSIS+1)Xy (15-21) with $=1andocouesponding tothe sates, ‘What webaveshown ithatthetrary ofthe foursates oftwospin1/2partiesmayberecombined intoatripletandintoasingleetotalspin.state.Foespns, theenodescriptions aecompletly equent. Tehowere, we havea physical sytem inwhich theforces depend omtheapi, thecg‘ionsoftheindividual spinsarenolonger simultaneous eigenfunctions ofHand, sy,Si,SipSe,Su,butcheymaybesimulancous eigenfunctions ofH,S*,§, S?,andSi.Thisismoseeasily seeninanexample. Ifwehaveapotential between twoelectrons thatdepends onchespin, sodue 1 VO=Vii)+eeSi-SaVa(r) (15-22) wecaneasilyseethatSy,andSz,donotcommute withthesecondterm,sothattheeigenstates ofHcontaining thispotentialcannotjustbesimpleproductsof ‘eigenstates ofSi,andSz.Ifweobserve, however, that SiS: =US" —S?—Se) (15-23) fothatciscer canbe replaced bythe eigenvalue, when acting onaeigen- function ofS*,8,*,and$;7,then ve)=vin+tran[ss+ 0-2] wan set =nott 0 (329wort(trio fir? 3-24) Such «spindependent potential isactualy observed intheneuton-protcasystem,Thebowedsatesan8={statesithedetetomburderetlanunbound5=0state,whichisonlypossibleifV(r)#0. Muchmore cnportant forfata eppliatios isthecombaation of«sia ‘withanorbital angular momentum. SinceLdepends onspatial coordinates and docs not,they commute {LS} =0 (15-23) eis eherefore evident thatthe components ofthe ota angular momentum 3, dened by ,J=4s (326 “TheAddition ofAngle Moment 247 willsatisfy theangular momentum commutation reltions, Wecannowaskfor linear combinations oftheYimandxxthaform eigenstates of baht (15-27) and S=(Ll+2L-8+$*) SLE SH 2S +LS +LS (15.28) Let usconsider the linear combination Vimar =Yims +BYinpix- (15-29) Is, byconstruction, aneigenfunctionofJ.witheigenvalue(w+3).Wenow determinewand8such chaitisalsoaneigenfunction ofjt.Weshall make wse ofthe fact that LyYim =(G+ 1)=mele+1)RY =(E+ w+ DE mPOA a LVig =(C= mE EE mI AY a Sixp=Sx0=0 Se =ting (15:30) Then FVints=ah?UE+1)Yins+2me+2m)Yeax- FUG my m+ DP Vian} +BELL+1)Vega $Yetx +200 +HB) Yawn +(GU—ml+m+1)!Vows} (15-31) q This willbeoftheform FAG+1)Vitara=BYG+W@%xy+BYinpix)—(15-32) provided thar aA 1)ED m+A~mym+1)=+DewAULAYDmtHalll—mtm+OP=G+0B (15-35) This requires that C=mlm1)LGD=~M+D2my) XU+D-+1) -d+m+1) 248 Quancam Physics which evidently hasewosolutions, <I +N M4 -t= (5:34) ! thar is, yaa in (1535) rey Forj=J+1/2,weget,afer litealgebra -(a ert 1330 a array "Nad1 (330) (Actually wejusgertheratio; theeaealeendy normalized forms). Thus _fitmth [Tm Pistiaapp Vite+afapppyYemane(13-57) ‘Wecanguess that thej=/—1/2solution must have theform im ipmsa Vie -¥BpVekafyoMina0538) inonder tobeorthogonal tothej=f+1/2solution. ‘Thesetwoexamplesillustratethegeneralfeaturesthatareinvolvedinthe additionofangular momenta: Ifwehavetheeigenstares ¥{%,ofLy?andLi,and theeigenstates ¥/2,ofandLa,thenweeanform(2s-+'1)(2,+1)product. wave functions hsmShy vik, (1339)-hS mS hy ‘These maybeclassified bytheeigenvalue of Jem bat be (15-40) ‘which ismy+ri,andwhich ranges from 2maximum value of4+ fdown to=h~hAsinthesimplecasesdiscuscedabove,differentlinearcombinations of fonctionswithchesame»valuewillbelongtodiferentvaluesofj.Inthetable belowwelistthepossible combinations forthespecial example offy=4,4=2.Weshallusethesimpleabbreviation (1,m:)forvive TheAddition ofAngular Moments 249 value ‘mm combinations ‘umber 4 6 (4.2) 1 3 (3) G2) 2 ‘ 0) Ga) @2) 3 3 4-1) 80) G1) (42) 4 2 (4,—2) (3,1) (2,0) (1,1) (0,2) 3 1 ,—2) (2,1) (1,0) (0,1) (—1,2) 5 o (2,=2) (1,—1) (0,0) (~1,1) (—2,2) 3 j “1 (2-2) (0.1) (“10 (“23 (32) 5 = (@,-2) (=1,=9 (-2.9 (3.1) (“42) 5 3 (-1,-2) (-2,-1) (3,0) (-4,1) 4 4 (—2,-2) (-3,-1) (—4,0) 3 3 23,-2) (4-1) 2 ~6 (-4,-2) 1 ‘Thereareatotalof45combinations, consistent with(2/,+1)(2h+1). ‘Thehigheststatehastotalangularmomentum/,+fascaneasilybe Fchecked byapplying Jo Yi2¥/2: PYARYB =at+Lit+Lads+Lael+Likes)YVR =FAN 1)+Ge+1)+abl YVR, =Rh+bt at) YRYER (5-41) Thisisj=6.ntheexample discussed intheable. Successive applications of @ jaket he (15-42) willpickoutguelinearcombination fromeachrowinthetable.Thesewillformthe13statesthatbelongtoj=6.Whencisisdone,thereremainsasinglestate withw=5,cwowithm=4,...onewithm=—5.Ieisextremelyplausible, andcan,infact,bechecked, thatthea=5starebelongs t0j=5.Again successive applications ofJ-pickoutanother linear combination from each row inthetable, forming 11states thatbelong toj=5.Repetition ofthispro- ‘cedure shows thatweget,aftercis,setsthatbelong to=4,=3,andfinally J=2.Themulkiplicities addupco45: B+Ut9t7+5—4 ‘Weshallnotworkoutthedewilsofthisdecomposition, asitisbeyond thescope ofthisbook. Wemerely state theresults. (@)Theproducs Yi2,¥f2, anbedecomposed intoeigenstates ofJ, with eigenvalues + 1)8,where cantake onthevalues fbbhhth—he hk (15.43) 250 Quantum Physics (b)Ieispossible cogeneralize (15-37) and(15-38) togivetheClebsch- Gordan series Ym=Ojm;holon)VERY, 3-44) ‘Thecoeficients Gm; /ymilum) ateknown a8Wigner coeficients,and have been tabulated formany valucs ofthearguments. Weshall useonly thecoeficients foefy=1/2,which wehave caleuliced explicil. Wecanverify thar themultiplicities check in(15-43): ifwesum the numberof states weget(> &) REF HN+WADE. EA-E =LRA-A+m+0 =@h+ Neh +D 13-43) ‘AGinal comment isinorder. Wenoted, when discussing identical paris, thacasytem ofewoelectons (oemore generally, twofermions) must bein state that isandsymmettic under theinterchange ofthe«woparis. Thisincerchange involvesnotonlytheexchange ofthespatialcoordinates, butalsoofthespinlabels.Forasystemoftwoidenticalspin1/2particles,the§=1 Atipletofsates ax? EGO 4QP) 03-46)VEO xx? issymmettic under spinlabel interchange, while the $=0(singlet) 1 FaPRE —XPXP) (5.7) isantisymmetric, Thusforatripletstate,thespatialwavefunctionmustbeantsymmetic, andforasinglet state, itmust besymmecic. The spatial wave function ofatwo-particle systemintheircenterofmasssystemisofthegeneral form le) =Runt) Yin) (15-48) ‘Aninterchange ofthecoordinates ofthe ewoparticles isequivalent tothechange nr) orete (05-49) “The Addition ofAngular Momenta 251 “Thus cheradial function remains unchanged. However under thistransformation Yinl) >Vine =0,6+2) =1) ¥en) (15-50) “Thus criplr scares must have oddorbital angular momentum /,andsingler stares ‘must have even otbtsl angular momentum. Weshall seeanapplicstion ofthis when wediscuss the states ofhelium ‘Apinteresting application ofthese remarks occuts inelementary particlePhysics.Oneofthefisthighlyunscableelementary particlestobediscovered‘wasthexmeson predicted byYukawa. This particle, which plays animpoctant role innuclear forces, comes inthree charge states x*,x’,x”.Itwas found to Ihave spin0,andchequestion arose whether thewave function ofapion—as this ‘meson came tobecalled—was even ofoddunder tefection, assuming thatthe known particle, theproton andtheneutron, hadpositive intrinsic patty. ‘The following experiment wassuggested. ‘Consider thecapture ofax~byadeuteron. Aslow pion inliquid deuterium F loses energy byavariety ofmechanisms, tillicnallyendsupinthelowestBohr ‘orbitabout the(pn) nucleus, andisthen captured through theaction ofthe nuclear forces, Inthe nuclear reaction rtdontn theangular momentum is1;thepion haszero spin, theorbital anguler momen: ‘umiszerointhelowest Bohr state, sothatthe only conttibution istheangularmomentum ofthedeuteron,whichis1,Thetwoneuttoosmustthereforebeinanangular momenturn 1sate. Ifthe toa spin oftheewoneutrons is0,then the orbieal angular momentum must be1.Ifthetotal spinoftherwo-neutton state is1,then orbital angular momentum 0,1,and2ispossible, since adding two angular momenta ofoneuniteach canyield 0,1,and2,andadding oneunit60 ‘wounits ofangular momentum canyield 3,2,and1.However singlet stateofewoidenticalfermionsmusthaveevenangularmomencum, andisthusex:cluded. Atriple state must have odd orbital angular momenturn, and this is possible iftheorbital angular momentum is1.Such astate, however, hasodd parity by(15-50), andhence thepion must have odd parity. Interms ofthe spectroscopic notation, which weshall use,where asae islabeled according to Ly (550) thetwo neutron states, from thesora class ofsates Ss,Pa,Ds, Fa, 8; Py, Py, *Po, Ds Dy Dy, MF Fe, sate esticted C0Sy"Ds. "Pa Pass bytheFermi-Ditac statistics argument, andofthese there isonly fone state, the#P, state, thac hasangular momenturn 1 252 Quantum Physics Problems 1,Work outthegeneralization of(15-37) and(15-38) totheaddition of orbital angular momentum L.tospin1 (@)Findtheeigenstates of$#andS,where 10 0 S=f{0 0 0 oo -1 (©)Ithesecigenscatesarelabeledfs,fyand£1,findtheectionofS, and ©onthese sates. (©)Calculate theeffect of Pa+S+20544S+LS, ‘oncombinations like Vint =Yat +BVingtbe+Vimo (2)Determine therelations between a,8,and obtained from THe =BGA DY 2.Findtheanalogof(15-46)fortwospin1particles,whichcancombine toformspin2,1,and0states, Usethenotation £9,£2,£2)fortheone-particle spin vectors. 3.Adeuteron hasspin1.What arethepossible spinandtotalangular ‘momentum states oftwodeuterons inanarbitrary angulat momentum state L? Donotforger thePauli principle 4.Aparticle ofspin 1moves inaceneal potential ofthe form Vie)=Vile)+SLAC)+(SLY) ‘WhatarethevaluesofV(r)inthesatesJ=L-+1,L,andL—1? 5.Consider thediscussion ofthedetermination oftheparity ofthe=~,Supposethe+~hadspin1,butwasstillapruredinanL=0orbitalstateinthereaction rtd ‘Whatarethepossibletwo-neutron states?Whichstatesaeallowedifthex~hadnegative parity? 6.Suppose thex”hasspin0andnegative parity, butiscapeured inthe wet don fom thePorbit.Showtharthetwoneutronsmustbeinasingletstate. “The Addition ofAngulat Momenta 253, 7.TheHamiltonian ofaspin system isgiven by aaBSS, utSe) # i Findcheeigenvalues andeigenfunctions ofthesystemofewo particles, (a)when both particles havespin1/2:(b)when oneofcheparticles hasspin1/2andche ‘other hasspin 1.Assume in(a)chattheewoparticles areidentical. 8.Consider two spin 1/2particles, whose spins aredescribed bythe Pruli operators 6,andé, Let@betheunitvector connecting thecwopasticles anddefine theoperator Sin=3(61-8)(44-8) —dy-dy Show thatifthe ewoparicles areinaS=0state (singlet) chen Siang =0 Show that fora tiple sate n= Sa +4)Xeiee =0 References “Thematerial discussed here isalso discussed inonewayoFanother inevery textbook onquantum mechanics. Many deaals eanbefound in 'M.E.Rose,Hlementary TheoryofAngularMomentem,JohoWileyandSons,Inc,1957. |chapter16 Time Independent Perturbation Theory ! Thee aefewpotentials V6)forwhich theSehodinger equation i‘exactlysolvable,andwehavealreadydiscussedmostofthem.Wemustthereforedevelop approximation techniques toobcuin theeigenvalues andeigenfunctions forpent thadonotledtoexact sale oats, Tntigsnr ne discus perubation tery. Weaster weave found heigen sed thecompe sexofeigetancion for+Haman he Hobe =Bad (161) tadweakforthe eigeoalucs andcigefunction fertheHamionian H= H+ hy (16-2) (Hy+Mh)oe=Ente (16-3)‘Wewilexpeshedesiredquaniisaspowerwesiesin,Thequestionofconvergence othe sis wlotbecnc: Pequny ontash thathe tries cant becongen andyettefat tse aes hl, to properly describe thephysical system. Wewillassume thatask—+0,Ey E,* andba ¢n Sinethefrmacomplet ewe mayexpnd Yinaseisinvolving E anstegn We vee ve=noofa+Eco«|(16.4) ‘TheficorNO)heretoallowsonormalisehe,Weavethefeedomchoot thepate of sawechoose hahdae eee ae 256 Quancurn Physics ‘expansion isealandpositive. Sace werequire thatfa, 48—*0, webave Noo) =1 Gul) =0 (46-5) More georlly, wehave Gud) =ACP+CHE . 166) and B=BSA NEE MED 4. (6) “TheSehdinge equation thenteads ‘ ttnfortZarda tBvette+..| =Weenersre+fatactorFrets] (6a) ‘Notethatthenormalization factor N(X)doesnotappearinthislinearequation. Identifying powers ofdyieldsaseriesofequations. Thefirstoneis HoChoe+thd.=ELEChoe+Ey 169) Using Hibs =186. weobtain te =Hide+B,8~Be)Con (16-10) wenowtakeasisproductwithanduetheorthonoemaliy condition Gul) =bu (say weobtain AED?=GINIba) (16.12) ‘Thisisaeryimportantformula.icstatesthatthesconderenergyshiftforagiven stateisjostheexpecation valueofthe perturbing potential ithatstat tthe ‘change inthepotential isof«definitesign,thentheenergyshiftwillhavethe samesiga. Theexplicie form of ap=farate)000)a0) (16.13) shows thetforthesifetobesignicant, both thepotential change andthe Probability density |¢,(r)|* must belarge. TimeIndependentPerurationTheory257 Ifwetakethescalar product of(16-10) witha,form36n,chenaltilda)+(Ex?—Es)C2=0 that is, (uInMh|4,) cy=Geldthldn) re=ee (1614) ‘Thenumeratoristhematrixclementof2;iathebasisofstatesiawhichHeisdiagonal. Thisformula isusedinthenextequation, which comes fromthe ‘identification ofterms proportional co: MeRon+neClee =BSECbs+EECon+Bt (06-15) Taking thesalar product with6,yields oo «y=5alHilde)elil.) BP=Baltlds)CR= EfBe = elton)?-Zhe (16-16) ‘ThelastLinefollows fromthehermiticity ofH:: ulHils) =Gulile)" (1617) ‘Tris, t00,is«veryimportant formula, especially sincethefstorder shiftfe-0‘quentlyvanishesongroundsofsymmetry. Wemayinterprettheformulaasfollows: thesecond orderenergy shiftischesumofterms, whose strength is given bythesquare ofthematrix element connecting thegiven stategytoall cotherstates bytheperturbing potential, weighted bythereciprocal ofche‘energy difereace between thestates, Wecandrawseveral conclusions fiomthe formula, (a)Ifdxistheground sate,thatis,thestateoflowest energy, thenthe‘denominatorinchesumisalwaysnegative,andhence(16-16)isalwaysnegative. (b)Allotherthings being equal, thatis,ifallthematrix elements ofHi ‘zeofroughly thesameorderofmagnitude (whichis thekindofguessone ‘would makewichoue morespecific knowledge), thennearby levelshaveabiggereffec onthesecond order energy shiftthandistanc oneshave. (©)Ifanimportant level"E"—imporeane inthesenseoflyingnearby,ot of(¢x|Hi|4x)being large—ties above thegiven level“'n,”*thenthesecondorder shiftisdownwards; ifiliesbelow,theshiftisupward,Wespeakofthis a5a _tendencyoflevelsforepeleachocher.‘Anexpression forCimaybeobtained from(16:15) byeaking chescalar 258 Quantum Physics Productwithdm,a»m,utweshallnotrequitethisformala,AlsoNi(\)canbeexermined from aide)=safe498lenin| =1 (16-18) Icistherefore1tofirstorderinHence,cofirstorderind,wemaywrite < Gn|MH\$e) vetDe (16:9) 4formula tha isometimes useful. ‘Theabovedevelopment needsmodification whenthereisdegenetacy, since,onthefaceofit,thedenominacor involving energy differences could vanish, Thedificuley isassociated withchefactchat,instead ofaunique ds,thereisafinitesetof6,allofwhichhavethesameenergyFy.Thisetcanbemade orthonotmal with respect tothelabel "i"because, aswehave seen in Chapter 4thislabel canbeassociated withtheeigenvalues ofsome other,simultaneously commuting, hesmiianoperators.Wethuschoosethesetof4!such thae ODP) =Gants (46-20) ‘Thenaturalwaytotakethedegeneracy intoaccountistoreplace(16-4) byanexpression thatinvolves linear combinations ofthedegenerate eigenfonctions ofHe = at? ” venofePENT CPDBaP+}(1621) “Thecocticients ai,i,.«willhavetobedetermined. Whentheaboveissub-stituted incotheSchrOdinger equation (16-3), weget,tofistorder inh, MoEREAe?+HEa? =EPLae+BOEAPYBe? (16-22) ‘Taking chescalar product with9'Pgivesthefirsordershiftequation %asl?|Mil?)=NEDay (16-23) Thisisafnite-dimensional eigenvalue peoblem. Forexample, ifthere isatwo folddegeneracy, andifweusethenotation PH AP)=hy (1624) Time Independent Pesttbation Theory 259 thisequation reads Icy +nse =EP Assan +hres =ESa (16-25) Both theeigenvaluesthere will,ingeneral, betwopossible values ofEf?—and thea,canbedetermined ftom thisequation, ifweaddthecondition that Ealt= (16-26) ‘Wedonotbother withthedetermination oftheBy,sinceweshallonlyuse F degenerate perturbation theory forchefirstorder energy eigenvalues inour ‘pplictions. ficsohappens thathy;=Oforij,chatis,thathematihis diagonal,thenthefistordershiftsarejustchediagonal elements ofthismatrix. ‘Thiswilhappen when thepercutbation /2,commutes withtheoperator whose‘igenvalues the"7"labelstepesent.Forexample,inchehydrogenatom,thereis4degeneracy asociated withtheeigenvalues ofZ,thatis,all m-vales have-the same energy. Ifehappens that [HiLJ=0 6-27) andifwechoose our¢?tobeeigenfunctions ofZ,,thenhywillbediagonal. Toseethis, note that with LA? =fim? (16-28) OPWL]19) =@P[HL. =Lith|a?) . =Kw? — hy =o (16-29) thais,(16-27) implies hy=0 form 2mi (16-30) Some ofchese features willbeillustrated intheexample below; others will appear laterinourdiscussion ofthe eelhydrogen atom. ‘Toillustaretheapplicationofperturbationtheorytoarealproblem,we Fwill consider the efect ofanexcernalelectricfieldontheenergylevelsof hydrogenlike atom, ThisistheStark Effect. Theunperturbed hamiltonian is n=k2 (6-31)mr whose eigenfunctions wedenote bydyin(). Theperturbing potential is Mh=e8-r =eb: (1632) where &istheelect field,Thequanti <5willplaycheroleoftheparameter 260° Quancum Physis Theenergy shiftoftheground state, which isnondegenerate, isgiven bythe expression B= e8l\slom) =8drlomi)?z 4653) “This integral vanishes, since thesquare ofthe wave function isalways aneven function under parity, andcheperturbing potential isanoddfunction undet reflections. Thus forthe ground sae theres noenergy sift tacislneat inthe clectcfeld 6.Classically system thathasanelectric dipole momeat d,will ‘experience anenergy shift ofmagnitude —d-6. Thus theatom, initsground se, hasaopermanent dipole moment. The argument given above may be generalized: stems innondegenerate states carnot havepermanent dipole moment ‘The statementofnondegeneracy isimportant:itisonlythenthatthestatesare alsoeigensates ofthe patty opetor andthen |g()]* seven, andtheexpecta: tion value of=vanishes, ‘Many molecules dohave permanent dipole moments, anditisoften said thatthisisbecause cheground states aredegenerate. Theexpectation valueofz inastate likeey+A¥_, where thesubscripts indicate theparty, certainly does notvanish, andastateliketheabove willbedegenerate withitsspace-inverted stateay,—Byifchetwostates y4,¥-have thesame energy. Thisexplanation isnoequite correc. Thereason ithatthe lowest lying sates arenever quite degenerite. Consider,forexample,amoleculelikeammonia,NHs.Itsstructure istetabedal, with thethee Hnude forming anequate tangle. TheNcan beatposition (determined bythe condition thatheenergy sminimum) eter“above”of"below"thetriangle.Theevenandoddlinearcombinations ofthesetwostates donothave quite thesame energy, though theenergy diference is‘verytiny(—10-+eV),becauseofthelargebustierbetweenthe“above”andbelow” locations Thus, strictly speaking theground saeisnondegenerate. However, if&¢,where =ftwtion=~¢fWoe2 (63) ismich age than thecnyspliting, then theenergy sift wilbenea itheclectcfelandchemoleculewilehaveasiithadanelectricdipolemoment(chFg.1664). Tetuslook atthesecond-order term. Itreads Lean zits)? ooget . Ba=eerypGale (16-35) “The mattx clement inthisexpeston is ants) =fdRal)Yb)e088Rael)Youd) (0636) 1secoursnplemodelofmolecule, and dodscuniod oap.98 ‘Time Independent Perutbation Theory 261 ‘where wehavereplaced 2(which appears with&pointing inthex-direction) by themore convenient [email protected] purtof theintegration canbecated ot, since Yu=OnVie [im cos = | yy 637Vs” 0637) Itistherefore 1 1 40Yil€s8)ezYu(6.8)=ban 16: fi(048)75Yul6)=Je (1638) bytheorthonotmaliy oftheYin.Thefactthatchem-value must bethesame forthetwostates isoutfrst example of selction rule,which canbesated intheform Am =0 (1639) Iefollows from the fact that U2) =0 (16-40) thatis,thastheperturbation commutes with Ly.One chus hastoevaluate the radial integral toobtain theanswer: R=f©srRes)FRio) (16-41) ‘This can bedone,* and theresule is 1Be—ays 2o|zldre)|* = Iat 64 1en|slo)=5Oye (16-42) which wewriteas/(9)at.Forchesecond order shift,thisgives 2=—aeee foBid=—00hDea=va 2884 nifin)wetUnd *SeeHA.BetheandEB.Sulpet, Qaim Mrbanis ofOneadToo-Etron dum, Antonis Pan, Men You 262 Quanturn Physics Onthefaceofit,thesumcanonlybeevaluatedtermbycerm.*The #=2cetm contributes 0.74cotheseries, andthew=3tetm contributes 0.10. Theconvergence isnorspectacularly rapid, andthesunactually addsupt01.125,Theftstermintheseriesdoes,however,giveusanestimateoftheorder‘ofmagnitude oftheeffect.Thedependence onay?is,ofcourse,automatic, onpurelydimensional grounds.Thefactor&mustbemultipliedbysomethingthatisa(lengeh)*,andtheonlynaturallengchistheBobrradius.Ifwespeakofahydrogenlike atom, rather thanhydrogen, wemust make thesubstitution a alZ, Ifwediffreniae theenergy shiftwithrespect tothe electic field,wegee anexpression forthe dipole momenc = pegsLO)a=—Se=ee (16-44) ‘Thisisproportional totheelectric fieldstrength, thatis,thedipole moment is induced. Thepolarizability, defined by a pat: (6-45) ‘an thus becalculated, Inmakingestimatesofsumsofthesorchatoccurin(16-3),onemaysometimes finduseful upper bounds. Forexample (102bate)(Gata:100}a Be 1TE’_Ee 0)2 nia||a0 46) <[Rei Beometa)atllt) (16.46 However, because ofchecompleteness ofthe states, wemayreplace |eGata > aca) ‘asargued in(14-17), sothat &,Gumlzltnin ata2|dm)=(012416x0} (06-48) ‘This, however, iseasyroevaluate, Since theground statewave function is spherically symmewic, wehave 1 @)==O) =5Ormitltm)=at(16-49) +Acraly thesecondordershiftcanbeevslasced inclosedform.Se,frexample, S.Boxowie, FundanerialfQnoninmMechs,W.A.Beniamin,NewYesk,1968,pp. huasso, Time Independent Perutbation Theory 263 whete thelaststepfollows from (12:31). From thisweindthat Lf) =1 (16-50) and therefore Smfln) 4s 4Zzseo.<5Lf=- 06-51) SeiS3% 3 The telation |Wrsal=leuw)|? =vel24160) (16-52) iscalled«somruleandisanexample ofrelations thatareuseful inmaking estimates Toillustrate degenerate perurbation theory, wecalculate thefstorder(linearin6)Starkeffectforthe#=2statesofthehydrogenatom.Foethebapercurbed systemtherearereallyfourm=2statesthathavethesameenctgyThese are =aay (2) ory foe=on¥a(— 1) tue=(Qa¥5¢)onyy * rar=Bag508()me (16-53) The/=0satehasevenparity,andthe=1stateshaveoddparty.Wewanttosolveanequationlke(16-23)and,onthefaceofit,fourequationsareinvolvedIfwenote,however, that(a)theperturbing potential (tharis,2)commutes with 1,sothatitonlyconnects sates withthesamem-value, and(b)parity forces ustoconsider only terms iawhich thepereusbing potential must connect F= 100 f=Ocerms, that is, @ascatl#ldayan) =0 (16-54) ‘henchematixin(16.23)isonly2.2X2mattix,Theequationreads (me|z]¢0) aulzhao)\/ox a ee=o (16-55) .aua|zl¢me) aie)z|¢20) oe ae 264 Quancum Physics Thedingoal element are210, ews ofpte, andtheofdiagonal elements areequal, since theyatecomplex conjugates ofexch other, andeach maybe ‘hosen coberel, We have slow) = eiraaysereSE(1LY, (omlelene)= [eiraarrenm GE(1 «|¢2¥alaes YoYom 1656) = andhence (16-55) becomes. 80—3ta\=o (16-57)—3ta,—B/\ay Theeigenvaves ofthis ae BY=t3ebay (16-58) : andthecorresponding cigenstates, when properly normalized are wan watval-t) =!vali) respectively. Thus thelinear Starkeffect forthem=2states yields asplitting of Gegenente levels ashown faBig161. ‘There aresome general comments hatcanbeabstracted from thecalela- tions jst concoded (2)Thestatesinthepresence oftheelectricfieldarenolongereigenscates ofL,since intheabove ese, forexample, wefound thtchesates tatdr ‘agonalize theperturbation were equal mixtures of/=and/=1,thoughthey ‘arestilleigenstates ofLy.Thereason, isthattheperturbation changes theHamiltonian, sothtitolongercommuteswithThiscanbeworkedoota deal, buttisrealy evden haheexeraa eldspecies peered dtecon, 40tharthephysical system isnolonger invariant under aroiayroatons. Tes stillinvariant under rotations about thepreferred axis,herethez-direction, and hence £,issil»good constant ofthe nation. (@)Quite generally, whenever there iaperturbation tatdoes aotcon: serve some quantity (forename, L?here), then thesates that“diagonalie™ thenew Hamltonea intoyapprotimaice, wesuperpositions ofsecs with difrent values ofthe previously conserved quantum number, and this de genet evels wl bespi. Time independent Perurbtion Theory 268 hme 10-1 oa mat a=Fes SN itesott Fig.161. Pate ofSak pliting ofhydrogen arom inw=2sae. Tefuse folddegeneracy ispariy lied bytheperurbacon, Them= 1mater remsn degcorte andaeoeSifted inthe Stack eet (©)Wemaysummarize theprocedure iadegencnte perturbation theory inmatrix language asfollows. IfHeisdiagonal, butH:isnot,then, since Hyand do noecommute, ixisnot possible todagonalze Hybyisl, withovr “undiagonalizing” Fa,One must work with Hem+mh, 182whole.Ifweworkwithasubsetofdegeneratestatesallofwhichateeigen sctesofHiwithhesameegenvla,then,afaastsesatesaeconcernedHa isnocmercy dingona, butiisproportional cotheunitmatin Since H,(and everything else)commutes withtheunitmattix, onemaydiagonalize H;by itself, without affecting Hy ‘Thehydrogenlike atoms considered here were somewhat ideaied. Aswe willseein Chaper 17,cheearsmall raiviic andspin-orbit coupling effects Ghatacrualy remove some ofthedegeneracies. Does thismean thtwenever really needtousedegenerate perturbation theory? Actually evenif,say,ganand én»doaothaveexactly thesame energy, itmaystillbesensible totakesome Kiser combination ofthem inthepeturbaion expansion, Ifwehave, for example Haome =(Es—4)dam Hyor =(Ex +3)dno (16-59) with4small,hentheSchrdingerequation,withhelinearcombinations, reads (th+a1)(asbm+extn+9EEGate) ~(cst+arto+AGos) (6.60) 266 Quancurn Physics Taking thesalar product withdnganddeo,respectively, leadstothefolowing equation toorder (""2~lawMision) Ganinn)\(°)(")=2 Canim) Bt+8—Geelgow)JNe Ss, (16-61) Ifwewrite (Gal|Gro)=(OrnABiIban)=ad (16.62) ‘wemust findtheeigenvalues ofthematt BY-a Mw (16.63) Me Bt td these are B=Bs Van pat (16-64) (lntheabovewehaveset(fuelHila)=(nelue)=0.)Weseethatwhen 42> ah,wegeta“quadatic” effect only. Thiscorresponds tonode- sgeneracy, When A&ahwegettheresult oftheform (16-58). Intheinter mediate region, theabove, more careful weatment isnecessary, Furthermore, ‘hen thenewlinear combinations ateused, theninsecond order perturbation theoty thete nolonger appear verytinyenergy diferences inthedenominator. ‘Wedonocdiscus cisindeal, buetisisnotdifcul toestablish, ‘Asafinalcomment wepointouttwoapparently contmadiciory facts, (2)Thepredictions ofperturbation theory concerning theSark effec areborne coutverywellbyexperiment, and(2)dheperturbation series evidently diverges, sincetheperurbing potential «6grows without bound aszbecomes verylazge,nomatterhowsmall«is.Thequestionarseswhetheronehasaayrighttobelieve intheaccuracy ofthefrstfewterms ofamathematically divergent seties, since itiswellknown chatamathematically divergent series canbe rearranged togiveentcly different expansions. Theanswer liesinthephysics ‘tnd notinthe mathematicsoftheproblem.Thereasonforthedivergencecanbe seeninFig.162,whichgivesaroughpictuteofthetotalpotential forx,)fixed Icappears that there is&baie erated forthebound electron, This bacir is‘ultimatelypenecable,eventhoughforsmaliisverybroad.Whatthemathe. iaticaldivergenceoftheseriesisrespondingtoisthepossibilitytactheelc- tronintheground state, forexample, has» Gnice (although very, very small) probabiley ofbeing sufficiently faraway fom thenucleus, where theextemal lect flissronger thantheCoulomb field, ndtheelection iscatried away bytheelect field. Thus thenew“shifted” energy levels ofthe hydrogen arom asenolonger stationary states, uteather metastable states, Ifthe fieldisweak, Time Independent Perbtion Theory 267 yer Fig. 16-2. Schematic pcre ofpotential energy a«fnction of#withxandy tlfred. Thedrt linerepesents sheCouom potent, thedashed inethe potential nergy duothe extra ela hesoliTnehewuprey homer,theytybesableonstimescaleoftheageoftheuniverse,andhencetheobsracions age perfectly withwhacthefsfewtems ofthe petrbionsctiespredic, 7 Problems 1.Considerthehydrogen som,anassumethattheprocon,insteadof beingapoinesource ofcheCoulomb fel,isuniformly chargedsphereof tadissRs thattheCoulomb potential isnow mosled to 5 (1 =- (wba) weed =-+ r>R Ceslte theenergy shift forthe w=1,=0sae, andfotthew= 2sates, caused bythismodifcasion, using thewave functions given in(12-25) *Acrlysnpebriepencuioseestonoftheypecae!tinChapter5 stowstatheneeh meteike1ens ofthe ule frkywoes +fat 268 Quanrum Physics 2.Calcalie teenergy shiftinthe ground state ofthe one-dimensional tatmonic oscil, whea thepetubtion Vane isadded v0 =Lo matt A—+h 3.Consider asquare wellinonedimension. Iftheedges ofthewellareroundedoffsshowninthefigure,whatisthechangeinthegroundstaceenergy? Chooseyourrounding.off parametrization suchthatfV(x)deremainsun- changed, 4,Thebottomofaninfinitewellischangedtohavetheshape Vix)=esin osesd Galelate theenergy shifts forllheeacited sates tofstode ine Note chat theweloriginallyhadV(x)=Ofor0<x<b,withV==elsewhere 5.Prove thesum rule(Thomas-Reiche-Kuhn sumrule) e ©E,~Bi(eleta)|?=5 (Hine. (a)Weite thecommutation relation [p,«)=fi/intheform h é z={ova)(olsla)—Gelstavialpient aye (b)Usethefactthatae i Chon)=(elmsn)=m%alltslln) inworking outtheproblem] 6.Checktheabovesumrulefortheone-dimensional humonicoscillator,> vith “a aken inthepround sate, 7.Work outthefirstorder Stark effect inthex=3state ofthehydrogen arom, Donotbother #9work outs hesntegas 6.Consideranelectoninasaewinahydrogenacm.Theatomisplaced 4 Time Independent Perurbation Theory 269 thetransmission coefficientthroughthebartermadeupoftheCofulombartac-tion rothenucleus. Itisenough toconsider aone-dimensional model ofthe problems 9.Consider «rwo-dimensional harmonic oscillator described bythe Hamiltonian 1Hm RE+Be+Amati +9) GeneralizetheapproachofChapter7toobtainsolutionsofthisprobleminterms ‘ofraising operators acing ontheground state, Calculate theenergy shifts dueco theperturbation V= Dey intheground stare, andinthedegenerate firsexcited sates, using frstorderperturbation theory.Canyouinterpretyourresultverysimply?Solvetheproblemcexacily, andcompare itwith asecond order perturbation calculation ©Gints. (a)Examine thesymmetries oftheunperturbed Hamiltonian, (b)De- E>compose themotion intocenter ofmassmotion andintemal motion] References There aremany examples oftheapplication offirst-order, perturbation theory inthetextbook literarre, andthereferences listed attheendofthisbook may serve asasource offarther examples. Fora discussion ofcheexact calculation ofche Sak effect see $. Borowitz, Fandameials ofQuantum Mechanic, W.A.Benjamia, Inc, 1967. a |chapter17 TheReal Hydrogen Atom ‘hediscussionofhydrogeaikeatomsiaCaper12wasbasedthe 2°Hamitronian ied ry2ee ,-B_# zhot He2, (7a) FP) tnamoreric ween, sevencometions mustbekenitoaccount |Fistofal,chegzpression forthekinetic energy oftheelectron isaltered when | ivisticconceoons tetaken iotcount theoii lection eto Lica pe,pe(4x)PF eo am1OM®am)2A,ae (inthecenter ofmass frame) by segmayPEnet4PLEOEpe (pee+me} tom = +2m 8met 2M ane ph LO? :ane oe (172) thereiced mis, bot thew now's cone term, 1 na) oe rs) hushouldbe added totheHamiltonian Hy,Wemayema themagicaof Gh)PY (wit?. (Ho) mit mit (ay ara) 272 Quantum Physics Forhydrogen thisisoftheorderof10-*,smallerthanthereducedmasseffects.‘Theexistenceofthelectionspingivesrisetoanothercowtectionthatisof| thesammeorderofmagoizude Iemaybequalitatively understood asfollows: iftheelectron wereatrestrelative totheproton (wearediscussing thisona«lasscllve),iwouldonlyseeanclectricfieldduetotheprotoncharge.ThisistheCoulombpotentialcrmthatappearsinHa,Becausetheelectronsmoving, thereareadditional effects.Intheelectronretframe,theprotonismoving, so thattheteisacurent prescot, andtheelectron “sees” amagoetic eld.Iehe felative motion wererectilineat, themagnetic field,asseenbytheelecton, wouldbevXEye.Thismagnetic flinteracts withtheapioftheelectzon,ormoreprecisely, withthemagnetic moment oftheelectron. Wemight expect anioceraction ofthe form —MB= SB =gaSYXE=—Spxvol) ‘ 14Topas PRE L 1a=maSkXP7bt) (17-5) ‘where 6(r)isthepotential duetothenuclear charge. Actually thisisnoccorrect.Jeturnsoutthatrelacvisticeffetsassociatedwiththefactthattheelectrondoes‘otmoveinstright line(theThomas precession eflec) reduce theabove byfactorof2.Thusthecorrectpercurbation is 1 1desir] y=Lg.p i 10 7 apaShy (7-6) Letusnowusefirstorder perturbation theory tocalculate theeffectsofHi andHsonthespectrum ofhydrogenlike arom. Wemayrewrite fintheform en 2 (py *8omic 2m\2m ==sa(a+2\(u+%) wm ifweneglectreducedmasseffectsinHy. Hence L we ze (ueldan)=—555(eum(i9Cag2)oun) ‘TheRalHydogenAtom273 1 : o{h ay(1 >galetee(2)+er),]’ -f(y _aznttat(Z.) netL2 atAa 2 !+OYedeval Largo [|__(Zat__ead ; $wea[es-a] ar In calculating theabove, wehaveusedexpressions for 3 1 fa 1 fa(Ean(eee) mtGB)=Geealoe) from(1231, Thespinofthe lection doesnot eteintothienergy hisnce PIB i,doesnoedepend onthespin. Hsdoesdepend onthe spin,andforourun- perturbed wave functions wemusttakeewo-component wavefunctions, since Tne wewent toclcuae isthe expectation valve of 4 gy te 2 gy j ieETde~amehe (rs) Fe weapn,wetaveansampleofdentepentonary,Foen ‘and |,there are2(2/+1)degenerate eigenstates ofHo,with theadditional => factor of2comingfromthetwospinstates,Thusthecalculation oftheenergy "9Dshift involves adiagonalization ofasubmatrix, asinEq.16-23. Wecansave ‘ourselves agreat dealoflabor bynoting that S+L=J (17-10) implies thac $+ 8L+l=f SL=sgr-u—s» ory “Thaifwecombinethedegeneaeeigenfunctions intolinetcombinationsbat areeigenfunctions ofJ*(theyalreadyareeigenfunctions ofJ,=Le+5),thenthese linea combiotions wilagonaline Hy.Theappeoptate linear combina tions were ebained inChapter 13,Eq,13:37 and 15-38, With these linet conbinstions we have I(p_ge|Ss. (PoE &)0% Sel(4 \r42)— ey2=e[(4Yee})meTe, Lay“se, om and SL =i[(-eo)wen] a) a ==EOD orca amAODra (73) foregivenJvaluethereate(2+1/2)+1]+[200~4/2)+1]states.Whar ‘ashappened istharthedegenerate sateshavemetelybocarearranged, butthe ‘twogroupsthattheyhavebeensplitintobehavedifferentlyundertheaction ofIEweealthelinetcombinations bonthen zee ys xfareal (7-14) forj=11/2,respectively, Withthehelpof 1) oz 1(),-3 PFVTE wv) Wegettheenergy shift :‘zoe Leta} ns EeOOTMD Wemust,ofcoursecombine theeffectsofHyandHs.Whenthisitdone,weobrinitersomealgebre os z4E=~1/2me(Za)'Ls12Fal (17-17), forbothvaluesof=j3:1/2.IeisnecessrytoworkwiththerelativisticDineequationt0showthttherelialoconectwhen=@ree, theproduct in(17-14) snot weldefined. ‘Thesplicing isdepicted graphicallyinPg.17-1Averyintrestingslitat {hecomeetons addupin&manner thatlenesthePesnalheaet “The Real Hydrogen Atom 275 x a NS, Ste N SpinomiteetSNS Phe : mom Fig.17-1.. Spliting ofthew=2levels by(2)thespin-orbit coupfing (which leavesthe§stateunaffected)an(2)theeelaivsticeffect.Thefinaldegeneracyofthepand*Pystatesisactuallyliftedbyquantumelectrodynamic effects,Theupward* ‘shi ofthe, sate iscalled theLam shift. degenerate. Amorecarefuldiscussion, usingtherelativisticDiracequation,does‘oCalterthisresult, In1947, averydelicate microwave absorption experiment Carried outbyLamb andRetherford showed thatthere was,indeed, atiny splicing ofthetwolevels. Themagnitude ofthespliting, oforder ‘me(Za)alogacouldbeexplainedbytheadditionalinteraction oftheelectron+>withitsownelectromagnetic Bld, tatis,asaself-energy effect. These maters «ateoutside ofthescope ofthisbook. Letusnowcumtothediscussionofchebebaviorofhydrogenlike atoms jnanextemal magnetic field,thatis,cotheanomalous Zeeman ofc.There is,of ‘course, nothing anomalous about theeffect; itisjustthartheZeeman effect ‘thatcould beexplained classically wasexhibited onlybyatoms instates in which thetotalelectronic spinwaszero. Forcheother states, forwhich ehere vwasnoclassical explanation (since thatinvolves spin), theZeeman splicing ‘pattern wasdifferent, andtherefore “anomalous.” Fortheunperturbed Hamiltonian wetaketheusualHorogether withthe spinorbitcerm. Thereason fordoing thisisthattheexternal percurbation may besmall compared withtheeffect ofwhat wecalled Hs.Thus ~-f_4%,12a 718) “Theperturbation nowreads = +28) (17.19) +Thefrstcermis,ineffect,theinteractionofthemagneticdipolemomentarising 276 Quantum Physics fiomthectculesing charge, andthesecond termisthecontibution oftheintsnsicdipolemomentofsnobjectwithspin --# 7Mime (17-20) with g=2, ‘Thechoice ofHedicate thatwecalculate theexpectation valueofthePerturbation ineigenstates ofJ*andJ,(15-37)and(15-38).Ifwechoosethe axisasgivenbythedirectionofB,thenweneedtocalculate 2 a ag|(Ls+28)yup)=na]5Ue+8)400) B=Fac5+(ByesSI4pug})(17-22) ‘Tocalculate thelastmatsxelement, weeatyoutthecalculation explicidy,usingtheeigenfunctions givenin(15-37) and(15-38). Thusforj=+1/2 ‘wehave imei [i=m inti (y!Ut+yYuate+ViaYawn\s|JatiYanxe [i= Afitm+1 lam *aryMann.)=4Wtzai) RRtm+ 1 hms “2Wad “a+ (17-22) ‘andforj=—1/2,wehave Te mei i|== Siptare—RT yyIs)femyy (aig1varenYee[S|yopYam ~fitatiy )-h(ton ieatt) apy “MPH aeUt. a+. ee “2 w4t Ut. (17-23) Inbothcasesweusedthefactthatmy=m++1/2intheabove,Inserting theaboveinto(17-21)yields AB 1 aeBhni(ve 35)jetey (17-28) ‘The Real Hydrogen Atom 277 1) tya |St/pr 12)(eaewee |4 i Js — \ Te . tte pre ws i |aeetan :| Fig.17-2. General cepescoation ofanomalous Zeeman eect. “Thesplicing isdepicted inFig.17-2.Theselection rule!forthe transitions issil :Amy=410 (07-25) butsincethesplitting betweea thelinesinotthesameforeverymultiple, we Fdonotgetjustthethre linesthatweobtainedforthenormalZeemaneffecia. ‘Chapter 13.Forexample, forn=2,the?Pystatesplitsintofourlines,withthe ©splitingewotimesaslargeasthatoftheewosatesinthePyeines(Fig.17-3), | REPTeeeextemal fieldisverystrong, 50thatthespin-orbit coupling canbene- RE secre, wemayusetheordinary hydrogenic wavefunctions simply multiplied+FE”byspinors,this,eigenstatesofL,£,,$%,andS.Ifwealltheeigenvalues ofHpLand S,mand ma,respectively, thentheexpectation valeofHin(17-19), SE widhBpointing inthez-direction, is * iBt (thy=Se(om2m) (07.26) “Thus the»=2,/=1statesatesplicintofivelevels,comresponding tothevalues of m=1,0, —1;m= 1/2, 1/2. Tnaddition tothefinesractare ofthelevels caused bythespin-orbit coupling, there isaverytinyhyperfine splising, which isrally «permanent Zeeman effect duetothemagnetic feldgenerated bythemagnetic dipole‘momentofthenucleus.Ifthespinofthe nucleus isI,then themagnetic dipole moment operator is =2xMeoe (727) whereZeisthechargeofthenucleus,Myitsmass,andgyitsgyromagnetic moot “The detation ofthis selection rule(endhen) wllbedtcussed inChapter 2. 278QuoaaPips now roe aie)a NZwap fen HITT]HittdyHHHALT|||| Fg173Zanefitindog, «wept teeapy/an. "Te sontwhch=1n=ineasgeee/a,Th OFtheunperabed sates icabyBes mo.Teerrpildeopitilsomcomgei ey A=—2onxvt (17.28) sothsthego lds M 11 1 BewxanTro+ivy)+(17-29) ‘hasthepertain i Me Mee -xsB ~imag [Lee 2]ars ‘TheReal Hydrogen Atom 279 . “Theexpectation value oftheterm onthetight canbecalculated verystraight. |forwardly. First, wenote thatthemagnitude ofthesplitin is 4 Few3(Zam)me(i)= 7 . SemMye ®a]oe(ayMw)e (31) thatis,cis factor ofm/My smaller than thetypical spin-obiesplitings. The Calculation oftheexpectation value of(17-30) inthestate characterized by : 1=0,forexample, theground state, issimplified. Wehave 3( 4 Py(jt221:fewa(S") ‘)#0)=SiefeHOS on £Becauseofthesphericalsymmettyofallthecermsintheincegrendexceptfor fathederivatives, theangular integration willvanish unless /=4,Allthei=&3 contributions willbeequal forthesame reason, sothattheabove yields :1 A . sufaterm :‘Thus, when inserted betweenJ=0states(andonlythen),wemaywrite ble yet '4 S-¥M-w) |=5Sa (732) Thus whatisneeded is 3 2-Zeign 1 ’ y= —an a _n=~2%gy(wt) (733) T° Weusethefarthat? X 1 riba (7-34) toobtain 1 mafSU) (BY [ge 1)=eenstne(*2)(2)[ania sent a () ye Oeodmes ()Rolo) (07-35) 1Only the eda par of"is relevant, Toshow hi, weprove that (1/#9d/de) 4 (RE/de)} /r) =0forF¥0,and that 9*(1/r) integraced overasmallsphereofradius«, inet «rele —Ae independent of 280 Quan Pics When chevaleofthe dalanction atbeeign iinset inttheabove,en 4 (ame)! nwo=4(Be) 0736 leads othe result 4m, tS (i)=3enjg,Camet+) (07-37) IfwetaketheroralspinoftheelectronandnucleustobeF, F=S+i (17-38) then STPS=P(Ae+D~5/4- 1040) ea 2 tft rerdyiti, fora (739) Forhydrogen, gw=gr&5.56,andtheenergy difference between theexcited sat, chiacesaed byF=landtheroundseofFsO's Soggy tk pe b=3636)1840(137)we) ‘Thewavelength ofthe nation conespoading totheanton between thePeliedP=Oates8Awaiden (07-40) sadtheFequeney! v=faummepeydes (ray ‘Thedation aising framthisansiion paysanmpostnt roleinastronomy.Inagusofnewtaoms,theF=1stecanoebeencedbyontopdene,becuse of«selection rlhatstrongly suppresses wnsitonsn wich thes noching inosbialtogulr momeatum Both he PYandtheFOsexee haveseoangolar momentum. Onteeet hand, theeateether machusoee theecanrte ctstons. TheF=1stecan,foreample becoeeed yp $e freal, Bate adSpe ‘Sts egang hones heoence ete nts inphys = Mzotspia00" Boe cys (hg) Themar are a sea donne ror botcnn ea) yo Saoa eae “The Real Hydrogen Atom 281 ‘collisions, andthereturn totheF=0ground statecanbedetected. Froman t ‘analysis oftheintensity ofthe21cmradiation received, astronomers haveFJearnedgreatdealaboutthedensitydisbucion ofneutrlhydrogeniincestella:space,aswelathemocion andchetemperature ofthegascloudscon- Btainingthehydrogen,Theaveragenumberofneutralhydrogenatomsappears |tobeabout1c?inchegalacticplaneneathesun,andthetemperature isof ! the order of100° K, }.Problems ;ES1.waeeffectdoestheadditionofaconstanttotheHamiltonian haveonEthewavefonction? q2. Ifchegeneral formofaspin-orbit couplingforaparticleofmass FiespinSmoving inpotential V(r)is a gp lhVO . zHao=aS ar Frebat ischeeffectofchatcoupling onthespectrum of2three-dimensional MMRharmonic oscillator? »HAMS 5,Consider them=2statesintherealhydrogen atom. Whatisthe EEE speccrum intheabsence of«magnctcfeld?HowisthatspectrumchangedwhenBPEtheatomisplacedinamagnetic Sedof25,000gauss? ee 4,Show that - ho—tetie) BY seteprocedure owedntheooo Fa.17.4 F 5,Consider4gasofhydrogenatomsatlowtemperature anddensity.At‘ whattemperature willtheF=1andtheF=0statesbeequally occupied? "ote, TheBolezmann factor | wm givestherelative probebilcy ofoccupation of«given starewithdegeneracy g ‘whenthesystem isinequilibrium, attemperature T.) +6Consider aharmonic oxcllaror inthreedimensions. Ifcherelativistic expression forthe kinetic energyisused,whatistheshiftintheground state ‘energy? yoo 7.Thedeuteron consists ofproton (charge ++)andaneutron (charge 0) 282 Quantum Physics jn2stateoftotalspin1andcotalangularmomentum J=1,Theg-factorsfortheproton andneutron are a=202.7096) bv=2(-1.9103) (2)Whatarechepossibleorbitalangularmomentum statesforthissysem? fitisknown thachestateisprimarily *S,,whatadmixture isallowed giventhat patty isconserved? (b)Writeanexpressionfortheinteractionofthedeutefonwithanex. ternalmagnetic fieldandcalculate theZeeman splitting. Show thatiftheinteraction withthemagaeticfeldiswattenintheform Vm poeB thentheeffecrive magnetic moment ofthedeuteron isthesumoftheproton andneutron magoetic moments, andaaydeviation from thatresule isdueto21 ‘admixture ofnon-S state rothewave function. 8,Consider positronium, ahydrogenlike atomconsisting ofanelecttoa andapositron (same mass, opposite charge). Calculate (a)theground stateenergy,andthatforthex=2stares;(b)therelativistickineticenergyeffectand thespin-orbit coupling; (c)thehyperfine spitting oftheground state.Compue yourtesults withthoseforthe hydrogen atomandexplain major differences, References ‘Themostdetailed discussion ofthephysics ofhydrogenlike atoms maybefound in HLA,Bethe andE.E,Salpeter, Quantum Mechanic: ofOne-andTwo-Blectron “Atans, Springer Verlag, 1957, ‘TheThomas precession isdiscussed in R.M.Bisherg, Fandaments ofModern Physics, Wiley, NewYork(1961) |chapter18 &. is ! IE The Helium Atom & .3 “TheheliumatomconsistsofanucleusofchargeZ=2andtwoelectrons,I sricn weuber aad2Eachceewon isaaced fotemcs, andtheemo Seton eelachother Weassume, andhiswilumotc Beconet, eatBeer Shectantheeleewomagneti ones(CoulombveryB04appO3-b mation), atenecessary todescribe thedynamics ofthehelium atom withthe Be ap otguannun mechanics, itn mules iplaced atthe origin, andifheelcton coors are Re ated esses thentheHarton fotheator (Fg,182)-4 Latpdpa My8q HePtP +in-7l (ae-1)Feemisheelecon aus.WeslgotheslectsconnectedwithheEmotion ofthenucleus," relativistic eect, spin-orbin effects, andtheeffectof .a tthecurrent caused bythemotion ofoneelectron, upontheotherelectron. The VRE Soretetndtoian maybewegen a a HeHOPHOT (48-2) = woutp 2a PO (83) a and a yet «3a inaalEEWeshallworkwiththenuclearchargeZandsetZ=2later.Ourworkonthehydrogen atomprovides uswithacomplete setofeigenfunctions forHi!”and Ie|commhepamiennoeclairancponarohaBi. SMU RMEEIIGTGINES Tits, Mccain Cooe, 284 Quantum Physics Fig.18-1,Coordinaies usedintheformulation oftheheliumHamilonian H©,Thas,ifweweretoignoteVinthetotalHamiltonian, wewouldhaveasolution 10theeigenvalue problem forthetwo-electron system. Thecigen- fanctions would be wlrie2)=bet(F)Ondonlts) (185) fortheequation [HO +HO] wey) =Bleu) (18-6) andtheenergy would begiven by(Fig. 18.28) B= Ey+Ey (8-7) where Fy=—(me*/2)(Za)"/nt. Thus intheidealized model inwhich thewo electrons ignore eachother, thelowest energy is E=~28) =~me'(20)* =—1088 eV (is) Note thacthisis2X2?=8times thehydrogen energy of~13.6 eV. “Thefrstexcited stateisoneinwhich oneelectron isintsground state, sn=1,andthesecond electon israised tohefistexcited »=2state, Then B=B+=—6600V (89) ‘Theionization energy, thatis,theenergy required toremove oneelectron from theground sate toinfinity is Eos =(Ey+Ba)—2B=944eV (18.10) and, interestingly enough, theonset ofthe continuum lieslower than theexcived state forwhich both electrons ateithe»=2seate. Theenergy ofthelatter sate is B= 2,=2720 aa.) andicbrings upanew phenomenon: theexistence ofadiscrete state inthe ‘contiquum fortheHamiltonian 1°)+H®,Weshallbriely discuss theimpli ‘ations ofthis atthe endofthe ebaprer. . Yi Cy la ParanaOrthabetiim _po es 0% - 3 “lavu diloue: UdYjGY “Vu*2u ||em —"° “ofhe “2* 2 -36- ———1.2 a | s Zz . .][4 afontna Ea Fig.18-2,(4)Thespectrumofheliumasicwouldlookintheabsenceoftheee lecronelectioninteraction. ‘The2er0energypointischosenattheionizationourgy. (@)Thescl specu ofhtm fethesinge (paral) adpe {Grothatom) sate, Thelev abating asasuppcessed (i), 20chhelevel (2p)isapproximately described bythe(11)(2p) orbital. 205 286 Quantum Physics Sincethetwoelectrons areidenticalfermionswemustmakethetotalwavefanction anisymmictic undertheinterchange ofspaceandspincootdinates of theelectrons. Thus aproper description oftheground sateofthis idealized model is so(rat)=dude)él)Xena (12) ‘Thespatialpatofthewavefunction isnecessarilysymmetric,andchatswhythe statemust beaspinsinglet state Lg =yt Kanate=752X2 —PB) (18:13) Forthefrstexcited stare,wehavetwopossibilities, which, forV=0,aredegenerate inenergy, These are 1 ) a=Fluke) dunled) +damle) Oo) Kaeser (08-14) sndthespaceantisymmenic, spinsymmetcic 0 1 )a0=7[oinkdnialts)~data)dlte)]Xander(1841) where ae 1 . Keine=175OPR® +xOxP) (18.16) 2% isorthogonal £0Kaa Thepresence of¥,theclectron-clectton Coulomb interaction may,in firsteppronimation, bewearedaaperturbation. Letusfrstcompute theenergy shiftofchegroundstatecofirstorderinV.Wehave aefendnsties) ieear(ron) (1837) Sincethepercurbation doesnotinvolve thespin,weneedonlyconsider e a=fdnanigiaied Troe emlodl?—— Gs8) ‘Theintegral hasasimple physical interpretation. Since|ya(e);?is theproba-bilitydensicy offinding electron 1atry,wemayinterpret el@nodr,)|* asthe charge density duetoelectron 1.Hence ue)=fentiensealt -Gsas) i‘The Helium Atom 287 (Be istepore atductothe hrge ditibution ofeleton 1,and | ae=[Preloaded *Ud) (1820) i 5istherefore theelectrostatic energy ofinteraction ofelectron 2withthatpo-(HI encal Theinceg canbecated out.Withgm=(2/-VAn)(Zia)"™ 2 weave -3 AB={ealas2fredvm[ota fea fener res) ~PRE10wccingthis,wewedtheseparation fore[onan andiolatedtheonlycrmthatdependsontheanglesbeoneenrandmy.Wehive 1 1 ;3 Fn Gn anncond* (822) Fwhereistheangleberweenrandx,Wemayproceedinoneoftwo ways. j (@)Most direly, wechoose thedirection ofx;aszaxis forthe dO > imegation, andget . 2 ops 1: 4, = eos¢) Zfoe [4[00Gamaw a =inte nan) 2 (18-23) ey ‘Theintegration overdfistrivial,sincenothingdependsonthatangle,sothat iS [ante (824PP andweateleftwith f Xint: « nan aaa 08.25) | 288 Quantum Physis (©)Averyuseful expansion, necessary when there isaddtional angular dependence inhenumerton itthefollowing, Forrh¥ny, Cetrtarncntyt rot(12ont) =LE(*) rteoso asa | Withtherolesofrandrreversedwhenr>7.Thus . forfenp =feefenSHracam(829 . wherery(re)isthelarger(smaller)ofryandry.Wecannowproceedasbe- |fore, using thefactthat 3f*gos)Pi(cos8)=bx (18-28) 81 special case of ft ju2[tose088)Paleo)=os (829) Tnanycase, (18-25) becomes a6=4e0a/agt rad {2[0ntdnerme tanfon degra (1830) “Theinceaals aressightforward, andyielddheanswer eaeae=2222(Lm) (4831) Thisis«positive concibution, since tatses fromarepulsive force, andits magnitude, forZ=2is34eV.When thisiadded tothenroonder resultof—108.8eVweobtain,coistorder EX -148ev (18.32) When thisscompared with | Bay==78975eV 0833) 4sable discrepancy issce0,Physically, wemayacibuce thisdiscrepancy to thefetthatinourcalculation werooknoaccount of"screen thtisthe ‘‘The Heim Atom 289 effectthattheprésence ofoneelectron tendstodecrease thenetcharge“seen”Saneerhercraton. Verycough fonearguestht,forexample lection T aoe eeesperwewn” econ3andshenus,thenhalfthetimeelectronaan spurgeZandbaltheteitscesa chargeZ~1thas,eflecively ia “theexpression ¥at(220-2 ‘ntatm—Seca!(22*2) 834) L@—1/2)shoulbesubstiued forThisdoesimproveagreement buttheGradeargumentadvancedisnotsufcientjustification forthechoiceof5076 for heprobability ofeffective screening. Wewilltunrothissubjectlerinthisaercene wediscussheRayleigh‘ vations principe fortheground seat energy.Wenext consider thefirstexcited stateofhelium. I¢willbesufficient 0 ‘aleane theenergyshiftwiththe singletandciplee =0sacslistedin(18-14) and(18-15), sincetheshiftiscausedby«perturbation chatcommutesithLeForsuchaperturbacion, theshiftmustbeindependent ofthem-value.Tein,becauseofthespin-independence oftheperturbing potential, V,wehave 1i ant?=befata[ertomiedole)ole)led)” 1Hey londae) el)fe 71 =2fan[enigeted|"iene!inom + gt 1+ofen|ensinedelo)Fpgymds) S 2(8.35) FI obatining thissimplified form,wemadeuseofthesymmetry ofVunder aeT camrysitiseentoconstofsoems:theSnaheaigWEAfounofanclectosatc interaction betweentwo“elecuoncows”dimou7ccordingcothewavefunctionsofthetwoelectzons.Thistermisjustasimplerund ofthetmdtwefoundferberoundsatenergyshiTeSera usaoclassical iterpeaion, 1sotgilisinthePaliprinciple,wertibedepends whether thesatehussia0ort.Thos,Bees ofthisRAE PEinebuton, thesingletandtipletermsarenolongerdegenerateFateegh weconsidered a=2bere,wehavequitegene ED =Ju—Kot . AE}=Jar+Kot (18:36) 290 Quantum Physics ‘Theintegrals canbeevaluated inclosed form (icisherethat(18-27) becomes useful], butweshallnotdothishere.TheintegralJj,ismanifestly positive,and itturns outthat thisisalso thecase forKy. For!=n—1thisisobvious: the ‘wavefunctions appearing in(18-35) havenonodes inthatcase.Thatthetciplet stateshould havealower energy chanthesinglet state, thatis,chat Jat—Ket<Jat+Kut thats, Ku>o (18-37) canbearguedonqualitativegrounds,ForthetiplestatethespatialwaveFunc- tionisanisymmetic, sochattheelectrons atesomewhat constained 10Sty away fromcachother. Thistends toreduce thescrening effect, sothateach clectron “sees” moreofthe nuclear charge, andicalsotendstomakethepal. sionbetween theelectrons lesseffective thanforthespatially symmettc singlet state. Aninteresting aspect oftisresult ischat,although theperturbing po. tential ¢/|x1 —ms]doesnotdepend onthespinsoftheelectrons, thesymmetryofthewavefanctiondocsmakethepotentialactasifitwerespin-dependent‘Wemaywrite(18-36) inaformthatexhibits this.Letthespinsofthetwoelectronsbes;andss.ThentheotalspinS=6;+#2,and Stemsit+se+281-82 (18-38) {weactwiththisontripleeandsingletstates(18-16)and(18-13)chacaealsocigensaesofi!andwr,weget Fg Fae SS+1)h*=4F+4F+28:+e shat, 1 {pe 2s;82/F®=(S+1)—37—2singe: (18:39) ‘Wemaythuswaite, interms ofthe #'stelated tothe spins bys.=(1/2) his, BayJap—5(UFBa)Kas (18-40) Weshallscethisphenomenon again when wediscuss theHymolecule. Usually spin-dependent forces between atoms arequite weak, Asillusuated inthe cvample ofspin-orbit coupling, thespn-dependent fores tend toarse fom selativstic corrections totheseaticforces. Inthespin-orbic example, these forces q 'Lo q ay .fone Fig, 18-5. Schematic sketch ofsping ofthestexcited states ofhtm aedown byafactorofa,whichisjus(9/6)?Suchforcescouldnotbestrong ‘enough toKeep theelectron spins signed iaferromagnet, except atun- "realistically lowtemperacures.? Thespindependence duetoexchange ismuch {stronger thanthat: theforce isofthesame order ofmagnitude astheelectro- F static force, and,asfstobserved byHeisenberg, itisresponsible forthephe- E —aomenon offerromagnetism. ' ‘Thespectrum ofthe istfewexcited states ofhelium isshown inFig,18:3"The notationusedforeheunperturbed statsisthatofobi,thati,thequan‘umnumbers oftheunpercurbed electrons. Thus bothelectrons intheground © sate arein=1,70sates, andwewete thisa5(151), ofmote briefly (1% Te-should beunderstood thatwhen wewrite (1(2p), aForchefistexcited state, this does not mean that one electron isinone state, and theother eleczon inthe ‘other, sincewemust write totally antisymmetic wave functions forthe elecrrons Another wayoflabeling thestateisbythe"Ly notation, which weusefor theperturbed states inthefigure, Wesecchatthesinglet states licabove the triplet states inagiven multiplet. This follows from chesymmetry (cf.our angurnent thatKy:>0)andisaspecial example ofoneofHud's Ras: Other ‘shings being equa, theaates ofhighes spnwllhavethelowest energy. Ifweexci helium from theground state byshining ultraviolet light onit ‘wefindthatthe selection ruleAL=1,which wewillderive later, implies anex +AelcumericalrelationithcinB—ATavepercueof500"Kconesponde soanenergy Bo1/406 292° Quancum Physics tationcothePsates,Furthermore, thetiselectionruleAS=0,chatis,onyttansitions singlet singlet andwiper -»wiplet areprobable. Heace thesaremoststronglyexcitedfromthegroundstateisthe'Psate.Theotherlevelsmayalsobecome occupied through other mechanisms, forexample, collisional excitation. Once occupied, theradiative transitions cotheground statearevery improbable. The#state, which maybepopulated when atoms inthe'P,state undergo collisions withotber atoms inchegas,canonlydecay tothe*S;state, andthacstateismetastable, since itcannot decay totheground sate easily. The facethatthete arenotransitions togood approximation, between triplet sates andsinglet states, ledatone time, cothebelief thar ehere existed two kinds of helium, orho-helium (eiplet) andpara-helium (singlet). “Thespectrum ofhelium thatwesawinFig.18-2b shows thatthe excited states (1)() have energies thatdonotdiffer verymuch from those ofthe hhydrogen atom levels. Thus thebinding energy ofoneelectron intheatom is246eV(totalbindingenergyminusbindingenergyofsinglyionizedhelium=79.0 —344 =246 eV). whereas theenergy that would beliberated ifonedectronweretoberemovedfromthe2stateisofcheorderof4—5eV,which iscomparable totheenergy 3.4eV(=13.6/s# eV)forhydrogen, Thereason for thiseffect ichacthe“outer” electron sesonlyauniepostive charge, since the “inner elecuon inthe(1s)obital ceads coshield thenucleus, leavinganet effectivecharge~Z—1.Thisisnotthecaseforthegroundsate,sinceboth electronshaveaccess tothe nucleus. Thus theground satliesquiteabitdeper thanthehydrogen ground sateIn ourdiscussionofthefstordecalculationofthegroundstateenergy,venoted thatthere wasadiscreptncy ofabout 4eVfrom theexpetimental value. Rather than attempt anestimate ofthesecond order result, which would beverytedious, wecurtoanenttely different method ofcalculating the round stare energy-—the Ritevariational method. Consider Hamiltonian H,andanarbittary square incegrable function ¥, ‘hich wechoose obenormalized tounity, sochat wat (8-41) “Thisfunction ¥canbeexpanded in«complete setofeigenstates ofHf,denoted byve ids =Ban 8-42) “The expression seads v= LG (8-43) *Scleaton cle will beducted iaChapter 2. The Helium Atom 293 Now WHI)=LLGWalHlve)Co Zzad i =XL Geka Halen) |4 =DIG) , 3 2BEIalt 8-44) i Since(1841) implies hae 7 Liat (184s) : Ye obtain theresul chat 4 Be<(IH) (8-46) [ Ewemayusecistesulttocalculate anupperboundonEo,Thiscanbedonebychoosing a¥thacdepends onanumberof parameters (a,as, ),calculatingEQeHI),andminimizing chswiehrespecttotheparameter. ‘Welastecheutilityofthisprocedurebycalculatingthiegroundstate ©exergy ofhelium witha¥chosen tobeaproduct ofhydrogenlike wavefunc-i©onsiache(15)orbitals,butcorresponding toanarbitrarychargeZ*.Wetake . Cera) =vaowles) voor) (18-47) where zeE ea-Be)nee) (18-48) with «=~(1/2) me(Z%a)?, Wenow need . a Jan,f1,Pe)Viale)ee+EBe +er} ort)Yl) (18.49) We have fanfPraoo)Volts)(=-z)Wool)Yawles) =fervnten (BE-FEP22) pales 294 Quantum Physis weeneformvotentt> z wet @-neZ+a-n0% =ObNE 2)net (8.30) ‘Anidentical factot comes fromtheHamiltonian forelectom 2,ndtheexpect tionvalue oftheelecron-lecuon repulsion hasalready beencalculated in(8.31),excepethatwemussubstitte2*forZthere.Addinguptheterms,wea 1 5 late)=~Fmt(220+«ance29-324) 1 5 =Ene(azz—220—220 aes) 2 4 Mining thiswithrespect co2*yields pagwaz-2 (08.52) which isan improvement omtheguess wemade eater (Z—1/2).Wethus chain Bs=fnew[2(2ay]--7138eV (1853) whenwesubsiute Z=2,Thisimuch betterthanthefstorderperubationrevue. ‘Thevtttional calculation canbedonewithmorecomplicated tilwave fonctions. Pekerist used 1075termwavefunction andminimized (H/H|®)on 4compute. The-esulting bound agrees, within experimental enor, withwhatismessured. Is,ofcouse,tuethatsuchacomplied wavefunciondocsnothave formthatisaseaslyinerpeetable a8(18-47), withispartial screeningeffectsItdoes,however,providestongsuppontfrthecoeenestofquaatunmechanics, andfordheassumption thitonylecromagnetic forcesaeFequited toexpan thescuctre ofatoms, Tnconclusion, webrieflyreturntoourobservation thatthereexisteigen- valuesofH®++H'chatlcabovetheioniratonthresholdandtheatenever thelessdiscrete. Thesateslabeledbytheoxbials(23)or(2)(2p), forexample, liewellshove theionization energy. Thishassome dramatic physical conse. quences, Conside, forexample, the(2)(2) state Ifthe electons form«spin “ThisdictedinBethandJc,he | TheHeliumAtom295 \ singleeseate,thenthiswillbea'P,state,anditcanbeexcitedfromthegroundstate bytheabsorption ofradiation, since theselection rues A=1andAS=0 atenotbeing violated, This state, once excited, need notdecay back tothe 4 ‘ground state (8)oftoanother stateallowed bytheselection rues (aDyste,HIBRE say),because itcangointoanother channel: itcandecayintoanelectron and (WBE singly ionized helium, He’, with theelecuron energy determined byenergy { conservation. This process isdescribed asaudsonizaton, : ‘The (2)(2P) state inthecontinuum willshow upvery deadly iathe5scateingofelectrons byHet ions, When theelecuon enctgy issuch thatthe cempuund satecanbeformed, avery dramatic peak willoccu inthescattering race.Similatly,intheabsorptionofmidiationbyhelium,inthevicinityofthe Fenergyofthecompoundstate(-"—He"),asharppeakisseenitheabsorption +BBEGig.18-4)Thereisabsorption atotherenergies,t00,sincetheprocess radiation +He>+He* ‘anoccus, buttheabsorption atenergies away from thecompound stateenergy willvary verysmoothly with energy. Wecandescribe thestate insillanothervraybycallingiaremansstate,Sinceitdecaysintoitsconstituents =+Het, ee30 708 70 Fig. 18-4. Resonance inthehelium absorption specrum above thecontinuum threshold; thefispeak occuts atheenergy conesponding fothelocaton ofthe{@)2p)level.(FromR.P.MaddenandK.Caching,Phy,Rer.Leter,10s316 *4963), bypermission) 296 Quantum Physics irdoesnotexistforever. Hence, bytheuncertainty relation, AE%F/As,it appears chatitsenergy isnotprecisely defined, which seems toconttadice the factchatche(29(2p) statedoeshaveawell-defined energy. Ieturnsoutthaifthecouplingofthediscretestatetothecontinuum stateistakenintoaccount,thestateceasescobediscrete, anditsenergy maylieanywhere inanarrow rangeabout theenergy as,calculated wichout thecoupling. Weshallreurn tothis ‘topicinChapter 23andinSpecial Topics section 4,“Lifetimes, LineWidehs,and Resonances.” Problems 1.Considertheheliumatomintheapproximation inwhichtheeleczon-slectron interaction isneglected. Whatisthelowese orthohelium (spin1)sae? ‘Whatisitsdegeneracy intheaboveapproximation? Writedowntheexpressionofthesplicing duetoelectron-electton repulsion infirstonderperturbation‘theory, andestimate itsmagnieude. 2.Galeulate theenergy shiftAB)(!=0,1). 3.Considertheloweststateoforthohelium, Whatiitsmagneticmoment, thatis,calculate theinteraction withanexteral magnetic fel 4.Consider FE=(&\H\w) ‘withanarbiteay eralwave function ¥.Show thatifVdiffers ffomthecorrect aground statewave Function yubyterms oforder ¢,then"E”differs from the‘soundstateenergybytermsoforder€.(Note.Donotforgetthenormalization condition(¥|¥)=1.)5.Usethevariational principletoestimatethegroundstareenergyofthethree-dimensional harmonic oscillator, using thetialwevefunction va New 6.Consider«one-dimensional cut-offharmonicoscillatoroftheform \ . Vos)=Fmate=a8)[al<a =o Isl>a Usethevariational principle tocalculate thebestupper bound totheground state energy using theexponential form Ne“*"". The Helium Atom 297 +7, Consider thebinding of«proton andaneutron (both withmet=938MeV,approx)bymeansof«potentialcn Vn =j On swiththesystem inanL=0state. Therange ofthepotentiisgivenby.Use the following procedure tocalculate chedepth ofthepotential required cogive thebinding energy Ex.(#)Calculate anapproximate value ofthebinding cenergy usingthevariational principle. (b)Intheexpression thatconnects che pprozimate valuewithrand thedepchofthepotential, insertheexperimental value ofEx.Doyournumerical evaluation using re=2.8X10"cmand Es=—2.23 MeV. (Donotforget thereduced mass.) . 8.Consider afinite-dimensional matrix H;;.Show thatthecondition for minimizing . Wine)=&aiHyas oh . subject tothecondition W)= Ddaet yieldstheeigenvalues ofthematrixH.(Hint, Usethemethod ofLagrange multipliers.) E45, Usethevaiational principle coshowthat«one-dimensional attractive potential willalways haveabound state, (Hint,Evaluate (|H|%)withaconvenient trialfunction, forexample, NO## and show thattheabove canalways bemade negative.) 10,UsethedataofFig18-4tocompute thelocation ofthe(2:\(2p) level a above theground stateofhelium andcompute thevelocity oftheelectron‘emittedinautoionization, iftheHeionisinitslowesttateattheend.WhatwilltbeiftheHetionisinitsfisexcitedsate? 11.Consider awave function Yen, a2, tq)forwhich only thede- pendence onsome prameters isexhibited. Thewavefunction isnormalized (lar,a2...) Yhanyan,---a4))=1 andthedependence ontheparameters issochosen that = Wan. )IHiven--)) isaminimum, Show thattheparameters aredetermined bythesetofequations Br)-#Won) Kon,MY (yaVeo ieee (v DA)ar #(ve Iae o 1,2 298 Quantum Poysic where 4is«Lagrange multiplie. LetHdepend onaparumecet »(eg, the near charge orsome distance, saycheinternuclear disance inamolecule). Themtheawlldepend onthatpanmeter. Prove that a anB=(Yen) We.) Thisisknown asthe Feynman-Hellnann theorem andisveryuseful inmolecule physics calculations JS12,Usethevariational principletoestimatethegroundstateenergyforcheanharmonic oscillator H=£bt Compu your result with theexact resue By=1.060"(£)"am References ‘Averynicediscussionofthespectrumofheliummaybefoundin H.A.BetheandR.W.Jackiv,IntrmediteQuantMecha,W.A.Benjamin,Ine, 1968 i3 4chapter 19 1i H _TheStructure ofAtoms j ‘Theenergy eigenvalue problem foranatom withZelectrons hastheform J Z ptZe é4(EeA+Ewa)Meise oot)=Bikateoe)(9) andis«pei dient xption in32dineasions oright atoms ii roe fleet op Weslbeordaca oftom satan Seer ak Aine empleo hum (2 2)both pl ad serratPncetepsconiningZindependentcewon Fo reeintotterteecworeleon mecaePeri :tiontheoryturnedoutsobeadequateforZ=2,butasthenumberofelectronsf EEBOUDOE Eicite shen naacto: bylwoder pean Fe eee iadvse inporan, Themartina pepe Escased PR Terhap inade ste ofmanetinng hesgl ce cts SARC ST sme dng single pile fanny ttnce the Trening coon Tr nina pin, tose thaheilwaeFc Meri ry, rz)=dilts) b2(rs) o2(rz) (19-2) ach ofthe fanton ismoma ony, fweele ®explention sue of Hints sate webia w=&fens (-Eve- 2)ated ed=ffaryeerode)|*(a9) “Theproceduse ofthevariational principle istopickthe6(r) suchthac(H1)isa adam iwc wer todhe ine) tobeydrogesie weve foncions ae 300 Quantum Physis witha diferent Z;foreach electron (and with ech lecton inadiferent quantumstaretosatisfythePauliexclusionprinciple), wewouldgetasetofequa-tions analogous to(18-51) and(18-52). Amote general approach ischatduetoHartree.Ifthe(x)werethesingleparticlewavefunctionsthatminimized (H),then anaeration inthee functions byaninfinitesimal amount bdr) —dre) +files) (19-4) shouldonlychange(H)byaermoforderX,Thealterationsmastbesuchhat [eeteseo +9ftealt= 1 a9) that is,coist onde in, frilgiledfled+dledf1le)]=0 9-6) etwscompute thetims liner in} hat aie when (19-4) issubstcuted into (19-3). Term byterm, wehave =,ia Ke &fnLwea(-Lee)nara+witea(-Eve)ovea] sf wags a HAZJtreefted |—5>veelied|+tea|—55weeded (9) “Toobstin hiswehave imtegrated bypars «wotimes, andused thefacethat ‘ls muse vith atinfity inonder beanaccepable vsation ofasqume {integrable function. Next wehave adfen[ie%sued+oe)2reo] (98) andfinaly MEEfernfae wtteaeen+seedeeationea +GD 610) +Hed SEDI! 099) Wecannot justsetthesumofthese thre tems equals 2robecuse thefie) sateconstzained by(19-6). Theproper wayC0account fortheconstraint isbytheoseofLagimagetulips,chatwemultiplyeachoftheconsuining flations(19-6)byconstant (the“multiplier") andaddthesumtoourthreecerms.‘Taetoracanthenbesetequ1zr,sincetheconstinsomthefe)are i TheSecureofAtoms 301 now taken careof.Wichacertain amount ofnotational foresight welabelche multipliers —e,andthusget 45 a a) 5 fan{ateo[- Eveeseo]- 1100oie} +eLEf]fon,xfeyOOoe) i -«fdrsEle)oe.)+(complexconjugateterm)=0(19-10) ! Indtiving thesecondline,frstweconverted thedoublesum33j5,Tyinto i (1/2) Sim Sopwhich iswaresercted excepe fortherequirement that#+j, : tndtheausedthefacthactheintegrand in(19-9) isymmeric injandj.Now : file)iscompletely unsestricted, sothatwemaytreatf(r)andf7(e,)xscom- | BRGiecly independene (euch onehas«real andanimaginary part). Furthermore, ther thanbeing square integrable, cheyarecompletely arbicrary, 50thatfor(9-10)tohold,thecoefficients offiir)andf7(e,)mustseparately vanishateach“4 Jointxsinceweareallowed tomake localvariations intheFunctions file)©andfi(@.).Wearethusledcothecondition that a Ze pars)|*. [-zu“oteLfenet]br)=edd)(19-11) sndthecomplex conjugate relation “This equation hasastraightforward interpeation: tisanenergy cigen- ‘value equation forelectron “i”located atri,moving inapotential vey=-+05feee (92) % ‘thatconsises ofanextractive Coulomb potential duetoanucleus ofcharge Z,landarepulsivecontibution duecothechargedensityofall the other electrons. ‘Wedonos,ofcourse, know thecharge densities pile) =elated |* (93) ofallcheotherelectrons, sothatwemustsearchforaself-consistent secof(ri),inthe sense thattheir insertion inchepotential leads toeigenfunctions that reproduce themselves. Theequation (19-11) isarather complicated integral ‘equation, buticisatleaseanequation inthree dimensions (wecanreplace the SSiabe ybys),and chatmakes aumercal workmuch easel. Anevengreatersimplification occurswhenV(x)isreplacedbyitsangularaverage Via)-{2Vite) (9-14) 302 Quancum Physics | forthen cheself-consistent potential becomes central, andtheseif-consistentsolutionscanbedecomposed intoangularandradialfunctions,thatis,theywillbbefanctions thacanbelabeled byn,m, 1,withthelastlabelreferring tothe | spin state (Sy=1/2). The tial wave function (19-2) does nortake into account theexclusion | principle. Thelater plays animportant role, since ifallcheelectrons could bein thesame quantum stite theenergy would beminimum wichallcheelecctons ia them =1,1=0"orbital.”Atomsdonochavesuchasimplestructure.Totakethe | exclusionprinciple intoaccount, weaddtotheAntatzrepresented by(19-2)the rule:everydecron mastbeinadiferent state,ifthespinstates ateincluded inthe labeling. Amoresophisticated wayofdoing thisautorotically 1scoreplace (19.2) byatrialwavefunctionthatisaSlaterdeterminant(ef.(860).Theresult- i ingequations differ from (19-11) bytheaddition ofanexchange term. Thenew |‘Harree-Fock equationshaveeigenvalues tharturouttodifferby10-20%fromthose obtained using Hartree equations (with therulestated above), andsincefcislideeasiertotalkaboutthephysicsofatomicserictureintermsofcheHarece pictute, wewillnoediscuss cheHantee-Fock equations. ‘Thepotential (19-14) 20longerhasthe1/+form,andthusthedegeneracy ofallsates withagiven nandJ<n—1is00longer peesene. Wemayexpect, however, thatforlowZatfeast,chespliting fordifferent £values foragiven1willbesmallerchanthesplittingbeeweendiferentn-values,sothatelectronsplaced intheorbitals 1,2s,2,3, 39,3d4,4p,Ad,Af... wllbesuccessively lessstrongly bound.’ Screening effects willaccentuate this:whereas: orbitals do ‘overlap chesmall rregion significantly, andthusfeelthefullnuclest attraction, thep,d,... orbitals areforced outbythecentrifugal bait, andfeellessthac thefullartaction. Thiseffect issostrong thattheenergy ofthe3delectrons is veryclosetothatofthe4reteccons, sothatthe anticipated ocdeting issometiraesdisturbed,Thesameistrueforthe4dand5electrons,the4fandGrclecronssndsoon.Thedominance ofthe/-dependence overthen-dependence becomes ‘mote importa’aswegotolargerZvalues,a8weshalseinourdiscussionof theperiic tbl “Thenumberofelectrons chatcanbeplacedinonbiealswith«given(nis 2(21+1),sincechereatetwospinstatesforgivenm-value.Whenallthese 2(21+1)statesaefilled,wespeakofthecosngofashellThechargedensityfor 1closed shell has the form 6ES|RatO'1 Vino? (19.13) Thenotionstesameashatusedforhydeogen.Ameresensiblecotton,wsed bymuclea shellsrectare physicist, ivoreplace themby wf which isusta indexsepreseatig theorderingof«pivensate.Thsisteatofstatingwth34sates,for‘aample itmight bemore sensible vohave thelowest dsae called theIdstate, and so0 ‘Weshall neverhelss cootinse tousecheconventional ntstion, even though shewvaluedoernothavemuchtodowithcheoreringofeelsforlangeZatoms, B ‘TheSuuctute ofAtoms 303 andthisisspherically spmmettic because oftheproperty ofspherical harmonics Ltt a3 Yate =2 6916) Levus nowdiscuss thebuilding upofatoms bytheaddition ofmore and ©mote electrons totheappropriate nucleus, whose only role, inourapproxima tion, isco provide thecharge Z. Hydrogen. Here there isonly oneelectron, andtheground state con- HBB gveacion is(1).Thespectroscopic description oftheelectronic state is"Six andthebinding energy, asiswellknown, is13.6eV. Helium.HereZ=2,and,25wesewinChapter18,thegroundstatecon- figurationis(13%,whichisashorthandnotationfor(15)(1:).Thestate,inthe E(LS)description is«'Sstate, andthetotalbinding energy is79eV.After one Fectton isremoved, theremaining electron isina(1s)otbit about aZ=2 charge, sothatitsbinding energy is13.62?=544eV.Thustheenergyrequired toremove thelest bound electron, theionization energy isthedifference, thatis, 24.6 eV(seeFig.18-28). Ieisalsointerestingtoestimatetheenergyoftheist excited stave, which is(1:)(2): thisis13.62?+(13.6/a8)(Z —1)?because of theshielding, chatis,approximately 38eV.Thus ittakes approximately 79~ 58S20eVtoexcite thehelium atom." Because theelectrons form aclosed | shell iischemically inere, property shared byallatoms whose electrons form |dlosed shells. Lithium. Here 2.=3,andcheexclusion principle forbids a(1:)*con- figuration. Thelowest lying accessible configuration isthe(1s)*2i). Since we aieadding asingle electron toaclosed shell (9), chespectroscopic description f Ofthestate is*Sysasforhydrogen. Ifscrecning were perfect, theadditionallecttonwouldonly"see"'aZ=1,andsince=2,wewouldhaveanenergyof 136/4 =3.4eV.Thesercening isnotperfec; infact,since theorbitalofthe extraelectron is(21),thereisareasonable overlap ofthewavefunction atr=6, andhence theeffective Zislarger than t.The experimental energy, 5.4eV shows that Z*=1.3. Beryllium. With Z=4,thenatural place forchefourth electron togoisintochesecondspaceinthe2orbital,sothattheconfiguration is(11)*21},and‘weagain haveaclosed shell, withaspectroscopic satedescription. ASfar2s theenergy isconcerned, thesituation isvery much likethatofhelium. Ifthe This isecde eximatt cht ignores theclecron-lecuon repulsion and exchange sift, The difecnce between the20eVnl the24.6 eVithe4-5 eVtha wll bereleased + Sihen theexcited atom decaysroitsgroundsate(Seeig.1824) 304 Quantum Physics | screening werepefecr, theonlydiference would bedatthelasteleton isin | tn =2sat, giving abinding energy of2ug/t =62<V.Sceming isot pede, andthe experimental value 0.3 eV.athough «sell iclose, the cearemaae29Ahoacoe thepresence ofanother elementarearangement ofelectronsmayyieldenough | egy0beakupthedosedshellHenceberlium snosiea8hea. Ingeneralthistypeofshellisnorquiteasstableatheselinwhich,foragivenavail theposible Fats aeled Boron.Aftertheclosingofthesecondshell,thefifthelectroncancither beputinthe3orinthe2porbital.Thelatteislowerinenergy,anditisthe2shellehatbegins tofilup,starting withboron. Theconfiguration is(1:)*(2#)(2p), andthestateis*P,/s.Thelastdeserves acomment: ifweaddspin1/2toorbitalangular momentum 1,wemayhaveJ=3/2ot1/2.Theseatesplitbythespin- orb iterion 1 © 1aspat82VO~2yg4y-rety-ss4 yt | (19-17)andtheformofhisendstothehigheJvaluehavihigherenergy,sincetheexpectation value of(1/)ldV()/drl, eventhough 00longer equal t0chevalue ‘given in(17-16), issilpositive. This conclusion maynothold when there are Bore econ insnunled sl. Theinition ney might beexpeced Co besomewhat smaller thancharofbeylium, sincethe2ateenegy issome. ‘what higher thanthatofthe2sorbital, becnuse ofthecentrifugal batter. The Cxpecinennl luc 83eV Carbon. HereZ=6,andthe2pshellcontinues tobefiled. Thecon- Siguraion is(103420). Thetoalspinmaybe0oFLandthecoraloxbia angular momentum maybe2,1,ot0(since weareadding twoorbital angular tremens 1)Since thewae function muse beute forthe wesee tons outside checlosed shells, sngler sate must faveeven Lyanda elt ‘oddL,sochatchere aeonlythepossibilities 'S,2P..,»and 'Ds.WenowinvokeHund’sale,referedcoinovediscussionofhelium:“Thestateofhighest spinhas thelower encgy:” Thus wemst havea *Psae, heele that henate i Pyfollows fromHund's second rule,thathasbeenabstracted fromspinorbie daleahcions 1heincompleteellisnotmoreeanhalled,eenthelowestvelhasJ=|L—S|,itsminimuch value.Iftheshellismorechanhalffilled,thenthemaximum Jvalue] =Lt hus theetergy Since takes sixloons tofilthe2pshell, weobnin J=0.Asfra the onion energy iconcerned, weaveimreased ZbyonSiac theSecond 2p The Structure ofAtoms 305 lectroncanstay“outoftheway’ofthefrstone,bybeinginadiferentm-stace,therepulsionbetweentheelectronswillbeoflessimporcince, andweexpectsomewhat larger binding. Theexperimental value is11.3eV. q Nitrogen. TheZ=7atom hastheconfiguration (1)"(2:)"2p)*, of, (2p)if,Forbrevity,weomitcheclosedshellsfromoutdescription. ByHund’s 5tule,thespinoftheground stateisthemaximum valve $=3/2.Thisisa ‘symmectc spinstate(thisismostevident intheS,=3/2stat,forwhich althe ‘spinsmustbeparallel), andhencethethreefiled3porbitals musteachbein2diferent m-state. Ofthetotal Lvalues of3,2,1,0thatcinbeobeained by vectorially adding chreeunitorbital angular momenta, theL=3stateisclearly |excluded. Onemust lookatthedetailed construction ofthestates ofindout ‘thatthetorlly antisymmetric stateistheL.=0state,s0thatcheground state is“Sy,Theionization potential might beexpected cobealitelarger thanthat forcazbon, sinceZissgain increased byone,andthethidelectron canbeput inthethitdporbital without significantly overlapping theothertwoelecrons in the2pshell,chatis,byreducing somewhat theeffect oftheelectron-clectton repulsion. Theexperimencil value is145eV. © Oxygen. (Z=8)Theconfiguration maybeabbreviated by(2p),*and theshellimorethanhalffull,anditappearsthatthedetermination oftheelectronic‘staceisverycomplicated indeed. Wecio,however, lookattheshellinanother way:weknow thatwhen theshelifiled, thatis,when theconfiguration isC29)"(Z=10),thenthecotastatehasL=$=0.Wemaythusthinkofoxygentshaving dosed 2pshellwithtwohaleinit.These holesarejustlike‘ant- clectrons" (choughtheyateaotpositrons!)andwecanlookatpossibleewo-hole | Configurations. Thesewllbethesameastwo-electron configurations, sinceholes alsohavespin1/2.Thus, 28withearboo, thepossible staes consistear withthe antisymmecry ofthetwo-fermion (two-hole) wavefunction are'S,P,"D,and 4 thefourelecttons that,when added (0thesegiveL=$=0,mustbeinsimilar : states. Thehighest spinis$=1,andbyHond’s second rule,cheangular mo- < ‘meatum, foramorethanhalf-flled shell,mustbethemaximum J=2.Thusthestateis#Py,When thefourth electron isadded tothe2pshell, itmust be putintoanorbital withanm-value chaialrady occupied, 0thattheoveilep between twooftheelectrons islager than before, Hence itisnotsurprising,‘hattheionizationenergydropscothevalueof13.6eV. Fluorine. HereZ=9,andtheconfiguration is(2p),thatis,wehaveone hole inaporbital,Thestatemustbe*PsisincetheonaximumofJ=1/2oF3/2 ust bechosen. Themonotonie increase intheionization energy resumes, with the value of17.4 eV. Neon, With Z=10,the2pshell isclosed, theground state isaSystate, +andtheionization energy, continuing chemonotonic trend is21.6eV. 806 Quantum Physics ‘Acthispoint, theaddition ofanother electron requices putting icinan coxbit with ahigher#value(w=3),ndthusneonmarkstheendofaperiodin theperiodic table, asdidhelium. Inneon, asin helium, thefistavailable state ‘ntowhich anelectron canbeexcited fas higher n-value, sothaticcakes quite «lorofenergy toperturb theatom. Neon shares withhelium theproperty of being aninertga, “The next petiod again haseight elements init.Fits che(3s)shell isfilled, withsodium (Z=11)andmagnesium (Z=12)andthenthe3pshell, which includes, inorder, aluminura (Z=13),silicon (Z=14),phosphorus (Z=15), sulphur (Z=16),chlocine (Z=17)and, closing theshell, argon (2=18)“Theseelementsarechemicallyverymuchliketheseries:lehiuin,.., neon,andthespectroscopic desctipcion oftheground states arethesamie. Theonly dliference isthat,since =3,theionization encegies aresomewhat staller, a8,canbeseenfromthepetiodictableatcheendofthechapter.1kmight appearaclestrangethattheperiodendswithargon,sincethe Ga)shel, accommodating tenelements, remains cobefilled. Thefacei,thattheself-consistent pocentialisnotofthe1/rform,andtheintrashellspieing hereissuffciendly largetharthe(41)statelieslower thanthe(3d)state, though notby much. Hence acompetition develops, andinchenext petiod wehave (4%), (AD, HBA), ADF, (49%3A, GNGM)*, (49d), (4), (4°), (4)Gd)*, 401d)”, (40)4Gd)!and henthe4pshellgetsfilled untiltheperiod ends with krypton (Z=36).Thechemical properties ofelements atthebe-Binningandendofthisperiodaresimilartothoseofelementsatchebeginning,andendofother periods, Thus pocassium, with thesingle (4)clecton, is analkali meca, likesodium with itssingle (3)electton oucside aclosed shell Bromine, withtheconfiguration (4:)*(34)!"(4p), hasasingle holeinap-shell‘ndthusischemically likechlorineandfluorine.Theseriesofelementsinwhichthe(54)states atebeing filledallhaverather similar chemical properties. The reason forthisagnin hastodowiththedeals ofthe selconsistent potential Jccurnsoutchathetaioftheseorbits*aresomewhatsmallerchanthoseofthe (4s)electons, sothatwhen the(4s) sell ifilled, ehese electrons tend toshield the(3d) electrons, nomatter how many there ate,from outside inluences. The same effect occurs witen the(4f)shell isbeing filled, justafter the(6)shell has been filled. The elements here areealled therare earth. Limitations ofspace prevent usfrom amore detailed discussion ofche petiodic table. Afewadditional comments ate,howeve, inocd,()Thereisnothinginatomicstructurethatlimitsthenumberofelements. ‘Thereason thatatoms with Z%100donotoccur naturally ischatheavy nucle? undergo spontaneous fission -Ifnew, supetheayy (aeta)stable nuclei ateever discovered, there willpresumably exist coreesponding atoms, anditisexpected icisunderstood thattisisjust&wayofeng about thepeaking tendencies of theehage dstbtion PERIODIC TABLEOrr eeTonization zElement Configuration ‘Term!PotentialeV !1 H 09) Sin 13.6 | ro 1S 246 2 i He)Gy ie 344 Be(Heya? Ss. 9.33 B He)" Poa 336 €HeyQNap)* we 137 N(He)@2)%29)* Sua 458 © (Hea*20)* Py 36 9 B (He)(297429) Pas 4 0 Ne (He)(2y¥2p)* 8 a6 E " Ne (NOG) Sue 3a 2 Mg (Ne)(39" s 76 3 ‘AL (N}31)°09) an 60 a 4 Si NIGPOP ” an. 6 P(Ne) ‘Sin 11.0‘ 16 Ss (Ne)G)*OV) Pe 104 ” NAGI) Py 3.0j 18 Ac(Ne)(3)109)* 38, bs19 KANG) ia 43 4 20 ce (AN) % 6aa Se(ANd) Dus 652 Ti (ANG) *F 6.8 3 23 vo (anasy°od)? te 67& ra ce(ANGE % 67 B Mn (AD)(4s)*04)* Sea 7A4 26 Fe(An) Da 13 a Co (ANAM Fan 78 2% Ni(ANH) a 76 :2» Cu(ADAG in 130 Zn(Anns ‘Ss 94 a Ga (ANAND) Pus 6.0 32 Ge ANUS) ~ Bt B As (ANCA) Sa 10.04 Se(ANtss)*(34)"(49)" a 9.835 BeCANA) CHp)* in ne36 Ke(Ancannadndp)* 4s 14.037 Rb (KGH) i 42 38 se ORG 's 37 39 Y KG) Da 66 40 Ze (KOGHUA)* F r0 Comin) . 307 PERIODIC TABLE(continued) | Tonization | Z—-Hlement Configuration ‘Term Potential eV | a Nb(KC)(44) Din 68 |a Mo K)(3)(44)" % 12 | ® Te (KGa) Suz Nocknown 4 Ro (KIM * 1s6 Rh(Kas)(4d os cad4 Pad (K(4dye s 83 7 Ag (Ke)Gn(4H Sin 16 « ca oni ae ‘s 9.0 ® In AAO) a 38 30 So KCI)" % 73 >” Sb (NGS) SS 6 2 Te (KN) (5)* Py 2.0 3 1 eGSNAdM()» Pa wotst Ke(KNGuNAM)"(p)¢ 8 a3 Cs Ke) Sa 3936 BeXe)? Ss 32ca 1ae}G)XSd) Dux 368 Ce ENEMA) on 69x” Pr(Key? ‘tan 38@—g{Na Kenora % 63El PmOren thus Nocknowne sm (evant % 36 & Bee cera Sa 376 Ela Kenanapron Ds 62 6 Sm Koaran ts 67ra3Dy—(Ke)(Gs)*C4p)" hy 6.8a He (KeNenrapr Siva NotknownAder Gxetaynepe 4.Netkaown® Tm (Ke}G)UpH Fn Notknown70 Yoe}(@yapye s 62n La Ke GNEMSA) Dus 302 HEKe)(@APMA)* % 352B TaOke)(6)ANS@* fn 9* W e)EEAMA)* Dy 0 i) Ree) ‘Ses 79%6 OsOke)G)AGa)" D. 877 Tee) GGPOa* faa 9.2 m= Pe ke)CGEPGMY > 9.0 »AuKeNGN(EN Si 92 *0Hg CxeNlapranncd s wosa TENE IAPMSAN*G) ——Pin 61(content 308 ‘The Scucture ofAtoms 309, PERIODIC TABLE—(continsed) Ionization zElement Configuration ‘Tecm!PotentialeV q m Ph(KEMGNANA) Po 74 %BiKe)? Sax 73 if#4 Po(KeNGNANA) Pa a4 i % Ac KeyG)AUAMA%G)?——*Pan_—-Notknown | 6 Ra Ke) 10.7 j a Fe (Ra) ‘Norknowna8 Ra(Ra)? % 33» AcRn}(76) Dye 6990 Th myMC of% PaRallS/)*Gd) Kus2 BW Ro}CHHD/UG!) Le 3 Np Rn)(7)45P46d) Shue 3 4g] Pu RN)ONNSAE % 95 EfAm Rolnyrisfy "Sin 96 E]Cn RoI) ‘Ds 7 <P RMA His98 CFROA” “te» Fs (Ro)(H}GA Shoe4100 FmRn)" Herot Md (RAPA Rs 102 No a)(nyrf' 'sa a ‘Team designation isequivalent 1specoscopic description Echac theirstrucure willconform totheprediction ofthebuilding-up approach ‘outlined inthis chapter. ; (b)Wewent roagreatdealoftroubletospecifytheS,L,andJquantum ‘oumbersofthegroundstaresofthevariouselements.Thereasonfordoingthis istheeinspectroscopy, thequantum numbers ateofparticular interest because (ofthe selection rules as =o 4. b= 41 Aj=0,41 (200-0) (19.18) ‘thacwillbederived later, andthatmay then beused todeteamine thequancar ‘numbers oftheexcited stares. Thespectroscopy ofatoms, once wegetbeyond hydrogen andhelium, isverycomplicated. Coosider, asarelatively simple example, thefistfewstates ofcarbon, which areformed from different con- +figurations ofthecwoelectrons thatlieoutside theclosed shellinche(zp)* | | 310 Quancum Physics ‘orbitals. Asalready pointed out,thepossible states are", *Pze and"Ds.The *Pystate lieslowest, buttheother states atestillthere. The fist excited states maybedescribed bytheotbirals (29)(3). Here $=0or1,butL=1only. \ Since thea-values aredifferent, theexclusion principle does notrestice thestatesinanyway,andallofthestates3P,*Ps.0arepossible,whiletheexcited‘states thararisefrom theorbicals (2)(3p) canhaveS=0,1andL=2,1,0, leading toallthestates *Ds,"Ps," Ds, *Pa.aa, and*S,,Even with thete- stictions provided bytheselection rules, there arenumerous tansitions. Needless tosay,theordering ofthese levels septesents adelicate balance be-‘weenvariouscompeting effects,andthepredictionofthemocecomplexspectraisverydifficult.‘ That taskisnotreally ofinterest tous,since themain point thatwewant tomake isthatquancum mechasics provides qualitative, and sometimes quantitative, detailed explanation ofthechemical properties of | atoms andoftheir spectra, without assuming aninteraction other than the eciromagnetic interaction becween charged particles. Weshall have oceasion 10return tothetopic ofspectra, Problems 1.Lisethespectroscopic states (intheform **L,) that canatse from | combining S=1/2,b=3 Se2b=1 Si= 1/2,=b=4 SSR Las Si 1/2, =1/2,L=2 Which states areexcluded, among thecworspin questions, iftheparticles are identical? 2,Consider thefollowing saxes DP, AEG, 3D ‘What atechepossible Jvalues associated with exch? 3.Consider thesates 1D,"P,$5,5G,,8S.Given thatineachonechesate consistsofrwoidenticalputiclesintheielargesepossiblespinsate,whichofche statesaredisallowed bythespin-statstcs theorem? ‘Thespecuoscopic nbeliogchatwehaveusedhasitsimiations.Beinditvalidity listhephysical picture eat thetora orbital andspn angular momenta aceserustly conserved, except fortheperturbation caused byspinor coupling. Ths couplingSealfortheiheelements,butforhagZicannetbetestedat+petcbaion.Tereeit ‘bec tecouple theJad »foreach electron form sj,andthen cnsier te|coupling Denied consideration ofthis isbeyons thescope ofthis book. TheScuctueofAtoms317 ” 4.UseHund’s rules tofindthespectroscopic description oftheground {stares ofthe following atoms: 4 N(Z=7),K(Z=19),Se(Z=21),ColZ=27). Figure outheelectronic configurttions asfarasyouateablero : 5.Lisethepossible spectroscopic sates thatcnariseinchefollowing =electronic configurations: (1),(29%. (29)% (20) BA?, (2p), Gays. Take into account theexclusion principle(Nar.Rememberchatifallthespinsororbialangularmomentaarepointingin thesame direction, thenthestare issymmettic; alsoremember about holes in ‘dosed shells.) 6.Plottheionization potentials given intheperiodic table egainst Z. Observe thepeaks indicatingtheshellstructureofthearom. References ‘Anexcellent introductory treatment ofacomic structure maybefound in PG.Herzberg, AtomicSpectraandAtomicStructure,DoverPublishers (1944).‘Aaadvancedtreatmentthatisdefiitiveis 1.1.Sobel'man, Inreduction othe Theory ofAtomic Specra, Pergamon Press, New York (1972). This isaveryadvanced book. |chapter20 Molecules Jascastomsaragregtesofeconndinglecessooleseen dacs ulseend uci, Molecules inhihoweseycere anirkeracoainamountoferydisociedemintoaaaa ncedasocaion ofmolecules itoscons isthemestom- =monoccurrence whenenough energy istransfered cothesystem, wemaycall en hound saver ofatoms, adhough wesalscetatthsdescription hidesmuchofwhatmakes upthestructure ofmolecules. Thepurpose ofthis chapter, andthenext,iscoshowthatquantum ‘mechanics issuccessful in<eptblagthepopesanabebvorofmeeeHeaee rolecales atethosethatinvolve woauc, dhedaomicnoe eestheymcmotecomplex systems thanams Bectse feheCenterofmassisfixedinspace,chenucleiarestilfreetomove.Thisleadstoannereaseinthenumberofdegtcesoffreedom.Thusforthesimplestofsllmole-‘aules,theHytmolecule, consisting oftwoprotonsandoneelectron, thereareStillsxdegrees offreedom lef,threefortheelectron andtheeforthe relativemotionofthetwoprotons.Aswithatoms,afrontalattackontheproblemofthe Tema Shinui, taea,«uma solaion oftheSchrodinger equtioninmanydimensions ispossible. Forourpurposes cruderbutmorephysical Ffpponces wllbemoreeokgbening.. rosin yeumacs ofmolecules cnbeobtained fomwsofdeteam atetepetdalotemasive thandons (M/m >>10°and3) thustheemotion is«greatdealslower.Onemayviewthemotion oftheelec-FMR TNetbe mullweefueiaspceThemoton ofthemadontheobernd, isinanaverage fieldduccotheclectons. ForagivensetofmucearGECoordin,therewlbeaHamionianfortheeecons.Thelowereigenvalue FeofthatHamilronianwilldependonthesecootdinates,anditsminimumvalue “GSUMGEntepotionstthenerThicremubemodedeScie aedocinnedy musa, sincetheycaalsmove, Thtmodion i learns, bureyonlyee”aocage charge diuutoner oucnofteceeonsandvosespproaimaton, heyBove a3 St Quantum Phys Jinaharmonic potential about thelocations deteemined bytheminimum inthe energy oftheelectons."Wecandestbethesituationmathematically asfollows.TheSchrodinger ‘equation describing thenuclei tadtheelectrons hastheform (Te+Te+Vir,R)] ¥(r,R) =BAR) (20.1) whereTristhesumofthekineticenergiesofthenuclei,T;isthesumofthekinecicenergiesoftheelectrons,ndV1.8)isthepotentialenergy,whichcon-sists ofcheCoulomb attraction oftheelectrons tothenuclei, theelectron- Gectron repulsions, andtherepulsion among theauclei. Consider fistthe Hamiltonian describing theelectronic motion forasetoffixed {R} He=Ty+Wir.R) (20-2) ‘Theeigenvalueproblemcan,inprinciple,besolved (T+ VOR) aleR) =o(R) ele8) (203) Both theeigenvalues andtheeigenfunctioas depend onthevalues ofR,which bere playtheroleoffixedparameters,Siacetheus(rR)form«completeset,we sayexpand W(r,R) incms ofthem VER) =ZLnlR)tolR) (204) “Todetermine thecoefciens a(R) weinsert thisinto (20-1), andobtain, using (20-3), theequation TeX a(R) Hal(e.R) +SLem(R) bu) Har.R)=EXdal)sin(rR) (20-5) “Thefrsttermwillconsistoftemsofcheeype a(-at jza(R)Hal)=X(Trbn(R)]talRD —FE wnbl)-¥a nal) —3g Vebol Ve tal =BS ote)viele) — (20.6)2M“TY ios ve ‘Asafistapproximation weneglectthesecondandhiedtemsontheeightside ‘ofthisequation, thatis,weassume thatthecigenfunctions a_(7,R) areslowlyvaryingfunctionsofthenuclearcoordinates R,aleastitheregionofthesolutions forthe minimum electronic energy, Rodefined by Vx&(R)|eats=0 (20-7) | Molecules 315 . ‘This approximation : TeDba(R)talR) ~LZ[Tava(R] Mnl:R) (208) istheBeststepinasequenceofapproximations bywhich variations withrespect toRcanbetaken intoaccount. Theprocedure wasdeveloped byBorn and ‘Oppentimer, andthefrstapproximasion isgenerally goodone,Ifwenow raechescalar product with#(7,R), anduseorchonosmality [ULeste mtr=Bm 209) 4 then(20-5), together with (20-8), reduces to Tara(R) +ta(R) bel) =Bea(R) (20-10) "ThisisjustcheSchtdinger equation forthenuclear motion inapotential ¢w(R), theelectronic energy. ForRclosecotheminimum points Ke,wemayexpand ; sn2ty+beewas(FE)+ ou) Afonly thefirstewotermsareimportant thenuclei moveinharmonic oscillator wells. Thos thequel: wilundergo vibrational morion.' They wilalsoundergo Toational motion, since Trinvolves angular coordinates. Wemayestimate che Emagnitudes ofthe various energies.4 Tetusassumethatthesizeofthemoleculeisoforder.Then,bytheuncertainty principe,theelectronicenergyisoforder F wt(3) (20.12) an Na f—Thefrequency ofthevibrational motionofthenucleusis,according to(20-11), given bytheformula 1MAR) :a 2 (2013) [Given thepotetial energy ia(20-12), «dimensions argumene givesus < ee .yh OR~mat othatis, a a(2hbs ox¢ed (2014) +The motion wilbesomewhat more compliaed ifwedoor Smit cuseles co |. woter expuoion of(0-1) Butt wlslavedheuaatveproperien tatwesf decunicg 316 Quantum Physics “Thsthecatioofthevibrational energyofthenucleicotheelectronicenergyis Bow fo n\"nwfea(2) e019) “Themoleculecanalsorotateaboutchecenterofmass,Typically Pap¢RavLapeer any Bra = (20-16) “Thus,afara8molelar racers isconcerned, onemayneglect cherotational andvibrational degrees offreedom. Nevertheless rotational andvibrational teneegy levelswillexist,andinmolecular spectroscopy therewillbe(a)Elecronic tranitios. W-ehe dimensions ofthemolecule areofche order of1A,thatis, ee an then eB meat bearhe cha is, irkNTaeX137x05A ~350k @o18) “Thustheradiation emitted inelectronic transitions liesintheultviolet. (0)Vibrational sransrions. These atetransitions becweea different levelsintheapproximate harmonic oscillator well.Thetypicalenergies wilbeofthe‘orderof(or/M)*?«chatis,thewavelengths willbeoftheorderof(M/m)!**~50timeslarger.Therangeofwavelengths, ~2—~3X10-*cm,liesintheinfrared region.{@)Therotational spectra willbecharacterized bywavelengehs M/m~ 10°—10°cimeslargerchantheelectronic optical wavelengths, and®~,0.1—1cmistypicalofchemicrowave tegion."Toseewhatform«(R)caotake,letusturtothesimplestmoleculeofall, theHtion.Afterseparatingoutthecencerofmassofthetwonuclei(weignoretheelectron indoing this), wearelefewiththeenergy eigenvalue equation Boe Be fe =(Sie Beelam ent ome 019) Molec 317 E sagf \----4 ------2--- =oe ath a Fig.20-1, Contributions to“nuclear potential.” TheCoulomb repulsion and “hes verrepresents thei ecg ofherotons with ~ - ote (2029) Mo et Me 2» theeed asf hwo pon pt, Thesecond theint eniy of thecaecum, Thenoxwoene sesen heation Dewen thedon tndthtworons owned s/t #nd“1/2, anhesteneps te Teplice between teororotons nepal bydance R=[The qualitative features ofthesolution withRheldfixed, andtheproton kinetic ieiy temabset shows infp20BorRey getheeon wlbe bound cooneoftheprotons, andtheenergy ofthesystem is—13.6eV,the tocray ofingle gen sc, When Re0,adweleaveoutthecon pecan plone decom wlbeSound oa =2ues, ede binding Sgy milhe vi36 ~~344e¥. Thedecsonc energy54foncionofF imetuunes smoatly becwen these point, When thethegy ocepuion @/Ris added cothis,thecarve «(R)results. Thiscurve has«minimum forthe + protec &mim oesnoaways exis 39hatsome toms donoe 318 Quancum Physic form moleciles asweshall soon se.Thelaconic eigenvalue equation, which hhastheform anticipated in(20-3), Huw(eR)=(22—# 4)ts =(SS aai-wewat FO =eo(R) wo(r,R) (20-21) canacealybesolvedinellipticalcoordnaes, butwewllgetmoteinsightfrom ‘sing thesariationl prin wthtalwavefunctions eitreflec some physical inition abou thesyste “Aeasonable tal wave function isinst combination of WCeR)=(4)"e™ "— (2022) and en)=(LY cenaven=(2) (20.23) representing theelecron bound tooneothe other proton. Since theHamil:fonlanissyrmmecicaboutreflectionsintheoxigin(p.—>Px>—F,R—>R), wemaytakeasrialwave functions evenandoddcombinations ofchese* FER) =CRACK) +Yate] YalOR)=CARER) —daleRO] 2024) “The nonazaton factors tegiven by een Wehhkh) =242fernie vier) (2025) “Theintegral appeating above iscalled theoverlap integral, andicanbecaleu- lard, The alcltion of Stk)=f#eviteR) YD) =P gayremain gesim=ife -afer me erin (20-26) rai "Theabatingishiovie:"sdsfor“pend”whichcasevenioGea, sd “sf “angele” |Molecules 319 is sigheforward, though cedious. Theresalei ; R BY na=(1424F)om 2)so=(1+4 *) o2n | |Teexpecarion val ofZi theewosates i 1 4 q(DeaFagyHeeMheH) 4 1=p ttlya)+GalHel)ilHal¥a)Galald Sy Wt)+alt) Osu&enlvo : _uiHolya) &GalHalds) coo - 12 5®){ao28) rheusehasbeenmadeofthesymmezyunderR'—»—R.TheevoteminFthenumentor anbealulated =wor(22.#4 ) Egnitew=fete(2-1aaial te XwR) 4pat efayeR!entSafa eee (2029) “ThefirstcermisjusttheenergyofasinglehydrogenatomEy=~13.6eV;thesecondtermistheproton-proton repulsion,andthethirdtermistheelectrostaticowen energy detotheelecion charge diszibuson aboutoneproton beingFlancted cotheotherproton. Thelstintel canbeevaluated, sochatfay (hlthlya) =Bt(:+2)one (20-30) Siiasly ‘i olthion=fanvee(n+$-aaah :» WERE +ey RP aan efgyBERLER) ’ =(6+2)s0- 0foFERRE? coon Heretheastetmistheexchangeintegral,whichanalobeevaluated,yielding HER We A(R) om 2fetes f(+4). (2032) hen allofchsitpurcogeher, cheresulting energies canbecalulated as + functions ofR.Figure20-2showschecalculatedenergies. 320 Quancum Physics, i F | i 1| i 1 en) i co i Hwree|1105Arg > , ey Z i us|(i a Fig.20.2, Result ofvaiationa calculation forHet “Theexact solution, which according tothevariational principle must lie below thecurves obcained, differs lice from theminimum, Inourapptoxima-tion,weseethattheevensolutionyieldsbinding,whiletheoldonedoesnot-*“Thedifference between theevenandtheoddsolutions ischarintheformer, the‘dectzonhasaighprobability ofbeinglocatedbetweenthetwoprotons,whetetheattractive contribution ismaximized; fortheoddsolution, which hasanode midwaybetweentheprotons,theeleetontendstobeexcludedfromthatregion“Theexperimental separationbeeweentheprotonsis1.06A,andthebindinenergy is~2.8eV.Thecalculations outlined above leadtoaseparationofie and binding enetgy of~1.76eV. Thusourwavefunction isnorascompactasitshould be.Thereason istharwhen Rissmall, chewave function should approach thatof«He*ion,which(20-24)doesnor.Onecouldimprovethecal Culation byintroducing aneffective charge fortheproton andminimizing ie)e withrespect tochatpatametet inaddition coR,asinourillustration in- volving thehelium atom. Since wearemoteinterested inaqualitative wnder- Scandingoftheproblemthaoiimprovingthevariationalcalculation,wedonot pursuethis idea 2One might worry ththetuefs), yng below se(He=BAR) carve, sldips lowe andpines sesket bound sate: Dead elealtons show thai doesnot Molecules 324 “Theorbitalsthacwehaveconsidereddonotdependoatheazimuthalangle abouttheaxisofthemolecule.SincetheHamiltonian iinvariantunderrotations F bout cheaxis, wemayclassify thesoltions bycheangular momencum com ponent along theexis.Ifwechoose Rtodefine thez-axis, oureigenstates will fesimultaneous eigenstates ofLy.Thesolutions will,ingeneral, havethependencec=?with=0,+1,2,....Thesearelabeled6,x,8,.««in §<analogywith5,P,D,....Thereisalsothelabeling"g”and“u"whichis |applicable toalldiatomic molecules forwhich thearoms arethesame(hame- ‘ucear molecules). Thus inourexample theground state could belabeled Ley,andtheantisymmetric starecould belabeled 110,*,theasteriskindicating, thatthe state isunbound. Excited states oftheHy*molecule maybeformed BB with higher orbitals. ‘Wewillnotdealwiththerotational andvibrational degrees offreedom of themolecule except tonotetheirrolesintheewotopics chatwediscuss next. Firstwewillbeconcemed withtheeffec ofthePauli Exclusion Principle on homonuclear molecules. Consider, forexample, theHymolecule forwhich thetwonucleiareidenticalandexchhasspin1/2.ThusthetoawavefunctionmastD.-be antisymmetric under theinterchange ofthetwoauclei. Thetwoprotons in thisexample maybeintheantisymmetric spinsinglet ($=0)state, inwhich ease therorationl sate mustbedescribedby@spmmetrcfunction,sochathe sngular momentum iseven, Ifthetwoprotoas arinthe symmectic spintriplet(Gen 1)sae,theangularmomentum ofrotation mustbeod.Inags,collisionsamong theHzmolecules willrandomize thedistribution ofspinstates, and {ssuming chattheyhaveequalprobabiicy, chenumber ofmolecules ingiven spinstatewillbeproportional rothedegeneracy (25+1).ThestherewillbethretimesusmanyoddLmoleculesasthereareevenLHmoleculesinthegasThis willmanifest itself intheintensity ofthespectral linesassociaced withthe ©erasitionsbecweenrotationallevels.Moregenesllyifeachnucleushasspinthenthespinstates21,2—2,21—4,...andthespinstates21—1,21~3,.-» ‘willhaveopposite symmetry. If,forexample, Iisaninteger, thenthefrst eves ofspiastates willbeassociated withevenotbital angular momentum, since thenuclei arebosons inthiscase.Theictotalnumber is se ¥por +y=artyet —FY BS =r+NU+) (20-33) whereas theremaining q Qr+1%— @F+ 00+) =G+ yr (20-34) szateswillbeessocatedwithoddorbitalangularmomencam. ThusfrintegraI,thetioofevenLtooddLincenstisforagivenLis(I++1)/1.Forfermions+ that ratio isinverted 322 Quantum Physice “Togood approximation, cheenxgies ofthe rotational states ate _FLL+D) 2,- FH(2035) ‘whereistheromentofnetaofthehomonucest moleculeunderconsides-Ton,Transivont berween adjacent Lvalue (oconform withthe selection rule AL abtalc bedeed) yield tdiacioa withfequencies ottisn= ket nes Leto =kaen (2036 FromthestudyofrationalspectonectideifytherotationallevelsandfodthetLales. Theimenstes thengive wayofdiscriminating becween trenandoddpia,History, «study ofsherotational spctum oftheNz ‘oleae edtothe conclusion thatitssinwaseven,Thiscould notbeunde stood onthebasisof«nudeat model inwhich thenitrogen nucleus consisted offoureen prorons andsevenlecfoas; such2nucleus would haveodhalf tirepal spotTedacovery ofthe ncuron andtherelzaton thatthenitrogen ticles consisted ofseven protons andseven neuttoos removed chedificult. "Theexstence ofthehierrchy ofexcation energies, rxatina, vibra sions andelectronic manifest selinteformofchespecificheataxconstant Nolume a3fancion oftemperatte. Wetakefrom stadstical mechanics the following fcr. (a)Thespcic henatconstant volume igiven by 2» Gr=MED 030) whereF(T)istheavengeenergyof«moleculeinequilibiumatempersuceT,udNis Avognd's umber {b)Theaverage energy canbecalclated ftomtheBoltzmann disbution sen=fargoe/ [anoe a-arfRr wm|meen7faege« a ane=trogfate (2038) vere (8)isthedegeneracy ofsates withenergy E. Molecules 323 (©)The average energy canbewritten a52sum ofcontributions from independent degrees offreedom, sothatwecanwrite ECL) =ExranslT) +Eyo(T) +Evie) + (20-39) Forthewanstionl contribution, where isthe kinetic energy, wehave! -ear_[@Ppasar foeco=[2R.. (20-40) sothat fdeg(E)2=core (20-41) 4 ascanbeseenfiom dimensional considerations. Thus (20-38) yields 2 a a2j GrM2(erZeger) j 3 =ine 3 -te (20.42) where R=Nok=1.98calorie/mole K°,ThisisjusttheDulong-Pesit result, Foc the rotational coatribution we have [arene ¥gensnemenat —Goay where gis thespinmultiplicity comesponding tothe given L,and(2L+1)i[the usualdegeneracycorresponding toagivenvalueofL.FrtheHsmolecule wwehnve thespecial sitation mentioned before: theexistence ofpas- and —ortho-hydrogen, forwhich chenuclei areinthespinstates 5=OandS=1 respectively.orparschydkogen, Liresicedcotheevenvalves,L=0,2,4,..-andf=: foronbo-hydrogen, Lisodd, L=1,3,5,..-andg, =3.Atlow tem %—peracures—and here “Jow’” depends onthemoment ofinertia, sothatforHsFtherelevantnumbis? ; Bauer 4 H= ms (2048) Tora dcusionoftheepeeofatesappiabievFemotion,eeChapter2 . “he momen ofnea onbedeveined fo: thespuciag inrutin spec. q see Quan Pi —-aheL=0statewillbeprarilyoccupied,thatisthegaswillconsistofpara-bydrogen.' Atroomcemperature thedifference becween evenandoddL's 1drcowtesinsignificant, andtheratoisdeterminedbytheratiooftheg,thatiyitis3:1orho- copare-hydrogen.inanevesofthediscussionwesal,forbrevity,ignocethecomplicacion ofthetwoformsofmolecules. Attemperatutes wherecherotational degrees of fieedom become excited wehave BE LY LL+1)MT Eros=penepew DART (20-45) “Thiscanbeevaluated numerically. Athighcemperatures, thelevel-spcing i30 Simll,compared toKT,thatwecanreplacethesumbyaninegrl, anduse (20-38). Wehave fae ~faatewom OT (20-46) sothat,using,(20-38) and(20-37) weget,forlargeT, 2(1221op0")= (Creo=BNoor(rarisCc1)=k (2047) [Achighertemperatures thevibrational statescanbecome excited. Theharmonic oscillator potential inwhichthenucleus moves needaorbesym-tmarmer Ifthelineconnecting thetwonuceiisakeninthez-direction, itis Fsusile thatthe poten wallswillbesteperinchex-andy-direction chania Theediretion andthusintheexpression fortheexergy Bmfaalns +B)+hele +B+Rane +4) (20-48) thefirstexcitation willbefromtheground stateEy=Hite+hay+Huxco By=Ex fayThus Me Bly BeBF Ee earnorb Eat Bot =etite awa(itSemr)(i- eo) =Fy+fa,MT (20-49) «acallywansionbewsentbeorthoandpce-satesaetslow20bacooling ge anatemakehype Unprecineweaa Molecules 325 otaa Sa §nan a ‘9 a To 7eA0 Fig. 20-3, Specific heatofHTgasasfancionoftemperature Hence ? «BEN=ge(Be)?never £ (Coen=SENa=Nok(9)‘ (20-50) %FotHethevibrational effecdonotsetinuntil600°K(Fig.20-3).Ontheothetand, forCl,ioehasquite alowvalue, andatroom temperaate, thecontribu- tion ofthefstvibrational leva is~0.5R.lagener ayy arequite abit larger thano,fordiatomicmolecules,andthereforethehighenergycontribution ‘ofthevibational excitations computed from 4 fiEM dyLDctoeeeroar xfaensinir scmgost) {Ras fortherotationallevels.ThisisshowninFig.20-3showingCrforHs1. Theelectronic levels onlycontibute atexcemely highenergies 326° Quantum Physice | Problems | 1.JaHCLaumberofabsorptionlineswithwavenumbers(incm) |sone oscars Noeretehaefede eae |Sibrational oeroutional cansitions? Tftheformer, what isthe characteristic | Frequency? Ifthe later, whatJvalues dotheycotespond to,andwhaische moment ofiaertia ofHCL? Iathat cise, estate theseparation between the nucle. (Inradiation thequantum nsmbers change byoneunit.) 2.WhatistheratioofthenumberofHCImoleculesinastatewihJ=10tothequmber inastate with J=0,ifchegisofeolecules iaaempenture ‘of300° K? 3.Thefrequency ofvibration oftheCOmolecule initslowest sae iv=2X10!HzWhatisthewavelengthofdheradiationemittedinthelowestVibrational excitation? What isthe probbilcy chatthefstvibrational state is ited, clave totheprobability thatCOisinitswibational ground sate, sven thetemperature is300" K? 4.Considec thevibational androtational energy of«moleculeiathe sppeonination BR)=bmt(R—RP+keene Findtheposition where theenergy is2minimum. Ifthemoment ofinertia of themolecule iscalculated using thenewinternuclear separation, show thatthe rotational energy canbewitenintheform By=AN+1)+B+DP+Detemine thecoefficients 4andB(theler istheeflece ofcensifugal discorion) References Several elementary books thatcreatthemater inthischaper arelisted the ndofGupte 2 |chapter21 Molecular Structure Inthis-chapter wediscus, bet onlyeulatvly, howthedecwonic structure ofmolecules determines theirshapes andother properties. Webeginapuedeanddacsinsomeeabecauseeeewoeleceo{incontrasttotheHsmolecule)andtheexclusionprincipleandelectronspinFoeins saketheseappearance, Pothhseaadinthe setofthe ‘pater thenuclear nouen wilBeleTiitfeowebedBandtheewoelectons“and (Gig. 23) TheHamieouan asthefxn ‘nemtis 242 aa where dependsonlyonthecoordinates ofthe election jrelative tothenuclei. Wewill fagsincompute anupperboundtoE(Ree) byconstructing theexpectation value 3 f=Ht, as) : ‘@ 1 Pa 2 a Raa =* GA sag20-1. Comte bsinhedacs ofthe Hamolec & 327, y 328 Quantum Physics | aejustHamilconians forthe Hs*molecule (Eq.20-21) itissuggestiveroakeas | fourtilwavefunctionaproductofewoleyfunctions(Eq.20-24)fortheHs |molecule j 1 Yleuts)=a5aayNACE+Vale]Wale)+YallKine 21) “Theelecuion spinsatei«singlet, sincethespatial partofthewavefunction is taken tobesymmetric. InthstialwaveFunction, eacheleron isasciaed with dhprotors, eatis,thetialwaveFunction issuidtobeaproduct ofmoleularrbita,Thedescription intermsofmolecularoxbialsissometimescalledthe‘MO method. Theaculaion of(Fo||v) yields e a),@,e A =Ran)+(Ras)+(v2i)ius en)—2 # - =Ran)—(|A|v) (2) where«(Ras)istheenergyoftheHs"moleculecalculatedinChapter20.The firstoedr electron-lecton repulsion conuibution canalsobecalculated, and when the(oralenergy socomputed isminimized withrespect tothe separationanitsFoundthatthebindingenergyandinternuclear separationazegivenby Ey=-2.6609 R= oak a6) ‘Theexperimental values ate Fy 450 R= ok an Evidently theapproximation isnotaverygoodone.Wenoted inourdiscussion oftheHymolecule chatthewalwavefunctions (theMO’s) areinaccurate forSeallproton-protonseparations, andthefacthattheMO'saretoospreadoutinspace shows upinthenumbers above. Thetialwave function alsobassome Undesible facutes folargeRan.Theproduct in(21-4) mayberewritteniathe form ated) +voledIC ated) +baled] =(ales)vale)+vals)vale+(Yalevoles)+Yale)vale)e) Molecular Suucture 329 “Thefirstermiscalledan"onic"term,sinceitdescribesbothelectronsbound©‘oneproton oftheother. Thesecond term,the“covalent” term,isadescription fnterms oflinear combinations ofatomic orbitals (LCAO). Ourtialwave fonction thusimplies, sincethetwoterms enterwithequal weight, tharfor G—Aarge Rusthemolecule isalikelytodissociate intotheionsHand Hr,aiisfintotwohydrogenacoms,andehisisparentlyfalse.4 “ThelastdificultycanbeavoidedwiththeuseoftheValenceBond(also‘alld Heicet-London) method, iawhich linear combinations ofatomic orbitals, Este used. Thesinglet wave function usedasarialwave function inthevaria tional principle istaken cobe bi oeSven) =li+;saat{alesYates)+Vales)¥oCed)]Kenge(21-9) (where, asbefore, cheYa(n) arehydrogenic wavefunctions forthe FhelectronES—Thoueproton.Wecould,inpeinciple,addatripletermtooutvariationaltialwave function. However, atriplet wave function must bespiially anti symmettic andhaslowprobability forthe electrons beinglocaced inthe region—berween cheprotons.WesawinourdiscessionoftheHs"moleculehajasthis Configuation ledtothelowest energy. Although icisnotimmediately obvious 7 dhaedheacraction issillarges inthis configuration when thereaetw»elec tronsthatrepeleachotherinthesystem, itisinfatso.Theresults ofvaria- Sonal ealclation waththeVBrialwave function is 4 y= ~3.14eV R= 087k (21-10) | Thisinocsignificant impeovernent overtheMOresults, forchesimple reason tduttheinadequacy ofthetialwave functions forsmall Rapcartes moreS-—weight.Thereshouldbenoquestionaboutthequantitative successesofquaa-e tummechanics inmolecular physics. More sophisticated crialwavefunctions u Ihavetobeused:forexample, a50-etin trialwavefunction yieldscomplere| agreement withobservations fortheHymolecule, butitdoesnot,astheMO [and VBfonctions do,givevssomthing ofquale flingofwhatgosc=z herween theatoms. Inwhat follows, wewillexplore therelevance ofthese approaches coqualitative understanding ofsomeaspects ofchemistry. “Theexpectition value ofHfortheHsmolecule intheVBapproach bas thefollowing schematic form 4 w= +. GY)=Sayaolee+VasdanlHlHoan+Pastor) 1 | e @¢ @¢ @ 380° Quantum Physics +t“)[vata+Vestn) whereT;isthekinetic energy ofthei-thelectron, andsince i (n-2)enBen : sndsoforthiscanbesimpleto 1 | é @ é é 4T+=((vate2Tnoe+Te+len) é é é eopsp|2,—Se—pk ENva +(bat|fean+ale)) 1 ua “| ~285a}2pm)+0ffPotten” +eSfvilvettad ery Inobining tis,bea usehasbeen made ofsane, Thecams thatin mae this expeion more negativeat1 Ss{2|) ‘Theformer isjusttheatcraction oftheelectron cloud about oneproton rothe oeherproton; chesecond itheoverlap oftheeoeecons(oeighed with Mfr) thiscanbelarge, sheewlBebinding, Thetwosecon8 canonlycecapsignal,however,theipiasareaniparll;hisis acoasequence ertheexsionpail.Thegionofoverisberweenthetwomudand theetheateactoncotheavegenerallyovercomesthecostarepullonbeeen the clecons, TatheMOpicts, ro,iioverlap tern—the lasttermin(2:31)—diweicrucialtobonding,sadagin,bondingoceursbeaseteelectonhacg disbuton ilageberween theauc. Thus, hough htetheoxi Belong tothe whole molecule ater thaneoindia atoms, chephysic reason forbonding itheme "Wewilldices some molcses ates ofthese woapproaches othe descition oftheelectoni charge dsuibuion, Animportant spliation ‘cea because weely donorned 0takealeleceon incaccount. Intheonanofebbelenceotmolec,olytheouste008, Molecular Sacrue 331 + )-@ +» G)-Ot fig. 21-2. Illustration ofwhypaired dlectrons donotgivetisetobonding. (a)If patil electrons exchange, wave fonction ixspaially antsyromeuic. (3)TFanti parle clecrons exchange, oneerm inthe wave function haselecoas inthesame spin sae, which may require promo:ion toa higher energy orbital 0c inclosed shell, ehac is,theso-called valence electrons have &chance ©con- teibute tochebonding. The inner clectoas, being closer tothe nucleus, areless|afferedbythepresenceofanotheratomintheviciity?Furthermore, notallvalence electrons coatribuce equally: ifewoelectrons ateinaspa 0state—we callthem paired eletrons—tey will giveret» bonding. Toscewhythisi90, consider what happens when anatom wich asingle valence eleezon isbrought ‘eat amatom with twopaised elecurons, There aretwoeases tobeconsidered pg. 21-2) 4 (@)Ifehewoelectrons thaareparllel exchange (ie, areputinto «formsuchas(21.9)wichasignbetweenheterms)thentheymusbeinatriplet state, andhence thespacial wave fonction ofthispairmuse beantspmmeric. This reduces theoverlap, andicturns outthat cheexchange integral gives a E_sepulsive contusion totheenergy.(b)Whenthetwoelectonsthtateantiparallelexchange,thenoneatom finds itself some ofthetime with ewoelecrtons iathesame spinstate, The "1 may happen imatoms Bat even thevalence eleont acemer close tothe sucess. This isthe eeforthe neents Aconsequence ofthe atha thester eleonsFinidandfsbocloseathatthervcxsecheryeseativethantetn+donmeats2=20~20)+ originalstomicstacewillfrequentlynolongerbeapossibleone,andoneofthe‘lecttons willhavetobepromoted incoanother atomicorbital. Sometimes this 'traycoveverylleenergy,butusuallyhisisnottecase,andagainbondingis I‘notachieved. Chemical activity depends onthepresence ofunpaired outerelectrons. AD ‘lectronsinthe1sstate;promotion ofoneoftheminto22rstatecostsalotof | areinert,Nocallunpaired electrons ateofequalsignificance. Asnotedbefore,theunpairedd-andf-electrons inthetransition elementstendtobeclosetothe t ‘alledthe“sacuration ofchemical binding forces”: oncetwounptited electrons | j | o* @ Molecular Seructure 333, fromdifferent acoms formasinglet state(andcause bonding), theybecome paired; anelectron from 4chitd atom must findanunpaired electoa elsewhere, thatis,participate inadifferent bond. Another consequence ischatmolecules have spin 0inmose cases. letusnext gothrough 2process analogous tothebuilding upofthe electronic shails inatoms. InFig.2-3weshow pictures ofatomic orbitals, in | particularthe(Ys)orbialandthep-orbital.Forthelatterthelinearcombina- tionspon+Yara)andpy(¥n—Vie)areplottedinaddition0p(Y) A Thecontesponding orbitals, dry,dendysdn,004dre~dyyatenorshown, because thedelectronswillplaynoroleinourdiscussion, Figure21-4representsPGE whachappens when stomic orbitals arebrought together andexchange occursThus, ewo1satomicorbicalsmaycombineintospatiallysymmetsicMO(hencewithspin0)ointoaspacialantisymmetricMO,whichisantibondingsincethe [wave function beewcen thenuclei issrall. Similtly, theformation ofbonding sndantibonding MO's with p-orbitals isillustrated inthefigure. Note chat (@)theparity “g”oF“u"canbereadofffrom thefigures, since these indicate thesigns ofthewave functions; thedistributions chatchange sign upon re-flectioninthex=plane,heerepresented byaverticallin,areod;(8)sincethe F)— peeaad pyotbicals havemy=31,themolecular orbital formed from them isa —-orbiaal. Ieshould bestressed chatinthefigure wearematjustbringing two E_charge disuibutions together, butaewying tosuggest theprobability amplitude thac resuks when wave functions ate combined, that is,the MO's such as Yuta)Yule.)andYap(ta)-EVay(ta)foeRaplargeandforRepsmall :‘WecanusetheMO'stodiscussthepropetiesofafewdiacomichomo-nuclear molecules 1H,Thismoleculewasdiscussedinsomedetail.Wemerelyrepeatthatthe ——ewo electrons cangointoatsa,MO, andsince thisorbital hasalowerenergy thanthesepatacedtsatomicorbitals,cereisstability. 4 Hey Ofthefourelecrons, only ewocangointoabonding ise,orbital; theother twomust form anantibonding Lw.* orbital, Thenetenergy isgreater > thanthatofthe separated Heatoms, sothatnomolecule isformed. Incerms of | theValence Bondpicture, bothazomshavepiedlecuons andtheconclusion4 isthe sume. [agenera, electtons inbonding orbitals andinantibonding orbitals tend ¢0cancel each orher out. Since there ate two electrons involved in«full 4bond,wemayspeakof«bondnumber,givenby (Bond)at(Electronsio Eleceronsin a number) ~2|(bonding orbitals) ~(ancibonding oxbital, This number vanishes forHes Lis, The atomic structure ofLiis(13)%(2i). Thus, che 2selecttons are+unpaized,andtheycanform2s-bondingorbitals.Wethusexpectthemolecule © © Of 6i re%UD . @ w X. — ‘ a /6 6 dP % * " Fig.21-4, Molecular osbitls resulting when woatomic orbitals arebroughe together, (2)Two4otbitalscombinetoformthespaallysymmetricMOaethat fBvesrisecobonding: (6)Two#otbcals combine toforthespatially ancisym- netic antibonding MOes,(2)and(d)show bonding andantbonding withpy omic orbitals; (2and(7)show bondiag andantibonding withporbitals, The ‘axisisalongthelineconnecting thenucle, which aepresented byblackdos 3a Molecular Sauerue 338 2 * > a p ODOO coexist,butbecauseofthe#=2valueoftheorbital,wewouldexpectthe binding tobe significantly sale than fotheHymolecule Ben, Here cheatomic seuctute is(12%; there anounpited cle. trons, andhence weexpect nomolecule twexist. This isindeed so. : By,Theatomic sructue indies tthee is2unpsited 2pelecton in cach ton. Jecanbein anyoneofthe stats 2p,2pyad2h.They maycom bine cither into&2pr. orintoa2pe, MO.The former hasalower energy, 30 thatheretheground sate iipkt Thiiinagreement withHund's Rule ‘Thestate with highest mulipliciry haste lowest energy ‘Theresson why2pc, hasahigher esergy isthatthere exist 2,orbitals. Whenever there aeses thee bave thesame quantum numbers, “mixing” crus, andsexes thataealmost degen tend totepel cach other, Thesate thatislugely 2pe, ispushed up.Webegin tose theeppearnce ofcomplica |thansimilartotheonesthaappearedinoutdicusionofatomic sure! F Cs.The atomic steucrure is(15)*(2)%29)", that is,each atom haswo unpaired electrons, Since eachelectron canbeinanyoneofthreepstates,two=boningMO‘eanbeformed.TheMOdesaipionturnsouttoBe(pe)pe) 336 Quantum Physics Nu.Herechesicuation isverysimilar tothatofCxexcept thatthteebond. | ingMO’scanbeformed. TheMOdescription turnsoutrobe(2pe,)(2px.)* OxHere things get little mote interesting, because cheatomic structureis(1)*(2)*2pY, thatis,ehecearefourvalenceelectrons.fntermsofmolecslarorbitals, three bonds, asinNa,canbeformed, butthisleaves twoelectrons that cannot possibly formabonding orbital. Whatistheleastharmful antibonding foxbiaal? Thecwoelectrons should avoid eachother asmuch aspossible, andthis ‘anbedonebymeuns ofaeile sate,withtheelectrons inorthogonal orbitals, forexample, oneinaps,cheother in&p,state, withthetwospatially ant- symmetrized. InthiscasethespiaofO,is1,anexception tothestrong tend- cency toward zetospinthatwasmentioned cutie. TntheValence Bond picture, twoofthefourvalence cecttons inoxygen :mustbepaired,sothattwobondswillexistorthogonal toeachothe,aspist0‘puforexainple, Onemayseetheeffectofthisdirectionality in«molecule ike 1H,O. EachHusesuponebond, andwewould expect theshape ofthemolecule tobeanL.with90°between theequal length arms. Actually, theewohydro- ‘gennuclei repeleachother, andonemight expect cheangle tobealitelargerthan90°.Experimentally iisaround105°tisthedirectionality ofthep-orbitalsthatexplains theshape ofsimple molecules, Lestthereader feelthatallofchemistry canbeunderscood withthema-cetalathand,wewillpointoutjustafewofthemanycomplicasions éhatshow thesubtle soutces oftheincredible variety inthestructure ofmatter. Forex- ample, acarbon atomhastwovalence electrons, andonemight expect aCH molecule, withashape similar totheHsOmolecule, toexist. Actually, €tuins touttobetetrvalent (four bonds) rather thandivalent, sothatitisCHchatis actually formed. ‘Thereason isthatalthough theground stateofCis (as)*25)*(2p)% theexcited state(1:)*(2:)(2p) fers verylitdefromitinenergy. “This sate, however, hasfourvalence electons, andthemolecular bonding with fourbonds issuficiently stconger chanthatwithtwo,C0compensate fortheelectronicexcitationenergy.Morepecisly,theneardegeneracy ofthe2sand2p‘states inthearom allows theformation oflinear combinations thatallow larger overlaps. Figure 21-5shows thatalincar combination ofans-andsp-orbial {givesalopsided waveFunction allowing anincreased overlap, This“mixing” iseuallyquitecommon.Wemaygive«moredetaileddescription ofthewater‘molecule byworking withAybrid orbitals thatinvolve s-andp-orbitals. Ifwe ignore thedifference beeween the25und2pelectrons inoxygen, thenwemust really dealwith chemore general states x9=abn+BPbn,+Ban,+BEbare Qu) swith thecoeficients constrained by iP) hs 21413) Molecular Stucture 337 oS) Fig.21-5. Combination ofsand f,orbials leading tounsymmentic wave func: q 4 rather than with da»,day»dry404ap,2ndmolecular orbitals orvalence orbitals 3 coostracted foruse48tialwave functions inthe variational principle should be tmade upoutofthex's.Theminimization oftheenergy willdetermine the coefficients aand6,FortheH,O molecule, ittus ov¢thatthefouronho- F normal combinations are 4 1 42°7OutOntdn+don) 1: 2°=iOntten—ombn) 1 X°=7Ou—thn+in—tan) 1 x8=7Ou—bon—din,+nd (ard) ‘ Consider now the10electtons inHyO. Two Lselectrons remain strongly 3 ‘bound cotheoxygen atom. (One could, forconsistency, describe themincerms eofiss,and 1374MO's, analogous totheHetmolecule, butthischangesnothing.) By, x(0.Thegeomet shapeofthemolecule canbedetecmined fromcheform ofthese hybrid orbials, Ifwetryrodaw what xLooks like, weseethat : thelastthree terms areallnegative intheoctant (x,y,z<0)andthere theyeffectivelycancelthesteam.Roughlyspeaking,x°lookslikeafacigarpoint-3ingfromtheorigintothepoint(1,1,1).Similarlyx®poinesfomtheociginto©thepoint(1,—1,~1),andsooa.Theshepeisatevahedon,andsomesimple [geometry shows thatcheangle between thebonds, @satsies sin0/2=2/3 2 sothat @=109°, Without hybridization wefound theangle tobe90°,andthe = truth liessomewhere inbetween. Iamore accurate calculations, (21-14) is somewbat modified, sothatonly tee ofthe fourorbitals aresopronouncedly “pike” andtheremaining oxbea! haslessditectionality, that, ismore "ike"“Thismoredewiledpictureallowsustounderstandsomeoftheproperties +ofwater.Ifwethinkofthewatermoleculeasoxygen,withfourtecahedeally 338 Quantum Physics | oriented armschataxecharged, withprotons, fromtheH’sattached cowoof | thenbyCoulomb fores,wefstofallscechatthewatermolecale maybe ‘expected tohavealargedipolemomentcomingfromthetwonegativelychargedarmspointing swayfromthepron, Wecea(cf,Chapter 16)thataground Statefanonlyhaveaaclecic pole moment ifisdegenerate. Weexsily sce thatsincethegapchange signunderreflection andthegx.donot,thex"are noteigenstates ofparty. Thereflected orbitals, forexample, 1 (00=Fou=bn—bm— an xwe Gan,—bay—bn) 1-15) haveexacly thesameenergy andthefefected shape,Theground stateisthere- foredegenerate, andadipole moment canexist."When wavermolecils getclosetoeachotherinaliquid, chenegeivelychargedarmsofonemaycomeclosecotheprotonofanother.Theelectrostaticfanction between thereowillower theenergy, andcheewillbetendency forchewomolecules tobind,Thebond isfairlyweak (0.2eV)andiscalled « Iydrogen bond.Eachwatetmolecule canbondfourotheronesatonce,andonethusexpectstofindlargeclustersofwatermolecules intheliquid,effectively‘molecules oftheformHs,Ox. Thisleadstoastrong temperature dependence of theviscosity ofwater, Incoldwate, chelargeclusters easilyangle together: heating thewaterbreaks themupandreduces thesizeoftheclusters andhence theviscosity‘Manyothermoleculesalsoformhydrogenbonds.Thisishowtheprocess ofdissolving works: watermolecules foamhydrogen bonds withthe substance,findthemolecules ofthe substance would racher sticktothewater molecules thantocachodher, thesubstance dissolves. Oilsdonotformgood hydrogen bonds and thus dontdissolve inwater. ‘Notallhybrid orbitals areoftheform(21-14). CH.istetrahedral, andia facethebondangleis109.6, butinCyHychemolecae hasaplanar serocur‘Tetumsourthatforthismolecule, thecarbon orbitals arehybridized asfollows: x =Om owt. fi xWVou+NBom weston -seeet te »etn -ete : XW=yt—gtitm a1) “Thefistonepoints along thez-axis, andthelstcheeareoriented at120° ineris snthe27pline ‘Thefourouter electrons inczbon [specificallyGa'OpApoincetheseobials.Given«wocarbonatoms,thetwoelectronsio Molec Sucre 339 \ Fig. 216. Bonds intheGa mole thexandx!cris bind todhebydogens, while tos inthe xand ‘states form bonds totheother carbon (Fig. 21-6). Theelectrons inthexstare form 2¢bond berween thecarbons, while theelectrons inthexstate, being trig anges (othe bonding an,formabond. Isthee bonds thaforce thetwoforbitals tobeparallel, andthusmake cherestofthestructure planar Tat Gal comment wenote tht inthe so-called “aromatic” compounds, foecanple bonne weearnet speak ofsoch welllocaaed eras, The carbons arehybridized asinCsH, andtheyform aplanar structure, with thep, — uroftheplane(Fig,21-7).Theo-orbitalsformacore,butthex-orbitalscanbe | paired according to(12)(34)(56)ot(23)(45)(61),bothofwhichhavethesameFaagy.Aawaloatcombinationsofthesedegenetepolilowertheneg, so tebeste desnotaveeine cba Tinal‘exchangeeffectstronglyaffectsthephysicalproperties ofthe compounds, but« discussion ofthese would carry ustoofarafield. Formore information, the trae hos tum obooks onGut chem Ed '" i en KR % —k: (oe jFig.21-7.sehenpurofBenzenemole(CH). | 340 Quantum Physics j References i “Thischapterowesmuchtothebriefdiscussion ofmolecules inG.Baym, I Lecures onQeantim Mechanits, W.A.Benjamin, lnc,NewYork, 1969. The iinterested readershouldconsultthefollowing booksformoreinformation: IM.KarplisandR.IN.Porer,AtomsandMolecules,W.A.Benjamin,Inc,NewYork, 1970. M.W.Hanna, Quantum Mechanic inChemis, W.A.Benjamin, Inc., New ‘York, 1969. U,Fano andL.Fano, Physics ofAtoms andMoleals, Chicago Universiy Press, ‘Chicago, 1972. G.W.King, Specrecapy andMolecular Sracue, lt, Rinehart, andWinston, Inc, New York, 1964. “There ate,ofcoutse, hundreds ofbooks onquantum chemistry, molecule structure, andmolecslar spectroscopy, andthesampling listed above isjustthe tnefamilie totheauthor. Borabetter reference list,ehereader should consul anyphysical chemist. |chapter22 |TheRadiation ofAtoms 4 Inthestudyofspectra, thatis,thestudyofeansitions between atomicEvelcompanied bytheemissionorabsorpicaofradiation,neisinterestedFj theinertron between atoms andthecectomagnetic Bld.Since theria Eton feldoscillates, itistimedependent. Itistherefore necessary tostudy the effect oftime-dependent perturbations. 4 A.Time-Dependent Perturbation Theory g “Theproblemis,giventhecompletesetofsolutionsto :Hobs =Folge ay Fewselvefoeyt),whichobeysteequation : 02 +volvo (22) The seandard procedure istoexpand $(2)inacompletesetofstates: ww)=z=6 oy (223) The time-dependence associaced withchegyisexplicitly inserted imtheex- F pansion, sotharifV()=0,thea(#)would beconstants. Theexpansion © Reecints i)sais asetofequations thmaybeobsined bysbstiting(Ga)seotheimedependentSchrédinger equation(22-2),Weget£20)5pacegleg,=Hye) =[apFBate) RMy=BHO) i -=[»+aweaomoy aan \ | 342 Quantum Physics ‘ shat, : BEBO 6AEVOaeM&(224) | “aking thesear product with6andusing theorthonotmaiy ofthe $m. . (alts) =Bow 23) yields,afterthefactor¢~*="’*isdividedout,thesetofequations HAO AEyO OnIVOlb.) (2) ; Weshall solve these tofstorder inthe paramere 2Asan intl condition atP=Owetakethesystemtobeinapacarstate,30tha(0)~duthai,us(0)=Bua e027) Sincedepatues fromthesevaluearatertimeswildepend on’,wemay,fora fascorder cluation, substitute theabove intothenight sideOf(226). This yields thediferent equation (fom 8) 90@yg0-2O” WVOI60) (28) which scaly solved RPhgygeomeny.y eG)=Bia HOMGIVI) G29) “Theprobabilicy chatatalatertime1,thestare¥(/),isaneigenstateofHowithenergyBtthatis,Batdy,According0theexpansionposulte Pal) =|aldODE =ai? (210) | “Thisgener result canonlybemade mote specie ifV()isknown. Thepet turbation pilbespecified next! B.The Electromagnetic Interaction ‘The Hamiltonian describing theinteraction ofanelectron in&static pocenial ¥(?)withanclectromagoctic iddescibed bythevector poreatial IA(ed) isgiven byfalUONODEyy ea +Acrly, motecanbeftiknowahater7)aeseysowoxchanges ery py congue! totegpa euendea (eg E0/a) inhesyne. Sec TERRE Salen” apposite doce incare edvanced ems ‘The Radiation ofAtoms 343 faswesawinChapter 13.Thus, ifwewaite 3 weFave (22412) wefind that : WO=©Aeo-p (22.13) {Inobtaining thelastexpression wehavespecified thegaugesothat V-A(rd) =0 @214) ‘Under these circumstances, p-A=Ap, andwehave dropped theream quid “FBRxcinAe).AFweeat¢,theelectronchargeastheparamererofsmallness\,then the A?tem isasecond-order term, We will seethat the A*term will contribute tothescatreing oflight byanatom andcothetransition with the emission oftwophotons, butitwillnotcontribute tothetransition accom-paniedbytheemission(otabsorption) ofasinglephoton.Theprobability fora ‘tanstion involving twophotons involvesafactor(#)$,whereastheone-photon Ftransition probability isproportionaltoe.Recallingthactheappropriatedimen sionless number involving #isa=ei/fc &1/137, wearejustified inconcen- eating ontransitions thatareaccompanied bytheemission ofasingle phocon. “Togive&realjustificationoftheassociationofexchA(r,)withthe ;emission otabsorption ofasinglephocoa—sothathigherpowersofA(ry) ~imply thepresence ofmore photons—one must teattheelectromagnetic field —quancum mechanically, chais,teatthefields ateachpoinc rasoperators. Thisis fundamentally notteriblycomplicated, butitisoutsidethescopeofthisbook.©Thereaderwillhivecotakethefollowingassertionsonfaith,: 1wewaite : Ale) =ASle) +Aale) (22.15) ‘thenintheemizion of«photor, onlythefstterm, withthetimedependedce , istobeincluded inNV(), whereas intheadsorption ofaphat,onlythesecond term, with ehetime dependence «™, appears. This isaconsequence ofthe ‘Beneral association ofAg(e) withtheCreation ofaphotonandAc(r)withthe annihilation ofaphoton, undthetime dependence isjustwhat onewould ‘expect from theharmonic oscillacor (7-51). Theresemblance totheharmonic F —oscilcor problem isnotaccidental, since iathequantization oftheelecro- [magnetic field, what isdone is4normal mode decomposition, according to‘whichonefindsthatthefieldireallycollectionofsimpleharmonicosilators;these arechen quantized. The “occupation sumber” #thalabels theharmonic ‘oncillacoe statevectormaybeassociated withthenumber ofphotos, henceAS Ftises thephoton number byunity andAslowers thephoton number byunity. é+ "Themote quatitaive description ofAj(e) andAd(r) aced not, for- 344 Quantum Physics “eunately involve thefallmachinery ofquantum electrodynamics. Wemayuse fcomrespondence principleargumentstofindthesequantities,andtheajusttae ithequantum mechanical modifications. Awayfromthesources,theelecro- 4‘magnetic fieldhasaverysimple spatialbehavior. Ifwelookbackat(13-12) | =FAM)=Fae =o 0216) } ‘whose solution is | alr)=Age (22-17) with i w= (2238) H ‘Thechoiceofgauge(22-14)impliesthar k-Ay=0 (2219) Theeectic andmagnetic felds conesponding tothisvector poten are Bm—2A8go5complerconjugute : B=V XA=ikXAye +complex conjugate (22-20) Nowtheenergy density ofthe electromagnetic fieldisgiven by 1 ifje oy . Aeth if:=AwAS+201ADAD+titiens| (22-21) Ifweaverage overtime, sothatthe osilating terms dropout,andmake useof thefact that with (22-19) (kXAdeXAS)=AoAS (22:22) and that =ut/, weget 1 tye ALAS 5.aEBD Aes (22.23) Ifthesystemisenclosedinaboxofvolume,V,thenthetoulenexgyintheleccromagnetic feld is u 2)Hage (2 ferdons ny=SEide (22-28) The Radiation ofAtoms 345 IEthisistobecarriedbyNphotons, cachwithenergyfic,wehave f Wi,rary‘AdlNho (22-25) “ThedirectionofAeisdeterminedbytheplaiation oftheelecticfel,andwillbe dened bytheunitvector ermust satisfy went ES tk=0 (226 he Wetherefore obcain - Ate)=(=) oe (227) ov The quantum electredynamie medication isthefollowing: Fortheabsorpcion of@ {Tighe quancum by2charged particle fomaninitial satethatalrcady hasN Fphotons offequency a, : Ap)-(2EM)" cameo (2228) ow Rortheemissionofalightquantumbyachargedpatticleintoafinalseethat © hasN-+ 1quanta, thatis,fromaninital statewithNquanta offrequency w, 2a 4Ales)=LoW (22-29) Heace fortheemission ofasingle photon offequency fom «sae thathas0 photons, wehave, according to(22-13), E w=£28)" epee (2230) a a oy s cals)==(9)mieeviefe (22-31) ‘andthus theprobability ofuansition from theinitial state &cothestate wis 4 Biven by Re Qaet 1 aPrat=ZOEaale®™eian|t|[a2 [ese ‘Thetime-dependent factor is at|y- |e2sigAt|?Agigs oo[fiee| l=[zasoAtsataos) & 346 Quancuin Physics | xe | \ i a - oF OF OF Fig.22-1.Plotofthe function 1/a? sin?/2 versus a A where : a=Pethe (234) % | Figute 22-1shows thebehavior ofthis function Forlage¢ibecomes stronglypeakedat=0,andawayfrom&=Otoscilacsveryrapidly.Thisisthekind(ofbehaviorthatweassociatewithadelafunction,Infact,iff(A)is«smooth Functionof4,then,forsarge <4 Saige &[tof BasfofaAsine8 =uf)fapmy=2aif0) (22-35) that is,fr lange sine B2081a)=arte +eB) (236) “Thus theuansition probability in(22-32) grows linealy with tine, and hence ‘TheRadiationofAtoms 347 thetransition probability perunissmeis ; PrP Je e-plg)|?Ee?—Ba?— Tien=2HEEG[Ube"ecpldad]?HE?—Bat—hs)(2237) Forgencral purpose, itwillbeuseful toremember thatifthe time-dependent5)perturbation isoftheform : Va)=ve (22:38) then theformula fortheeansition rate is a ThanFl(Gal¥*|¢u)1*MEs*—Ba?~fs) (2239) _ Asthings stand, thereader undoubredly feelsswindled. First ofall the manipulations involvedin(22-33)-(22-36) certinlyatenotstraightforward. F-Theyinvolve vaguenotionssuchas“slag.”whichcannotbetakentooseriously,since ataasition beobebility chatgrows linearly withtimemust sooner otlater exceed unity. Second, theylendtoanonsensical formula, according towhich « perfectly reasonable quantity, likethetransition rte,isproportional roadelta E fuaction. Needless «osay,thedifficulties areconnected, andwewilllater F outline amore satisfacvory discussion. Atthispoint wemerely notethatthe — fauleliesintheuseofperturbation theory, andthatboth (22-37) and(22-39)=arecomect,ifproperyused. 4 orthis,wenotethatCy-aqi8 rally thetransition probability perunittimePSfefheanmangrationfomthattealacpaney e emia ofaphan ofenergy es.Thedeltafunction, unappealing asitis,doescellfstharenergymustbeconserved,chatis, fo=BY BP (22-40) |The deleafunction isactually integrated over, ifwetkeintoaccount thatthe phocon energy fedoesnotuniquely specify thephoton state. Thephotoa willingeneralbedetectedinsomemomentum interval(k,k++Ak)inthevicinicyof Fil =w/c,andcheeanstion ratechatismeasured isreally Ren= Elie (2.4) ‘summedoverallthepossiblephotonstatesinthatinterval.Notethathevationsfinal saces incheinterval Akareinprinciple distinguishable, sothat icisthe * probabilities thataresummed. Wewillsethatthesum(22-41) iswelldefined,esince,ineffect,icinvolvestheintegralofadelafunctionandasmoothfunction. i4.Teetummation willbewereithenextsection. 4 C.Phase Space | Wewillnowcalculate thenumber ofphoton satesinthemomentum | inerval (ky +Ak), chais,thedensity ofphoton states, Forthepurpose at td,wee theetnpenta Qe)inheform | 1 t Ate)=beaf+complexconjugue . 0d)=Fe +complexconjugs (22-42) ‘where Visthevolume oftheenclosure inwhich theealelation isdone. This ! "box" isustacooveniencecosaveustheeoubleofworkingwithwavepackets forthe feeprtcles (thephotons, hete—cf. Chapter 4.Itsshape andchecondi-tionsattheBoundarymaybechosenatwillbucitmustbelarge.Artheend,wewill take V—»«©,We will find itconvenient rotake thebox tobeacube of sideL,andtoimpose periodic boundary conditions, thais, z Nxt Ly7.28) =Ales 752) (2-8) andsoon.Thisimplies,justasinthesolutionofaparticlein2one-dimensionalbox, thatthe wave numbers, that is,themoments, aequantized. The foun (22-42)requireshacethaqth=ghay (22-44) tharis,chachewave numbers beofthe form q a t Fa 4ban beEm ewEe (22-45) t "where ms,andmateintegers, Wealsohave Ak=AbeAb,ah,=(2)nA,Bt (22.46) and 2x = Ihde = at+mht ma? (2-47) ‘Whenwecaryoutsumlikethatin(22-41),wesuoverallvaluesof(Ay) intherange specified by(22-46) consistent with theconstaine ofthedelta fonction. Thus Rew YePaw =fentin he Radiation ofAtoms 349, =fEIT iw (x) ve : | ~[hn (aay Iathesecond finewemade weofthe faethasasFbecomes lege thestases ‘ecome verydense, andthe sucanbeconverted ntoanintegral inthetied . line(22-46) wasused,andinthelastline,therelation ree p= ik (22-49) vesused, Theiategrson isoverthevolume inmomentum space defined bythe E Caperimenalamangement. Iwewrite op=apy~a,(*)a(*) (2230) srhere isthe Sliangle diferenil wefnhatheenergy conserving deta Faction simepated overandtheresult We ate rea % Rew=[EElose eplot alps . poMh) see pe% xieoR a(ES—Eat—fs) E 2 1 ter * ;=faySone,aleeles 2-51) wher : Bo -bet 5 nn (22-52) Bee experimen apparatus doesnotdsctizsinsreberween thepolation sates ofthe photon, theratecalculation coos include wsumoverthose two «HE independent finalstates. Furthermore, thesumshould alsoinclude allthefina!Statesoftheatom.Thiswillbediscussedfnatesecon. : | ‘Thepase space ba vea en=De (2253) Sig notesuicied tophotons, Ancectfon that ficeidecibel bytheplane trie function 1/9°°"andicwllhavethesaredensityofsate.Theonly f difference ischattherelation becween energy (which appeals inthedeka func: © ion) andmomentum isE=p/2m [orreais, E=(pre >me)" ainscadofE=pe. 350 Quantum Physics Ifwehave several fee particles inchefinal sate, thedensity ofsues is theproduct VaryTlGay (2:54) ‘Theexpression (22-48) combined with(22-39) thengeneralizes to ] Va nye fTEER myiss(ae+Emae)e259) ft+(xh)? 7 ingen | where Ay;ischematrixelementoftheperturbation between theinitialandfinal | sates oftheunpercurbed system. The deloa function again expresses energyconservation, thatis,theenergycarriedoffbythefreepatticlesisequaltothe | energy change inthesystem, andshingration isovrindigent mena Ths, | ifsystem decays intothe parties, there aeonly twoindependent moments, since thethied oneisdetermined bymomentum conservation. Note, however, thatheproduct offactors in(22-54) isoverall theparticles inthe fal sete,thatis,iinvolvesV*iftherearemparticlesinthefinalstate.Equivalently we could write (22-55) asanintegral over allmomenta, with adelta function that ‘includes astatement ofmomentum conservation. The reason that such adelta function didnotappess inourdetivation iswearedealing withatoms thatareso ‘much mote massive than thephoton (peccisely Mxion >>fa)chattheatomic fecoil never entered intothecalculation, Atanyat, theresue afTve a=2fvemshee ximaltd(er—2e- D&)a(w—w- Dv.) @259 whichcouldalsobeabbreviated by a Ray=MallB) (257) with p(B) called thedensity ofstates, isafundamental result, andhasbeennamedtheGoldenRulebyFermi'Note thatthevolume ofthe boxalways drops ou.Fo frepaces in chefinal state cheteisaV*fromthedensityofstates (phase space) anda1/./V foreach ficeparticle inthematrix element, coming from heir wavefunction enNoy 0258) “The Radiation ofAcoms 351 ‘There ace#ofthese factors, andchus theV’dependence ofthesquace ofthe matt element justcancels theV*from thephase space. Wewillhive further ‘occasion tousetheGolden Ral, butathispoint wecura totheevaluation of the maczx clement foetheradiative wanstion. 4 C.The Matrix Element and Selection Rules q (Oenext taskistocalculate nie" e-ploa) (22-39) ‘Webegin byestimating ismagnitude. For«typical atomic usnsiion ep~ [pl~Zmca (22-60) 4‘Wealsoneedtoestimatetheexponent,sinceitisanascillacingfactorandcouldchange theresult sigoificanly. Wich A ~ (261) and fo AmeZa me “mwheAma me iggy aBAe 5(2a (2.62) we have dw Wa (22-63) Heace, forZa&1,theotdet ofmagnitude ofthematic element isindeed Ema, thus Reon~2aaZa~yafay?G2” ~ala) ~2X 100Ztsect (22-64) Tesimpliies masters chat intheexpansion oar EP ay (22-65) thesuccessive tenms areestimated todecrease a8Za.Thus, tootdet Za, a nie*"pits)=nle-pigs) (22-66) 352 Quant Physic Wemay write thisas fGa|Plde) =me-(nlde/dt|dx) =Fe-Galttisits) =imOEBeaairign) | =imeealt|de) (22-67)“Thusweaeiotrestedincalcltingthematsclementoftheopeorandthatitonereuson forcling theapproximation (22-€6) theclectic ile | ‘resination. IFhein sate dis ahydogelke sae characte byte“inital” quantum sumbers ny,aod mand thestateduheGalstate, bythe quantum number ny, andmy,chen what needs cobecraluted i Caolerisay =fedefatal) Yui)eR)Yel) 7 | =V1drRafe)Real) XfVn0.0)CFV) (22.68) | “Thera imega wlbediscussed fora special caseinthe extsection, ere | wreconcentnte‘on cheangels integral. Wehave ef=esin8con6+asinsinb+66088 andmaking weof VeYell,o=co28[ZeVial6e)=Fainvee”—226) lie agebe yields era ratio atig ): =f(ot AErtSa) wm ‘Thus theangular integeal in(22-68) involves /BOY(68)Yi.nl0,8)YimO.#) (2m) “The Radiacion ofAtoms 353 ‘Letusfrstconsider theazisnuthal integration. Ityields fdy0GtP=28Bom @7n) Wethusgeethefirsselection rule my—m=m=1,0,-1 (2.78) “ThiswascheselectionrulethatwasmencionedinourdiscussionoftheZeeman Fs Bilec. Specifically, ifwedefine thez-axis toliealong thephoton momeacum direction k,thenthecondition (22-26) implies that =Oandhence m=-k1 ‘only appeats, sothat y= mm kh 278) Asa special case,wenotethatifchefinalstateischeground state,withJj=Dy=0,chenme=—m.Forexample,ifm,=1,thenmg=—1andhencetheF ——polariacion vector fortheradiation is(ey+iey)/-/2 Theimplication is © thatifcheatom intheinital stateispolarized along thez-axis withmes=1,4thenin-adecaytoastarewithzoangularmomentum, theconservation ofthecomponent ofangular momentum demands thatthephoton caythisoff©The photon musetherefore haveicsspinaligned along thepositive z-axis, chati, icmusehavepositive helicity (helicity =+1),of,equivalently, iemustbelef — cicculatty polarized, Thisijustwhattheterm(ex+fey)/V7 indicates.r “The6integration givestisetoanotherselectionrule.Considerfirstthepeeved =0SeYau=W-/x,theangularintegration (22-71) involves 1 y=x VefRYnlO8)Yell)=zsbisiim279) :whichimplieschatheintialsatemusthareI,=1.Iniydrogen.thedominanttmunsitions t0theground state willbe»p>Le q ‘More generally, when /,and lydonotvanish, westilgetaselection cule. ‘Thedetivation, beyond thescope ofthemathematical knowledge about special fanctions assumed inthisbook, makes useoftheaddition theorem forsphetical harmonics, which reads Yi)Vi@0)=|Ba)Glsmemftemm)Yimsam(0d) F(22.76) “The coctficienes C(L, ms+msihytymyma)atethesame Wigner coeficienes +thatappear in(15-44). Thepossible angular momenta ontherightsidearejust i 354 Quantum Physics thosethatcouldbeobtainedfromtheadditionoftheangular momenta Iendly. ‘Substitution ineo(22-71) yields ‘at fAVI69),BoLyw+miLl1)Vintne(O9)=0 unless |Wm ht hy 0 (22-77) ‘Thisisthegeneralformoftheelectricdipaleradiationselectionrale i Al=1,0,1 (22-78) with theobservation, obvious from (22-75) that there arenezero-zero sransisions. ‘There is«furcher constraint thatcomes from patity conservation. Since Fisodd i ‘under reflections, there isanadditional selection culefortheelectric dipole erasitions: ‘Theatomicstatemusschange (22-79)parity Since partyisgivenby(~1),hisimplischatcheEvaluemustactullychange. ‘Thus,forexample, 3p—2ptransitions arenotallowedtoorderZa. “Tocheextent thatthe only penurtacion isthe coupling pace) (22.80) thereinospindependenceint,andhencethespascannotlipinthetransition.“This leads oche addtional selection rule as=0 (280 mentioned eater aconnection with thespectrum ofhelium. “The selection rules sated abovearenocabsolute.Theconservationasof sogular momentum andpasty (frelecromagnecc proceses) areabsolute, but(22-78) isonlyapproximately true.Transitions between states thatinvolve& changeoflagerthan1canaotkelacethroughthelecticdipolemechanis, ‘They ansllake place, provided there is«nonvaishiag mattx element @|e°** e-plge) (22-82) ForAl=2,thefirstpowerofk-rwillgiveanonvanishing contribution, We smay wate kere-p =}(epler +exp-k) +}(e-pk-r —e-rp-k) =depker +empk) +4x0:Xp) (22-83) “Thefitofthese tems icalled anelecic_ quadrupole cer, andthe second iscleatlyreited oan LBtem, andi called amagnecie dipole team. Forthese The Radiation ofAtoms 355 transitions, whose matrix clement weestimated tobeZrines smaller than the leading term, wewillhave A!=2,and, since theoperators in(22-83) areeven, there willbenopasty change between theatomic states. Transitions becween 3d—1s,forexample,cannotgoviatheelectricdipolemechanism,butcango viatheelectric quadrupole mechanism. Actually, iftums outtobemuch mote probable thatche3dstare decays fstinto#2pse, andchelater thenunder: {oes thefavored 2p—»Lstransition “The spin selection ruleAS=0(00, isnotstced. Inaddition tothe‘coupling(22:80)thereisthecouplingdiscussedinconnectionwiththeanoms-Jous Zeeman effect { We=cadSBE) (22.84) ‘Thematrix element forAS#0cransition-inducing term canbeestimated. We compare itwiththeelecic dipole matrix element - (eg/2me) fkXe|_Rik} fasme*(Zar)* : (eglamoBikeehlllhemelee 7,an. (2e/me)\p-e| |p| plemc(Zat) 6) and seechatitissuppressed, justlikethemagnetic dipole matrix element, whichitstronglyresemblesinform.Asanexampleofasitationwherethecoupling(2284) plays animporcne rol, weconsider thenuceat process ofphoto- Aisineegation ofthedeuteron 4 ytdantp (22.86) The deuceron, toaverygood approximation isa*S,state. Anelectric dipole transition mist involve thefinal(x—)syscem ina'Psatesince AY=1and AS=0,Ittutns out,however, thatjustabove threshold forthereaction, the ‘vonucleons azeunlikely tobein«eelative Pstate. Ingeneral, particles will©beinatelatveangularmomentum 1.statewithanyappreciablepobabilcyonly if [paz AL (22-87) whete pistherelative momentuun and4aethedimensions ofthesystem. For thedeuteron ittums outthatfor7'sbelow 10MeV inenergy, the(w—p) system isunlikely tobein«Poste. Theaddtional coupling, —aigote+0):B (22-88) ‘an, however, lead coatransition berween the*S,sare, and theunbound 'S, face. Theintetction may berewrite inthe fotm bi. =sghGo+allFD+A —Bll—AB 2.09) 356 Quantum Physics | ‘Thefirseterm issymmetric under them+»pexchange,andhencecannotcon- | tribute co.ntanstion between symmetsc andansncayrameric spinsate‘Thesecondtmdoes,however, contibutes Thecovets areacyquite | lange,sincegp&5.56andga=—3.81. | Tharsis oneselection alecatisscred, andthcischeoneforbidding 2ero-rero transitions (referring tosotalangular momentum j=0)inone-photon |frocence,Agenerwayofatguingteabsoluenes ofthisselectionruleithe |Fetiowing: Themata clemene, 2salar quaniy, mast involve thephoton fobabaton Tnety andmusetherefore beoftheforie-V,where Vissome Perottateneriotheproblem,Iteinitalandfinalsaeaej=0ses, Thatihavenodzecdonaligy associated withthom, thensheonlysector isk, thephoton momentum. However ek—0,s0thattteis0wayofcon” Seruting »rat element, Temusetherefore notexis? D.The 2p 1sTransition Letusnowspecialize tothetransition 2p+1sin(22-68). Weneedto evaluate theradial itera, farPR)Ral) - EN aye\f1Gsene [oeole(2e la (@)"« L(z\ fr-1(2 deh eRevale) 1(zy2s)f° 24(y=~ (2) (24) [awe Are @vale)Gz)ieeevel)a=ea “The angular inept oad1fz( retieYoceF Yim=<j|AD ‘atFe Mia JarierinsSe/$(oat ee setigoy) Yo =(tee tOntvaline ye) (22-91) sheion«ok~0indepenenofthehiceofgut,and+sateen out tent acctomayie fell. Schspunens "byerent aaa i dhcnar pure a, nh thetec so ey tao, “The Radiation ofAtoms 357 [Now thetbsoluce square ofthe product of(2250) and(2291) is 962)"(SYfleet+aoa+mer+aG20 SY’(GYfeet+Gantmeme+0): ‘sothatthetransition rateisforagiven m-value ofcheexcited atom, wy8gt(ey torofoto)sae(2) XBacet +4Gast det +oP) 2293) F where LLLsae(1—2 3m os: “Ek(Za) (22-94) Bis chefrequency oftheradiation emitted inthetransition. “Theangular integration in(2.93) isoverthephoton directions, andthisisaot | trv since «isconstrained «0beperpendicular tothephoton momentum Bidirection. Theintegration isverysimpleiftheinitalp-scateisunaligned, thatis,{——icoccus inthethree possible m-stces (w= 1,0,—1)wichequal probebilcy The nates thea ii Ran=5SRepu) (2295) * Since ie a2Bane+HOa+ba+GD]=aGPtat=1(22.96) Fe incegrand becomes independent ofchephoton direction. Thisresultshould {alto bemultiplied by4factor of2.Thereasonisthietherearetwopossible 4 polarization states forthephotoa, andwearedetecting othofchem. Amore& ‘carefulwayofwriting(22-51)wouldhavebeen, fatemeE,[Gole™ pla? 297) . with \denocing thepoarizttions. Thetwopolatization sates ateorhogonsl, so chat we hve . 22 =by (22-98) 358 Quantum Physie When allofthis isputtogether, weget aism, ya ktPee Ca (a's =EMEalta)206X10Zee" (22.99)‘Thisdiffersbyafactorofabout25fromtheestimate‘madein(22-64).Thus | ‘ered factorsinthemaceselementsaeimportantandguessescannotreplace i SGleultion,Nevenbles, dimensionconsidentions andapropercounting ‘ofpowers ofadogiveusanorderofmagnitudeguidancetohowlargephysical Gquandhis iaatomic physic az. “Theexpression fortherate aynBeESS eee (22.100) maybecranslated intoaformula fortheintensity ofradiation bymulkiplying itbytheenergyofthelight quantum fi,Thus é7 Oya ImByaD L(fleli-e™| (22-101) “This, however, ijusthecanal formula fortheintensity oflightemied bytnoscilatingdipole,ofdipolemorened=e(fleli) ™ (22-102) providing another ilustation ofthecotespondence principe. E,Spin andIntensity Rules “Theincisionofspindoesnoechangethingserymuch,eisrethathe inl ses andheSeal sates caneach bein a8"op" ora“down” spinstate,putsincteineratiniatomicexostosis spinindependent, only“up”—> "up" and“down” =»down’ tansiionsateallowed,Fencecheeansitiones ‘willnotonlybeindependent ofmz(aswesawinthelastsection) butalsoofm,, Tidhence, mi,With theinclusion ofspin-obit coupling, here willbe Sal{ontheseeofthe2p-~1senergydiference)levelspitings.Forexample,thefron andn2levelstructureschangedashowninFig.22-2,Thespel Tincomesponding tothevanstion 2p» 1ssplititotwolines, "Pua —> Syaamd Pan >Sin Frhespi sates, heracial itepal andthepaseSpaceatementunchanged.andbenefheraiofhinaofheleomatermfomthegularpartoftinalne,thaaryfromangersomeon neato “The Radiation ofAcoms 359 27h —— » He T8i0 ¢Fig.22-2.Thesplitingofthe2p~speclinebyspin-orbitcoupling 4 ‘Thetable below liststhewave functions forthestares inquestion, 3 ‘odparity even pacityJ J m= Tt ra 3/2 3/2 Yaxs _ - 3 V2 Vai Yot-ViS Yin 32 AR MAB Yom VIB Ya = a2 3/2 Yuax - . V2 12 VifYee—V2Yux Yous j 2HWV2/3Yin—ViVx.You a Inthesquats ofthe mass elements, theal pceaecommon talof them,Thus,inconsideringtheratesforPsjs—>Siswemustaddthesquaresof| thetransition matrix elements form;=3/2—+mj=1/2, my=3/2 my= $ 1/2, ....my= —3/2—9 mj=—1/2,whilethetateforPry—>Sysinvolvesthe ésum ofthesquares ofchematrix elements fotm,=1/2 m,=1/2... mj=—1/2—» m;=—1/2. This canbedone ditectly bytechniques thatsre 3 ‘ite sophisticated andbeyond thescope ofthis book. Onecan,however, wotk ‘outthese quantities indetail, using thefactthatthespin wave functions are onthonotnal r Pan Sus ma 2m =12 nie] Yee)|? =6 i sa —1/2 0 sincex}x- =0 ; a+ 1/2 AVFBYol-e|Yu)/?= 0 Qn=0)Y24-V2 IVA Yulee! Ye)|?=6/5 <2 2 |VAPYicalr-elYo)?=G3 =Ya—-1/2 VB YoleelYa)l*= 0(m=0) 32+ 12 ° . 32-12 1Wialeel¥e)i =¢ 360 Quancam Physics IfwesumthecerswegetDee’ (22.103) Similaly Pua Sue y= 1/2my=1/2|<173Yuoye-¥|You)|?=0 V2-1/2 {(-V2/3 Ynje-r| Yoo)|*=26/3m21/2|V2BYast¥eo)|"=26/3Hin-1/2VBYiele-rl¥u)|*=0 Again 4Ee-% (22-108) “Thus theratio ofthe intensicies is R(Pur—Sin)_80/3RiPun—Sim)_86/3_ an RP Sud)473? (2209) “Thereascnforsummingoveralltbeinitialsacesisthatwhentheatomisexcited,allthepevels areequally occupied, sincetheitenergy difeence is30tiny Compared tothe29~Lsencegy difference. Wealsosumoveralthefinaltates Sfrwepeafor anexperiment thatdoesaotdiscriminate becween ther,asiche ‘aseforaspectroscopic measurement, InOutcalculation ofthe 2p—>1ttansi-ionrate,weaveragedoverteiian-sates.Therewewereconcernedwiththeobenofasking“IfwehaveNatomsinthe2pstates,howmanywilldecrypersecond?” Theaveraging cameaboutbecause ofthefactthatundermostEfcamatances, whenNatoms ateexited, about N/3gointoeachoneofthe fn1,0,=isates, Here, thefactchittherearemotelevels inthePystate thanthere ateinthePyastateisrelevant. There willbealcogethcr sixlevels,ourwithj=3/2andtwowithj=1/2)andtherewillbeontheaverageN/6‘atomsineachofthestates. Thefactthactherearemore atoms inthe j=3/2subsetoflevels justmeans thatmoredecay, andthattherefore theintensity will belarge. Problems 1.Abydzogen atomisplaced inanclecrc fieldEl)thatisuniform and hasthetime dependence B= 0 <0 Fhe bo Tetalain toast ‘Whar istheprobsbility thatas¢—+ =,thehydrogen atom, ifinitially inhe 2.Repeat theabove calculation with chetime dependence oftheelectric ! fieldgiven by Bi)=Eye § [Hinz Asa firscstep, modify Eq.22-9 appropriately.) Discuss your resule when 7 3.Consider aharmonic oscillator described by : Ct eeae Fwhere by wf)=a+bocosfi and boKon. Calculate theprobability that atransition occurs from theground state, asa (|A]0)=h/2V2me forw=2 : Canyouderive thisformula using chematerial from Chapcer 7? 4,Suppose aparticleofrestmassMdecaysintotwoparticlesofrestmass mandms,respectively. Usetherelativistic relation between energy andmo-‘mentumtocomputethedensityofstatespthatappeatsin(22-57).{Hint There isonlyoneindependent momentum, sayp,andwhat isneeded is sve ) Jeet(towBe i 5.Consider theabovecalculation whenthedecayisoftheform i A>B+C4D 4 with particles Cand Dmassless. (Hint, These atenowtwoindependent momenta} 362° Quancom Physics ‘butdoesnotmake anytrnstions. Tobespecific, consider theground sate, sothat Hobs =Exbo LetV(0)=fi)Vwheref(@)isslowlyvaryingfunction,asshowainthegraph,IfthegroundstateofH=Hy+Viswo,thetheoremstatesthat (ely fo “Thestepstobecatiedoutaethefollowing: (9)Show that Lf peortennBfaypae-noeing (yy xi. 40>aa forcimes#suchchatf(t)=1.Usethefactthat fe), Fat BEP< TO! Either construct anexample ofafunction /()oFuseintegration bypars, che i,wre La wattew iinthe above. (b)Calculace Y)wsing (22-3) and(22:5). Compare tiswiththeformule (16:19) which here reads waeantL,Galblts)emtetBt and thus show chat [esl ¥0| 2 7.Work outthe2p—+1stransition rateforthethree-dimensional oscil- lucor, following thesep eatied outinthischapter. &Nocet sometimes decay from excited states totheground state byinternalconvertion4processinwhichoneoftheLelectronsisemittedinsteadof photon.Letheinal and final clear wave fonctions be ‘The Radiation ofAtoms 363 or(t, Be, ste) and pelt ts, st) where ri(i=1,2, «Z)desctibe theprotons. Thepereurbation giving tiseto thetanstion jut theauceuselecton imtencion ¥--Rien I where ristheelectron coordinate. Thus thematrix element isgiven by y-beftn...teadtGeoeoval) / Oe Bra (a)Whats themugnitod of,thefeelecaon momentum? (b)Caeulue theefor heproces for digee tami ies of a aa=Zfon. endbre, bymaking useoftheexpansion Lot ien Ct lye 2 ipo te ine, Theintegeleanbeevaanedusing F Seethediscussion ofthephotoelectric eect inChapter 25. References ‘Theadcon theorem that leds tothemore general dvvation ofselection ‘esis dicused allo themore adtaned textbooks ted ch en ofthis folume, andaso i ME Rose, Hemetry They ofAngular Momeni, Jha Wey &Sons, 1957. | Theta incegals fothemore geoel ease tediscussed in ;HLA.BetheandR.W.Jackiw,Inermedicte QuantumMechanics, W.A.Benjasnin,e198 HLA.Bethe and E.ESalpeter, Quanta Machen: ofOn: andTw Been “At Springer Ver. 1957B.U.CondonandG.HLShorey,TheTheyofAvieSpectr,Cambeidge ).Caiverty Pres, Combai, 1939 |chapter23 }Selected Topics in Radiative Transitions c A.Lifetime andLine Width . ; “thenumber RG») thatwelened calcula Caper 2etesene theputa frteeeotion of dvd bythe tneturing which the perturbation hasacted. Thistimeraustbelongcompared t0f/(Eq* ~Ex+fs) renee tearation ebbepropriato,betcaycannot seen ioe snkforherbably cateial seteeen In, wee Pa)=1—[Eres ay where thesum isover allfinal stares thatareaccessible. ‘This clearly hasno wa ioleageoongh times ince probable wepostive, Ieune ox tre deceaon ofthe tine devopment ofthe sem itdone mote ‘ately, theaitcanbeshown thactherightsideof(23-1) justrepresents an ‘Hrounten (lowes order iathepert) tothe cote eentpnonlyeeforongtesthat Pa)=ep[-1Swop| (232) Covemaythsspeak ofiftime ofthe inal sae 14 T=EN (23-3) e 365 q 366 Quantum Physics| “Thetotaltansiion eteRisthesumofparal taasiton matsinothepossible ‘hal tntheexample thatwasGscused indeta he2p—+1canstion in {Rydtogenice atoms,nootherchannels aeavaiable, sochattheierimeofthe {2psateis | 1=16X102-450 (4) “Thisisinexellent agreement withexperiment, Letuscompare thi(wetke 2=1)with thetine takestheelectronco"gooncearoundthenucleus.”The : velocityiae,andthedistancesoftheorderof3X10cm,30thatthechar-Meese time isofthe orderof1X10"sec.fncexmsofthisime,the2PState isverylonglived Sincehe2psatehasaiteLifetime, icshould, bytheunceretingy principle,haveanuncersingyintheenergy,ofmagnitude Bare * (35) “Thewayinwhichthismanifestsiseehatheintensityoftheline,a8afunctionoffrequency, isoecompletely sharpathesaluewe=(Eap~B)/Abut fetus «discibuton oftheform R2 Ne)«22 23-6 Me)=at RIA 8) ‘NotethatintheimihatR—0,hai, intheimihaperzrbation theory is srcly applicable, weget,abaconsequence oftheformals Lim——* =#86a ae) inooae > ear thelineshaperepresented bycheenegy-conseration delsfunction. Thewidth‘feeLine(25-6)8Randthissameasureoftheuncersingyintheenegy- “Tislineshape, sometimes calledtheLorentzian lineshape,inocwhacis geoenly observed sincetereareothereects thabroaden it.Theres (a)Colin Broadening. Ovedoesnotobserve singlearominisolation,butegosofhorstomaIntheguschetwlloccuolisionsbetweentheatoms Tfwedetne «cllsion tine7somemeantinebetween collision, aadif yen theninefce thefete ofthe saewllber.,andtheenergy un-—* ceeainy 8Tegeesroughestimate ofthecllion ateRe(=1/72),consider oneatom1ost,Ifisefccive acais(checollision csssection thawillbe “focused inChapect 24),theniwilbebitbyanother atomwithin 1sc,ifche casbads elf sideacylinderofvolumeov(ig.251)fthereatearoms/ tint theeunber ofcolisions willbe Rem me sec! ese) SelectedTopicsaRadiativeTransitions 367 (iweeeeere @: Fig. 23-1, The number ofcollisions petsecond forparicles moving with velocity» normal reatve tothe tape £Toseethedependence ofthisonchepressureandtemperature ofthegas,we ttethekines theory relation x mot=3hT (23-9) andtheideal gasaw nak . : £ (23.10) where b=1.381X107"exg/deg isBoltzmann's constant. Thus ifforin ® (23-8)wetake()"%,weget t ey" pat E neB(2 un) ® Tfwenowwrite m= 16 10MgmsothatMsthemolecularweight P= 0!pe where puis thepressure iatmospheres =X10"DEwhereDistheatomicofmoleculardiameterinAngstroms then =De 2 R=34x108Lo 2) ‘Thecollision catecanbedecreased bydecreasing thepressure, sothatinthe laboratory (incontrast tostellar surfaces) collision broudening canbecontrolled. H (b)Dapper Broadening. Even stlow pressures, the midating atom is moving quite rapidly (thegeoishot) anditsfequency isshifted. If risthe‘velocityoftheatominthedisectionofthelineofsight,thentheshifcis domo (2313) 368 Quantum Physics Incerms ofchetempertute, given by ett was) wehave | 203x104(3) (23.45) ‘where Misagain theatomic ormolecular weight (M=1forhydrogen, M=4 forhelium, etc.). Thus, thebestchatwecandoisobtain Baws 3.16) whereas forthenatural linewidth, thisis~3 10-6 B.The Mossbauer Effect ‘Anatom (ofanyother quantum system) cenactasaveryaccurate clock,sinceitsaansitionsatsignaledbymdiationof«verywell-defined frequencyIftheonlylimitacion werethenatural linewidth, anaccuracy of1:10could beachievedinatomiccasitions.Asnotedabove,theDopplerbroadening reducesthisto1:10 Onemight think thauseofaliquid ofslidsource would eimi- tutethis,butchen broadening caused bytheeffect ofneighboring atoms is justasharmful. Onemight examine nvclear crasiions. Anucleus suchx6 i lt” emits apray ofenergy ~100 keV, withalifetime of10% sec.This comtesponds t0 Be 8B Bir 107 0 .ET 8IK 16xToe EMIT HHT) ‘There will,unfortunately, beatecoil shiftofcheline.They-ray cacties off momentum hi/e, andtheaccleus, 0conserve momentum, mast recoil withthe same momentucn. This gives ritetorecoil eDe:By Pia1“ey and thusadecreaseintheenergyradiated,Thefractionalchangeinfrequencyis FeTea2Mc!~~2X940X191(MeV) ead (2319)“Theobservation ofradiationofthisenergycannorbectredovwithche Seecrod Topics inRadiative Transitions 369 conventional, extemely accurate spectorcopic methods, butmust utilize « detector that isextemely “well tuned” totheradiation. This isbest done by ‘using chesame material (e.g, rit!) asanabsorber. Theabsorption willbevery ‘much enhanced atthe“resonant” frequency atwhich theradiation isemitted, ‘buthere, too, there willbearecoil shift. The overall shift isthus Aw/w ~6X | 10, Ths, the“fine caning” does noework, since thelinisshifed byfarmorethanthewidth,whichisoftheorderof10-o.Onecouldaycocom-pense fortherecoil bymoving theemitter with thececol velociey. Ths is x Biven by 3 a Paoot fol fioif aPrastBalt9Be6x107 (3.20) ©thatis,v=1.7X10cm/sec. Thispresents techaical difficulties, butithasbeen F adhieved with anuluacenuifuge 4 "Amajor breakehrough came with thediscovery byMssbauer in1958 that Funder certain conditions there isahigh probability ofreciles emisien, Theemission isnotrecoilless, ofcourse,buttherecoilisnoctakenupbythenucleus, : ‘butinstead byalargepartofthecrystal thattheaucleus isimbedded in,SinceEthemassofthenucleusis10timessmallerthanthatofthecrystal,therecoil ‘energy iscompletely negligible. Togetsome intuition about what ishappening, Jeeusconsider thenucleus asmoving inaharmonie oscillator wel, with char- acteristic frequency we.The energy levels oftheoscillator are B=fay(=+mtd ) @3.21) E ‘Theharmonic well isjustanapproximate description ofthecrystalline forcesthaareresponsible forthepropertiesofthelace.Iftheforcesthattctheaucleus toitsneighbors atestongif the“springs” atestiff—then ayislarge;ifthe“springs” aresoft,then«issmall,Intermsoflevelspacing,a"stiffspring” P: hhaswidely separated levels, thatis,alowdensity ofstates, whereas a"softa spring”hasahighdensityofstates.Letusnowconsiderthematrixelementfor 1transition from anucleat state described by (ts,ts,....t) toanuclea.state Esdescribedby(es,#3,..rw),andwetaketheinteraction tobe 4 ~FiePeAwe) (23.22) “Thematic element then isproportional to ;-<ff.dey.eet en)Sepee Hea t8) 23:28) 370 Quantum Poy Ifweintroducethecenterofmasscootdinate R=(1/N)ID«¥.then(4)the |interstion tem takes theform | ee “See * (23-24) | where ¢¢=#1—Rand ()theeuler wavefunction decompons ito prod- Uredesingchefrenmotionandthemotionofchecleatceceofmass inthebarmonic potential We,-.BW)=Peano(TR)$4015---OW) (23-25) “Tos themattis element (23-23) becomes a xfee+Pew-rdyler-- odDepre™POC,---ex) (23-26) ‘Wemaywrite hisinchefoam M=Mans{RR) *%HR) oa wherewehaveseem,=0,sincetheiilstateisinthepoundstateofthe Ince, Theprobably duetheradiative transition leaves thenucleus inthe lnvice grown! sacs iat|ferneo) Pub)= -,{Miac* fons (RD =|fermen oor (23.28) lnthelaststepweteplaced thesuminthedenominator byunity, using com- pleteness.! Tocalculate this,weusethenormalized ground statewavefunction, +ThefrauproofsquicheWebreErwimmonee EelemianwieMey sing r=Dien oveses Oe“®ARO)=1 Selected Topics inRadiative Teanstions 371 q ‘oftheoscillator. Wefound inChapter 7thatcheone-dimensional ground statei wave function is. 3 (2)=(™)pomncrnn: vols)(2) & Hence, forthtee dimensions wehave i Yul)=Helo)Yul)Yale)=(2)“~-m (25:29) eh ‘Wethuscalculase * ‘Marion foopeven inn |®[(Sae)"fener where Myisthemassofthenucleus. Weget : ‘Muce\*| .q=(Ma)fcop,e-cemavannatnHHA : n=) fon | a weMatas =exp(—Sevil_enesay’ (23-30)=o(-Ee eal ‘since Prcoa =Mand fusyisthelevel spacing inthelattice. Thus, ifthelevel spacing islarge, thais,wehaveastifspring, recoiless emission becomes ore probable, Themodel ofthelatcice thatwasused here, chacofeach nucleus‘movinginitsownharmonic porential, istheEinsteinmodelofalartice,andthefrequency ayistheso-called Debyefrequency, sothatweshould really ceplaceoy byap,whichis related eocheDebye temperature Toby fuan=bTy (23-31) © -A-more accunae treatment oftheltice usingtheDebye model fritsdescip- tionmetely changes theexponent byafactor of3/2 2Teisnotquiteconecttosaytaethewholecryscalrecoils;instead,in 4 time +equal tothelifetime ofthetransition (1.4 10-? secfotFe"), only & | —_tegion ofthe crystal ofmagnitade : Lane where tisthevelocity ofpropagation ofalatce disturbance, (ie.thevelocity‘ofsound)absorbstherecoil.Now¢reasonableestimateofoisgivenbY . na 372 Quantum Physies; where aistelatce spacing. Thos| Ler; oe andwithup210!sec“,thenumber ofnucleabsorbing therecoil, ~(L/a)* issil enormous.I “Teaboveestimates, combined withtheuncertainy reason, maybewsed : toshowhacivisnoepossible rodetermine whether iis2singlenucleus cht rally” secels. Tomeasure hececilenergy#°H/2My takesaGimeofthe onder of: f &>TEM) “Thecondition fortheMéssbaver efecto occur isthat ie Fie<foo Hence 1 are Dating thatimechedisturbance willhavecavelled adissance dxnuw~ are thacis, overadistancecovetingmanynucle “Thequestion aisesofhowdidwemanage togetawayfromtheproblemofteceentmomentum conservation bytalkingabouttheenergystatesofthestews inthecrystal latice? Whece doesitsaychatthecrystal absorbed thesrsmestum? ‘Thequantm mechanical answer ischat,ifwewinetoelkabout ‘momentum, weshould workinamomentum repesearaion. This,however, i Romplated iceiisdficul todescribe checystal forcesinterepresentasamtapnatoamustdoistodecompose thecxysal mocion (thecrystal iusexdactescltors withneaest neighbor “springs") intonormal modes end Uquanie theeThequanta ofthelaticemotion, analogs ofphotons, arethereson, Recolesseossion thenmeans tanstion iowhichphonons arenotanced "Theresulting Formula isverysimilar co(2330). Under theseicun- Srances, chesecobroadening isinfnitesial compared cothenatu inewich ThunissllDoppler broadening because ofthe thermal motion butehiscan betandied bycooling theemitter andabsorber. TRecolesscmiters peovide uswithasuperb clock, andresearch sizing cheMossbauer fect hasbeendoneinmany felds, suchassolid-state physics Seleceed Topics inRadiative Transitions 373 andchemistry, Wewillmention justoneapplication, theterestrial measurementofthegeavititional redshift.Wenoted"thataphotonwillhaveitsfrequencyshifted by a boale) (2332) ifiefallsthroughaheightx.Thiscanbecompensated byarecoilofvelocity#,S wheret Pn ae (2353) (Ifchephoton andtheabsorber were tofallfreely together, there would be resonant absorption.) IFtheabsorber orthesource ateallowed tooscillate;rapidly—oneusesatansducer—and theabsoeption curveiscorrelatedwiththeoscillations itispossible cocheck thegravitational shift. Since thevelocity, fora[seperation x=20m,isoftheorderof~20m/sec,theexperiment isfeasible,andwascartedoutbysevealgroups.Withintheertors,theeffectisconfirmed. Forexample,forFechepredictedshiftisAw/e=4.92X107,andthe Fexperimental shift found byPound andRebka is(5.13 +051) X10". A simile experiment inwhich theenergy shife ofthe pray emicted byFe!accel- entedonrapidlyrotatingcumtablewasmeasuredagainyieldedresultsin Fagreement with theEquivalence Principle. ss C.Induced Absorption andEmission ee Inourdiscussion ofthenormalization ofthevectorpotentialappropriate#0theradiationofanatominEqs,22-28and22-29,wesawthatthematrixelement foremission wasproportional to(N+ 1)", where Nwasthenumberfofquantaiatheinitalstatesndthematrixelementforabsorptionwaspropot-tioaaltoNY,Sincethisreferstoquantaofaparticulareype,thequanticyNshould really belabeled bythemomentum fikand thepolarization state} ofthePhoton,chatis,NshouldbereplacedbyNi(k).WemayusetheN-dependencetodetive thePlanck Radiation Lav, thus providing @quantum mechanical justification ofPlanck's approachLetusconsideracavitycontainingradiation.Thewallscontainatomschatabsorb andemit radiation. Since there is«variety ofatoms, wich avariety of ‘encrgy levels, chere will beacontinuous spectrum offrequencies. Wewill concentrate on particular frequency, conresponding totransitions becween a +See theSpeci! Topice secon 2“The Equivalence Pile.”*1cisoneofthesubilevesofradiationitgraviaionaleldthasheDopplersie |istheraosifer’=x~#¥/)”%Onlyinthiswaywilthesicinanacelted BBGanebetheame,wheterthexbatbeiflingorsitingontheedgeofroxtingdis, +with he erie nthe center. 374 Quantum Physics particularpairoflevelsin«particularspeciesofatoms, thatis,wewilldescribetheatomashavingewostatesofenergy,iandEs,respectively withEx<Es‘When equilibrium isestablished, theseareasmany photons absorbed asthere suzephotons radiated.’ Theaumber ofphotons radiated bythewallsiseqcal (umber ofatoms intheupper state"2")X(transition matefor"2"~*"I"; thenumber absorbed isequal to(number ofatoms instate“1"") X(ceansition sate for "1"—>2"), that is, NiReaission =NiRadeorncion (23-34) ‘We also have Renivion =[N(k) +1Re 23:35) where Ro:istheemission rateincoastatewithonephoton. Weuse(22-57) to vite this intheform Lore, : RaepT Naat 2336) Herepstands forthe density ofphoton sates; wehavethesquare ofthe mattix clement, anditisummed overthefinalstates oftheatom, cati,the2+ 1ftagulermomentumstates,andaveragedovertheinitialstates.Thsisexhibited‘explicitly—the sumisoverinitial andfinalstares, andisdivided by2s+3,the umber ofangular momentum states forstate“2.”Theteason foraveragingCovertheiilstateischatwhenchesate"2"getsexcited,chenallthestatesthatonlydiffer bythem-value willgetexcited withequal probability. Onlyone ofthestates isexcited atatime, andthustheproper counting isdone when Iwwesumoverallofthe2f,+1statesandthendividebytheinumber.Notealso ithatwedenoted theperturbation byV*asthetermassociated withthetime dependence eForabsorption, wehave Rerworption =Nu(k) Riz (23-37) where RyoEY Jalvinylt (23-38) ee tlh ° ~ ‘Thedensity ofstares hereisthesameasin(23-36), sincewearedealing with only onefrequency. Furthermore, Tielvivlt= LL elvipelyiye =ZY alviayraiye2) = lawl (23-39) +One must convince oneself hati ispermisble coconsider equiivin forone leqneney tatine,we dohereThisbecomes pase whenweean thatheproba[SyatEnsoofwophoconsa+meismall20chattheunBeinthecavity Sa shape Bacar equetions Selected Topics inRadiative Transitions 375 “ThisassertionissometimescalledthePrincipleofDetailedBalance.Onthefaceof }ititisamidentity, butonecould imagine thatthepercurbation leading tothe ttansition "1"—+"2"isnotcheheanitian conjugate oftheperturbation that leads tothe tanstion "2" >"I." iawhich ease the above derivation wouldbreakdown.Iecusoutthatheprincipleholds,providedthatthecoalHamil-‘onian isinvariant under time teveril* The interaction ofcharges with the F——electcomagaetic fieldhasthispropery. { ‘Asaconsequence of(23-39) wehave i Rewincon MWFLYtre Ruswrnioe NAUK) feFT MM) +1 gs= SW+1 23-40)NAR) gs oe) ‘where gisthe conventional notation forthe degeneracy ofthe stae "i."Onthe ‘otherhand,weleafromstatiscicalmechanicsthatatequilibrium, theoccupa- i tion numbers oftheatomic stares N;and Njaterelated bytheBoltamann = facror Ne eM arNongetar™ gf (3.41) Heace Btpwn_Nt_Raosioe _Ni(6)_aa NiRenivon~NAOQHg bacis, . 1 ; S00=sot (23-42) The photon energy atthegiven frequency isgiven bytheproduct (number ofphoconscatesintheintervalda)X(numberofphotons)X(energypetphoton) |(@factorof2toaccountfortheewoindependent polarization stares).Thus 7 Vip the0)=Cane Vedat dh tho Ta do oY arh(w)) _v(2) ke eae . +Acca, inowes order perturbation theory, (23-59 almays does old 376 Quantum Physics “Togertheenergydensity,wedividebythevolumeofthecavityV.Mfweexpeessthis interms of»=w/2x,weget Beh 0)=Bat (23-44) | Inthepresence ofalarge number ofphotons ofagivenwavelength [N.Gk) large]transition tatescorrespondingtothatwavelengthwillenormously ‘enhanced. Thus ifmany atoms canbemised t0agiven excited stare, andcheproperenvironment ofthe“right” kindofphotons isprovided, thentheywill Aecay inaveryshoretime,thusgiving risetoanintense, coherent, andmono- chromatic pulse ofradiation. Thelaser(Light Amplification byStimulated Emission) doesjustthat.Under equilibrium conditions itisdificule toobtaina largenumberofatomsincheexcitedstatesfromwhichthetransitionsateto takeplace, because theBoltzmann factor «is verysmal, evenathigh temperatures, sothatspecial techniques must beusedtoachieve this. ‘Consider forexample, thehelium-neon laser.There, advantage istaken of thefactthatthe2'S.and245;levels ofhelium almost coincide with cercain sets oflevelsofncon,the(29)%5s) and(29)*4s) excited states, respectively (Fig. i23.2),Theheliumlevelsareeasilyexcited;anelectricaldischargeinthegaswill ifexcite many levels, andeheyallulkimately decay tothese states. Theexcited : helium atoms willcollide with unexcited neon atoms inamixture ofthetwo‘gasesandexsilyuansfertheirenergytochem.Inthiswaylargenumberofneon 2's cation NES ee | ie intl oer i wonton 235, cotson Geraniret Ser toon ees ee tare Denzstaton byfovea‘wunoF Fig.23-2. Schematic sketch ofrelevant energy levels inHe-Ne fase. Selected Topics inRadiative Transitions 377 exter ———==] Patera( Teom vehact ih voto Big.23-3.Schematicsketchoflaser. HB scoms findthemselves instares thatwould otherwise besparsely populated.z[Apopulationinversioniscreatedintheneon.Theseexcicedstatesdecayto(2p(4p) and(2#)*p)states,emittingphotonsofawell-definedwavelength. These 1photons aretapped bymirors, thus crating theproper environment forthe"nextroundofwhatisnowstronglystimulated emission(Fig.25-3).Intisway|,incense monochromatic andcoherent beams ofphotons arecreated. = 2 “Thetechnological applications oflasers aremanifold, andtheir develop- ‘ment provides justoneofmany examples oftheusefulness ofquantum heory Snot only fortheunderstanding ofnatural phenomene, butalsoasasource of ‘new, subtle technological tools References ‘The Mossbauer Effectisdiscussedinderailin Hi.Frauenfelder, TheMéshaver Effect(AReview with2Collection ofReprints), ; W.A.Benjamin, Inc., New York, 1962. *Aqualitative discussion may-befoundin V.B,Weisskopf, “Selected Topics inTheoretical Physics,” inLecures inTheo- ratical Physic, Vol. Il,W.B.Betti, B.W.Downs, J.Downs, Editors,Interscience Publishers, NewYork,1961. |chapter24 Collision Theory 4 ‘Atomic andmolecular structure waslargely explored through spectros- copy, When itcomes coying towaded mule foes sate be the |goin theincon ofdepen pes, theonly technic sua Shar caecing +my ofpues by sy oftages. asome sce | sgecncopy sso» fo of“seating” Thetom ine grocnd toe excited bysomeprojectile (iemaybeelectrons inadischarge tubeotcollisions 3withocher targee particles, asinheating upofthegas),andthenanoutgoingForniobecithchestorgoingintthegroundeteyunoxpony £.anotherexcitedstate.Wedonotusuallydescribetheseprocessesas"collision iprocesses” because theatom hasverywelldefne2 energy levels, inwhich it (stays fortimes tharareenormously longcompered tocollision times," sothat “Pics possible tosepucate the“decay” fromtheexcitation process. Iaparciculat,‘.thecharacteristics ofthedeciyarenotsensitive totheparticular modeof fxcinion, Innucle andasofacementary pies, he ent lvl, bo F —frequently thelifetime isnotsufficiently longtowarrant 4seperation into exciton andcea, epecaly sinceaccompanying the“roan” satetog threitssononesonse“bakground” setcegs aedtedacrangling of | thevoi eines completed. lnthistaper oewilele ds he proces as whole A.Collision Cross Section Theide waycoalkabout srerng isto female equations that deve cal wshapens ainact rie descbed ysoeace tpprcacies heages navepacket must beSaige sta dos Drspend sppecely dung texpeiney, tod maeBeagecompet 3 +Recallchatthelifetimeofa2phydrogen stateis1.6x10-*sec,whichislargecom. 380° oan Pays “withthetargetparticle,butsmallcomparedwiththedimensions ofthe labora- Toytuefiemamnotsimulancously overlap thetargetanddetector. TheIe imesions arin fat,determined bythe beam seithe acceso, ‘Tho followsanitersconwithhetarget,andSaywesetwowavepacket ‘onecontinues intheforwatd direction, describing cheunscatcered partofthe |fm,andhcoteisofatsoneangleanddescribesthesareparties ‘Theaumber ofparticles scarered intoagivensolidanglepetunittimeandunit incident fuxisdefined tobethedifferential sauering creisscion, Wewillnot , follow tisspprach dec? burwilincad wesomeofthe acti de- ‘eloped inChapter 11toobaiathedifferential crosssection. Wewill,however, |Keepthewavepacket ceament inindaweinerpetout formal ress. ! Tnoutdiscussion ofthecontinuum solutions oftheSchrddinger equation iinChapter11weconcludedthat:(a)AsolutionoftheScheBdinger equationin itheabsenceofapotentialistheplanewaveform&*",whichdescribes«fux ‘ i ik 1 i-limWw—9H")=a (24-1) 1 If-wechoose ktodefinethezaxis, thenthelargerbehavior ofthissolution itnaybewten(11)inhefunofaincoming+anoutgoingspecial | ie ney eet :PretBartnef AEpeony cen (b)Theconservation ofparticlesforcesustotheconclusionthaschepresenceof |Pail pocenal canonyaethisosfonction, whose aspnpotc Frm | be tly flees iwoodd atva[E" -saeT]Paco) 43) i subject to } 1s) =0 (24-4) “Theasymptotic form(24-3)mayberewritten,withthehelpof(24-2),as Hy wore s[Easne rm] cos cowrespondngtoplanewave+10ousgoingsphericwave!Notthatweate cranewltheeerieoncpieSchringereqion,sodgmche teducedmassand@isthecenterofmassanglebetweenthedirection ofk(theTEunGtadthetapmptopot,whee,rsualythecounterwilBesep. hii doevarysen Hai, Arian raofPh3,857962 theee an stn hoa Sense ach Cnn wy ld onong wine, + Callision Theory 381 When thetarget ismuch mote massive chan theprojectile, there isn0dis- tinccion berween thelaboratory angle andthecenter-of-mass angle, Thekine- maticsareeaslyworkedoutusingthematerialoftheSpecialTopicssection1. ENore also thatwecould,ofcoursehavesetupsolutionthathastheasymprotic formofaplanewave+anincomingsphericalwavesinceitisthefrsttermin (24-3) thatcould bemodified byacoeficient satisfying (24-4). However the solutionchatdescribesthescaringistheoneinvolvingtheoutgoingwave. "/-Letuscalculatethefluxfortheasymptotic solution(24-5), RUTmee Ow) Lae ay I=FyLPFOS] veLA10S|~comptesconjugate (24-6) €. where we have defined 0)=¥(al+1)fib)Pacos6) 47) Evin J (fuk) =(SB) —11/218 (24-8) Galealating thegradient gives ;Ffmypa facegyMOM ina {{-+70)[ae +e :+10(iae9~complexojvgne}a x litte) incon 2 :aRear tn 5ep3 ; freien aqpeta 4=&f0)———+SP5~complexconjunc] “where wehaveleftout1/+4terms, andwhere wehaveusedk-r=&rcos6,in theexponential factors, Thus theuxis p-Hy FR ahb= + alors . +=4[reerie giggir~] \+#+[re inion+49)soon] AteLyepiece. pacgygitco aeA[roe9—HO)6 + te(FC)jarcrcou_26ioe|7 +EAPO pone 320, (uo 382° Quantum Physics “Thisater involved expesion simples considerbly whenweconsider chat fo0snceonenever does4sateing expetiment directly intheforward ‘deccsion,! andthaa'smeasurement onesways integrates theHuxoveasmal butficeslidangle Thusinthelasfourternsofthis expression weshould replace 8° by [sinesnoaom (2410) where(06)issomesoreofsmooth, localized acceptance function forthecoanteeNow,strs©wehaveanintegralovetaproductof«smoothFunc- ! Simandancxsemely epily varying oat,andthisvanishes faster thanany : ower of1/rThis1whatiskoown incheeathematical literate a8theRiemann-Lesbegoe lemma, andthereader canconvince himself thatthisis indeed sobyworking outanexample, with«gaussian aceptance Fonction, 4. . ‘Thus, onlythefisttwotems remain, sothat jeBlo eau)ata Intheabsenceofapotential,onlythefstrtmischer:itrepesentstheincident ifax.Inawave-packet treatment, fk/mwould bemaltplied by«function chatdefines thelatent dimensions ofthebeam. Thus,ifweaskfortheradialflax, ; Tehenthartermgives+conwibutionNkt/m=BkcosO/m,butonlywithin 1dite regionofthevaxis(eeFig.24-1).Sincechecouarerisputoutsideof thategion, cisfisttermdoesnotcontribute totheradialfuxintheasymprotic \ region, s0that jo=HE OE »jue EG (e422) : “thsthenumberofparticlescostingthearehatsubtendsaslidangleattheorigin (thetarget) is jee=BOa ous “Thediferent cross secon isthisnumber, divided bysheincident ux, im, thai, do=|f@)\*da (24-14) feepote asspindependence, theremaybeanssimetbal dependence, 0 thatnore geneally, deyap + '\yee)| cus) «owcoaloeetseatedomsclpanies? CatonTheory383, 4 .erent Ww - credyyy KDW = Brig.24-1.Schematiclayoutforscaterngexpetiment,Theseaterogangeisthe Inca angle = PF) Theruacrosssection givenby ‘ :ae cox(®)=fnt 2416) Ifwenowuse/(0)asexpressed intermsof5),andexpressthelatterintermsofthephaseshift(cf.11-41)5:(4)=2",sothat : 10)=£3@h+1)AMsin544)Pleas) (e417) then = en-falie t+1)HMsia4a)Paces| i=P ALe ‘ : : [FEer+0%sntotrcs] tnd wting ‘ soett fervicos 9)Pr(cos #)=Wi bu (24-18) |wget a ou=FE,C+1)satnaay 419) 384 Quancum Physics eisaninteresting factthat Info) = +1)Ime sin848)PAX) =PEe+aso) =Fone 420) ] “Thisrelation iskaowa astheoc! shor anditisuveevenwhen inelascc proceses cuaoccur, astheydoinaucleat andparticle physics scattering proc-fuses leisavery useful teltion andinwavelanguage ifollows fromthefact | tharthetotalcosesection ceptesents theremoval ofloxffomtheincident‘beam,Sucharemovalcanonlyoccurasaresultofdestructive interference, and |thelarcanonlyoccurbetweentheincidentwaveandtheeasilysearteredsre inthe forwatd dizectioa, Thisexpuios why(0)appears lineay. Amore Geuiled examination shows whytheimaginary pais involved.* ‘Therequirement that|5)()| =1followed fromconservation offlux, 4Accu,inmanyscatteringexperimenttheeisastonoftheincidentbeam;theaugermaymerely getexcited, ofchange itssate,otanother paride may | emerge: Under theseCrcumsaaces ourdiscussion isunchanged except che 548)=nde)2 (24-21) i isco beused, with osuh) <1 (2422) because wearedealing withabsorption. ‘Thepartial wavescattering amplitudeH isnow S@®)=i(k)2—1nusin2,|,1—mucos2h fee)=SadMDE 1meats AMAA(a4) andthetoleaecrosssectionis casMEC+0) = Lta—2910826, =aD +n as (42 “There isalso cross section forthe inulatc processes, Since we40notspecify ‘whattheinelastic processes consistof,wecanonlytalkaboutthetotalinelastic crasssection, which describes thelossofflux.Ifwelookataparticular termin (24-3), theinward radial fluxcarried by im ZT Moan See 11Sei Pro The Ph, (Kes), Hs281930 Collison Theory 385 j Cle]ca(2m)* (cf. Eq,11-36 andthefactthatYip=Pi(cos 6)/+/%). Theoutward radial flux is(ib/m)(|SK8)|#4r/44, a0tha thenecBuxlosis (HA/m)(e/B(L —2°)) foreach Lvalue. Hence, dividing bytheincident flax, weget LB aa=$Yet ypwe] (24.25) ‘ Thus thetotal cross section is io fn=644+ae =FE(E+1)(1+a=2m608.2 +1—nA) q =EEw+0meos2a9 (24.26) a Iralsofollows from (24-23) that Igio)=3+2)tmfle) e 1=eos26 =Dt ye cam (2427) {a0 thattheoptical theorem isindeed satisfied.3 nd)=1,wehavenoabsorption,andtheinelasticcrosssectionvanishes,‘When mi(#) =0wehave total absorption. Nevertheless there isstillelastic f seating inthatparcial wave. Thisbecomes evident insatering byaBlack dc. The black discisdescribed asfollows:(a)ithasawell-defined edgeand(b)itis| totally absorbing. Since wewillconsider scatcring forshort wavelengths, thatis,largeA-values,condition(a)specifiesthatweonlyconsiderpartial (|waves |SL,where L=ke (24-28) and aistheradius ofthedisc. Condition (b)specifies thatn((&) =0forthe i selevant values of/<L.Thus God=FEED =Eaat (24-29) and we ‘ vaBE OtDmwet (24.30) ses ema |jredaudd ane | liMELEE ELL: Geet=Fel+inet=2ma® (24:31) spelen ips ene Thal eo rly ei pnt mh er soe re i ing i laaincidenebeam(Fig.4-2),andthisleadstoashadowbehindthedisc.Faraway, Se on i eh yn oe Ete ey mn hc cg nh an seine ei oi i ia eT Sama ie et woke oe toma,Theelastic scattering thataccompanies absorption iscilled shadow iisconfined canbeestimated from theuncertainty principle: anuncertainty in siete a ei nt me — - o~os~a (24-32) we|opt Xe % 1084 P: ro&3 ii P : Boot Di f B 38 w? ‘fit ww + {| ° H oF ee eweMOSae Keo Fig. 243. Angular distribution of1000 MeV (1Be¥) protoas sae! by 0ude The anguse dissibation shows thedips that characterize dracon saci. Theepartures from theshape ofFraunhofer scatering inoptics idue fothefactthat mild azeaotsharp, nora they tally absorbing. The curve is che res ofatheoreticalcalculationthattakestheseeesincaccount.PromH.| Paevsky etabyPhy Re,Lats, 18,1200 (19657), bypermission.) i ae 388 Quancom Physics B,Scattering atLow Energies “Thephaseshiftexpansion(24-17)maybeusedtoexptessthediferencia crosssectionintermsofthephaseshifts de an f | SmEOI1AesinBb)Pleos (2433) | ‘Weexpect, ongrounds ofcomespondence withclassical theory, thatthe angulat |momentum involvedinthescatteringisboundedbypawherepistheceaterof- |mass momentum andaistherangeoftheforces.Thusweexpectthat '| 15be (2434) i Withthesumin(24-33)limited,onecamty,byfittingtheditferenialcross | section measured atanumber ofangles co2foem like BaEAlcor (2435) | todetermine thephaseshiftsforafinitenumber ofJ-values. Thereareambi- | Buities, forexample, thecrosssectionisunalered whenallthephaseshits \Change thet sign, butthese canberesolved wihthehelpofeheory, continuity |fromlowenergies,andothertricksofthetrade.Thehopeisthatonecaleara \ somethingabouttheinteraction fromthephasesift,whichformempiticl | datasomewhat closer tothetheory thanthecross sections do. |“Theconnectionberweenthephasesift(4)andthepotencalV()isvia | theSchrddinger equation; cheradial equation willhavea solution thatasymp- totiallybehavesas |fe 24)~siaar—ztup| (2436) side from anamplicude factor infront. Thus, given V(), aseaightforward way tocalculate 6,(A) iscointegrate theradial equation numerically tovalues of¢ thatarefaroutoftherange ofthepoteatial, andtoexamine theasymptotic behavioe. This is,infact, what onedoes, butthisdoes notgive usanyinsight intotheproperties ofthe phase shifs, ToJearamoreabout thephase shifts, we ‘consider thesquare wellpotential. Wefound inChapter 11,Section that c san(8) == (437) where theratio isobtained bymatching theintemal totheexternal radial wave function (11-63) ; Collision Tacory 389 Gilee)__lhe)+(C/B)nite) He, i :fies)~*jdka)+(C/B)milks) Gee) Sinwhich 4 tn Es pa20ee Bern wat 2439) Ene ‘denotes diffcentiation, withrespect totheargument, sndVo>0foran atanctive potential, Thus ; san84)=ibefled)~de)js)A 3)=Fala)jaa)—enh)jflea)=OM) Fthisisnocaparticularly ransparent expression, busimplifies isomelimiting aes F(a) Consider thecasechat j kek! een 1)Wedonotinsistthatxa<1.Withthehelpoftheformulas(11-25)and(11-26) Heget a ers 14s Vee) —najpa) Sar 4)flea)+eaten) OP) aftealindealgebea.Onecanshowthatforlargefthisdropsfasterchan«evenithe1.Thebehavioe i tan(8)~Bt (24-43) forka~»0isnotrestrictedtothesquatewellpotential,burstrueforallreason- ably smooth potentials. Iris«consequenceoftheceauifugalbatier,whichkeeps ‘waves ofenergy fatbelow thebarrier from feeling theeffect ofthepotential. ()Poecerain values oftheenergy, thedenominator in(24-40) will ish, sochar atthese energies thephase shift passes eheough +/2, ofmore ‘geaezally through (n+1/2) x.When thephase shift isx/2, then thepartial teave cross section eld)=FEY nsaay ren)e \ thasthelargest possible value. Onesayschatwhen tan&,(8)risesrapidly to infinity andcontinues tising from —=, wehave reonant satering. Tojustify thisteoninology, andexplain when resont seating occur, ltusconsider ba very deep potttia,aad alsolarge, s0hat . a> 1ba (24-45) 390 Quantum Physics ‘Wemaythen use(24-42) foran6:2), andthiswillbecome infinite when 4 1)flee) +exifiee) =0 (24.46) Since «a>>|,thiscondition isapproximacely equivalent to EDcosa£44)~sa(uw-Ete)=0= 2 2 bats, tea(Eg) wt (2447) Siacetherightsidesverysmaltheresonoceconditionis ey ceett (2448) . : tit ere oo *)aooroxinae |perooe js! Fig. 244, sketch showing shesquate well poten with shecena! baer‘Thedashedlinerepresentthenergyeveinaninnitesque wellfrangend theappeorimete locaton: ofthesemerng resonance eres ateindicted ndeigheTheloweronewilbechshapeshantheupperove Collision Theory 391 [Now this isjust thecondition (11-30) forthe existence ofdiscrete levels in« three-dimensional box, sothatsesoniat scateting occurs when theincident‘energyisjustsuchastomatchaaenergylevel.SinceE>0,theselevelsaenotreally bound states. AsFig.24-4 indicates, these arelevels thaewould bebound states ifthe burriee wete infinitely thick. Iisnot,butaparticle being scattered jus chesigh energy sill "knows" chathere isavirtual level thet. ‘As(24-42) shows, thephase shift isverydiayforkasmall. Nevertheless, 1shechanges andgoesthrough theresonance, 1rsesveryrapidly, incressingbywr;thustheparcialwavecrosssection(24-44)willexhibitaverysharppeakat 4fe thesesonast energy. Thisbehavior (Fig 245) isverysimilac cothecross sectionforthescateringofeleezonsbyHe"atcheenergycorresponding tothe(2)*‘excitedstate(Fig.18-4).Intheneighborhood ofcheresonant energythephase Shiftrises through /2very rapidly. Wemay represent thisbehavior by * ay ‘one (2449) tis adsothepalwaveossetion Ar(Ql+ 1)cant, 4x(2l-+ 1) brea apmOEY 8 ED eoBV Fuoti BB Be) +bbe (2450) EE2 |J.' \ 3H ' ' H i Fig.24-5.Thepartialwavecossectioncorresponding tothephaseshiftsketchedintheupper insert. 392 Quantum Physics . “This isthewell-known Bret-Wignerformala forresonant cross'secions. Again,thebehaviorisnotapeculiarity ofthesquarewellpotential,butischaracteristic fallpotentialschahaveashapesuchthatmetastablesatescansimulatebound SatesaboveE=Oinit.Wejustnoeforcompleteness that uy, bien jyat een! 2k an albak =ope ;Hi—fran8)”B—Bra(le)(40) Ifthere isnonresonant scattering thatisappreciable, thenthescattering ampli- tude isoftheform fill)=PW) +FP) (2432) Aclowenergies, thescatering iprimarily inStates, sothatwemaycon- centrateonJ=0.Itssimplertoderivethephaseshifedirectlythantoworkout (26-40). The solution inside thewel tht isregular atr=0is alr)=rR(r) =Csin er (24-53) and this isto bematched oaro u(r)=sin(ar+8) (24-54) thesolution outside thewell. Theconcinsity of(1/)(da/dr) atr=implies chat coca=kot(a+8) that is, (d/«)a0ca—anbe 08 a) antanbe (2499) Nore tha ifwedefine & ange =Acanes shen eng=BEM‘1+tangatanke(gaa) stat is, b=ant(Anes) ~be (2456) Collision Theory 393 ‘Wehave, following (24-39), 2nVoet 3(ea) =(hays + (437) a with Vy>0foranattractive potential. Thus, atverylowenergies, using%tan.x5forxK1,wegetbs anna 3éWhencagoesthrough/2(weimaginethatweateslowlydeepeningthepo- : tential well), which isjustthecondition chatthewellbedeep enough foraboundstatetodevelop(cf.Chapter11,Problem1),thentanxa—»©and(24-55) shows thar dand = (2459) {Fecha is,&goes through 2/2, Insense, abound state atzeroenergy islike& A resonance.& ‘Asthewellbecomes«litledeeper,weagainhavetan8~0(La),andig ‘continuity demands thatthebranchissuchthat i ESb=te(SL—1)(ooboundsae) :1a BS bertte(S—1) (withboundsar) —40)¢[Asthepotentialbecomesstilldeeper,asecondboundstatecanappear,xagoesthrough3r/2,andwehave8=2x+hal(tanwa/ca)~1],andsoon.Thereisa‘enezal result known 2sLevinson's Theorem, which states e (0)—He) =Nex (24.61) b where Noischenumber ofbound states, andtheabove isanexample ofit..‘AtverylowenergiesthecrosssectiononlyhastheJ=0contribution toit,and iis oFos(ME1)=seer(HEA)cer thatis,icis«constant.Theewil,ofcourse,beacottectionoforder(ka)tothis result.Ifweconsider neutron-proton scattering, thenweknow thithepotential ‘must besuch 48togivetheright binding energy ofthe deuteron. Ifwelet, \ na—he on 394 Quantum Physics and [a cn (effectively &=—a'forthebound stateproblem), thenthemarching ofthe‘wavefunctionoutsidethepotentialu(r)=AcothesolutioninsideBsinwrattheboundary gives scone =a (24-63) For k&x,wehave (22) (22), at (2468) 4Iran\6Dewwcn a Thus oSAra!(+2)aS(+20a) (2465) ‘Thus making thelowenergy approximation expressed by(24-64) allows usto 1 bypass theproblem ofdetermining thepotential andrbencalculating thecross section. ‘Theapproximation only works when thebinding energy issmall “Thequantity 1/aisthe distance overwhich thedeuteron wave function spills ‘over, andthisisalways much larger than therange ofthepotential afora loosely bound system. Icis1/crand nottherange ofthepotential thardetermines theseattering coss section atlowenergies | Inthe1930s there wasgreat interest intheform oftheneutron-protonpotential, sinceitwashopedthatthiswouldgivesomefundamencal cluescon-‘etning thenuclear forces ingeneral, Rudimencay experiments atlowencigies werefittedwithavarietyofpotentials. Itbecameevidentafterawhilechatalmoscanyreasonably shaped potential would work, provided that onechose thefppropriace depthandrange.Itwasshownin1947bySchwinger(andsubse-‘queatly derived byBethe insimpler mannet) chatatlowencegis iisalways 4good approximation towrite toi borb=—eb5ro (24.66) whereAiscalledthescateringlength,andristheeffeciverange.ThectossSection atthteshold determines thescattering lengeh ceed (2467) andcheenergy dependence detecmines theeffective range. Therelation between these parameters andtheparameters describingthepotentvarywiththeshape, buea two-parameter fitcothedacaisalways possible. Thisefective range formula Collision Theory 395 showsthatifwewanttoprobetheshapeofthepotential, wemustgotohigher energies. ‘Thebinding energy ofthe deucron is2.23MeV. Thus emembering that Ein ourdiscussion misthe reduced mas, thati,Mg/2, ae 4 fe fitaN2mEVast iaVgE RE red Ee ae es tma Sreacoreya ter 5x soar AE25x 10emt~25bams Fh. more accurate decermination leds tothepredicion thatthe cross section at |) threshold is4barns. Themeasurement, cared outwithneutrons atthermalF speedsyields21barns!4 ‘Theexplanation ofthisdisagreement camewiththerealization thatthe spin oftheneviton andtheproton hadnotbeen taken into account. Ifthe potential were spin independent, then allspin sates would scatter thesame 5° way tac iewould notmater wheter thespins ofthe particles are“up” ofs“down.”Ifchepotentialdoesdependonthespin,apossibleformcouldbe as VO)=Vi)+65°61) (24.68) j Inthis case spin isnolonger «good quantum number, and thestates must be assed bytoe angular momentum andtor spin, hais,wth =0hefour states divide upintoa*5,tripletofstaces,andasinglet'.Theseneednoc scatterthesame way, sothatthere arefeally cwophase shifts, ;forthetriplec, and4,forthesinglet. There arenotriplet-singlet transitions, sincethetotal 7 angular momentumJmustbethesameintheinitialandfinalstates.Thetotal ‘rosssection isweighted bythenumber offinal states ineach case (checross section iavolees asumoverfinalsatesandiindependent ofthevaleofthe {>=-component oftheangular momentum), 80that aotq erica (24-69) \ Forspin independent forces, ¢=c1=ay Tedeuteron is15,sate, 50thithefourbara aterelly pediced foro This ipl ce : oy=bo—or=72bans (24.70) 396 Quancum Physies Sinceweareathresholdchisimpliesthat ax io* up=[PMI aax10ow oan A,V7 10" (24-71) “Theeae seul implied that reat jal=a 247x10en (e472) ra “Thequestion ofthesigns ofAandA,nowatses Attheshold wehaveBord=HS—1/Asothat5,=—Adandb=—AdThos,theasymptotic‘wave functions have theform sin(br+6,4)~sinKr—Aig)He—Arad47) | ‘ThetwopossiblecasesteshowninFig.2.6.Weknowtatfortheplete | thewave funtion euros overjustbefore cheedge ofthewell(since chre isa bound state), 50aicmust cortespond tothe situation A,>0 “ | wi i | ae I | = H | t a"|i ! | : | i * | Fig.246.Sketchoftheswavesolutiona)neardrshold,Outsidetherangefalfurr=athewavefunction hasthefrmC(r~A),[Tissnotinconewith(24-75), whichis anexpansion ofsin 8).Wecould equally welhave taken thefrm ofa) cobe(C4) snbr+8), lace thenormalnacon icitar. Ti {tar theincor wave function andthe postion ofAthat determine theslope [oftheline} Thesignof depends onwhether theineior wavefunction hasot hasoottamed ove eases (8and (o), espetvely|Sincethewavefunctionmuse turn over iftha isa weakly Bound sate (00tat itcan match aslowly fling ‘Hporcnal) sedsince onedoesnotexpect thewave Buncon inside theporn tole verysesitve tovations inBsbout nero oneexpets thatfor potenGhathosaboundsatewithEeamal,4>0 Collision Theory 397 IfA,werepositivetoo,onewouldexpectasingletboundstate,with | very much weaker binding, since theinteroal wave function tesonto amuch| flatterasympeotic form.Infact,thebindingenergywouldbe70keV.SuchFE—_boundstatewasnotfound,suggesting thatA,<0.! ‘Thischoice ofsignwasactually confirmed bythescattering ofneutrons Fe ofthe Himolecule. Asweknow, theHsmolecule canexist asortho-H,, with HE chespinsinaeripler state,andpara-Hs, withtheewoproton spinsinasingletstate, Forneutronsatverylowenergie,suchthatthewaveleageh ismuchlarger{chan theproton-proton separation inthemolecule, thescatering amplicade for neutron Hy;scatteting isjust thesumoftheamplitudes fortheindividual scatterings. Onemayshow thareheamplicade offpaca-H, isdiferent from the amplitude offortho-Hy andthese separately involve lines combinations ofA,fadAy.Thefactthatpun23.9barns,whilecag©125barnscanbeexplained ?inthisway.Thecalculation iscomplicated byanumberofeffectsthatmustbe takenintoaccount,forexample,ththeeffectivemassoftheprotonin@mole- E_ealeisdiferentfromthatofafreeproton,andchatthemoleculesarenotrelly >atrest,butatemoving with adistribution appropriate tothe(low~20°K) temperature. The large discrepancy between thetwo coss sections isnotchangedmuchbythesecorrections, anditcanonlybeexplainedifA,isindeednegative. C.The Born Approximation 2 ‘Athigher energies many partial waves contribute tothescattering, andit a isthetefore preferable toavoid theengular momentum decomposition. Apro- cedurethatleadstoveryusefulapproximation bothwhenthepotentialisvery q ‘weak andwhen theenergy isveryhighisthe Born approximation, inwhich we ‘consider thescatering process astransition, jutliketheasitions studied in (Chapter 22.The diference isthat here weconsider thewanscions continuum > continaum Iweworkinthecenter-of-mass system,wehaveeffectively2one-particleProblem, andthisparticle makes atransition fromaninital state,described by theeigenfunction 1a)=Layo ' \WO=eH e474) f tothefinal suc, described by L nro _ We)=pe (2475) ‘where piandpyatetheinitial andfinal momenta, respectively. Thetransition 398 Quantum Physics nce,following theGolden Rule (22-55) isgiven by a2[VPuying(PA2) ' eedPn) Coe (416 “Thedees function expresses energy conseretion, Ifthepats thatemergehveadifferentmasffomthosethatenter,rifthetargetiexcited,tatdeltafunction takes1somewhatdiferetform.1cwil,however,alwaysbeofthefoun alp?/2m)—E]whereBistheenergyavailableforkineticenergyofthefinal prutice, The matix element Mis given by Mn=woivio= faveOTe ] 1=| tre Ve) 7.yf vr) (4-77) | =r ishmmr renWewitematic cones Mi= +Ha) (24.78)wet : ‘Theictegral in(24-76) maybesewsitten intheform | afagVM Hay)(2-) RasfanCoievaleanne(Ee at dbs(é-ipal aoatfombt(22—2)vant 1a aye ©aeFfnmin oe) “Togetthelastline,wenotedchatpydpy/m=d(p?/2m)andcartiedoutthedelesfaneion sateration. Thus, ymust beevaluated atfy=(2m), andwemast aotforget that bereithereduced inass incheBaltate. “Thisexpesion hasanundesiable dependence onthevolume ofche quantiation bon,butthisinotrally suprising. Outwave Functions wereslomalizedtoovepartieiatheboxV0thattheaumberoftransitionsshould certainly godown asVincreases. Thisdifficulty arises because weareasking = Goeston thadoesnotcorrespond coanexperiment, Wha onedoesissend«fluxofincidentparticlesatexchotber(inthecenterofmassframe;intheIaboesery, onepeicle isstationary, ofcouse). Kwe wane«fx ofoneparticle persquat centimeter petsecood, wemustmultiply theabove byVdivided byfevlimeoscylinderwithem?ase,andtheravevelocityofthearies Collision Theory 399 inthecenterofmatsfameintheinitialsate,Thenumberoftransitionsforunitfluxisjustthecrosssection.Wethereforehave La - 7 de= (a)|? (24-80)| FR[ahHMMA) c Since inthe center ofmass frame thetwoincident parcices aremoving toward ‘each other with equal andopposite momenta ofmagnitude pi,their relative velocity is j tel=PeBoe(Ls) &(24ai Vel=nteaens) Eaesifm,andm;aretheirmasses.Thus,iftheinitialandfinalreducedmassesand éa‘momenta arenotthesame,wehaveFe 4Bnge,|e. FmGibmt|Eray] (2482) © When theinitial andfinal particles arethesme, te.mist|gay]? 3) ae) ce ‘When oneparticle is«great dea! more mastive than theothe, i, the sass ofthe lighter panicle. Whenwecompatheabovewih(2415)wesethat =te iy Fad)=—FieMia) (24-84) ‘Accally, todeterminechesign,onemustgothroughamoredeedcomparison withtheparal wave expansion, Wewillnotbother codothishee. ‘ASanilluscration oftheapplication oftheBorn approximation, wewillcalculatechecrosssectionforthescatteringofaparticleofmassmandchargeZsby&Coulomb potential ofcharge Z:.Thesource oftheCoulomb fediken § tobeinfinitely massive, sothatthemass in(24.83) isthemass oftheincident particle.Forgenerality (and,aswewillsee,fortechnicalreasons)wetakethe & ‘Coulomb fieldtobescreened, sothat y Vee)=Zidn8 (24-85) E—where aisthescreening radius. Wethusneedtoevaluate a)=2200farce (2496) 400 Quancum Physics Wechoosethediectionof&aris,andthenget(epee - =fore -[rafsina[edeeteet -xfef d{cos6)ao a a aeae eek eee 2(1_L)_ca= Ga \G/a) =ia” fa) +id)©fat) +8 (47) Now 1 L Peee Ferre Facon408) sotha thecross section becomes awh ieramaeBe27" RANG —cos+C/A -(gant y GPi?@/2) +O78, _2ze q~(Garant ws) oo) Inthelastlinewereplacedp*/2mbyE,andweused}(1—cos6)=sin®(0/2). “The angle€definedin(24-88)isthecenterofmassscatteringangle.Inthe seem otscreening (¢>=)thisredoces tothewellnowa RutherfordTonmula,Therenofini,andiisthesaeaschecsscaformula,HadweJeftourchescteeningfactorin(24-86)wewouldhavehadnill-definedintegral.Geoften evaluates ambiguous incegnls withtheadofsuchconvergence factors meet Bornapproximation basiliitations, Focexample, wefoundchat(a)waspurelysealsothatf(salsoreaathisapproximation. Thisimplies,bytheopalCheer,tatthecrossectionieo.Infact,theBoenApproxi-ation onlygoodwieneither (a)hepotential isweak,sothatthe crosssees isofsecond orderinsmallparecer; thiswould maletheuseofit ‘Consistent withtheoptical theotem, of(b)athighenergies forpotentials suchtne eressection oestoaero.Thisisuefrmostsmooth potentials. Is anne alpatsins dereitsems thattecrossSecuons Sayconst a aorghcargo, andonecannot expect theBornapproximation 0rea8dace crguile ofthe behaioe ofthesearing amplitude. sefacommen, wecere thafhe patel asa independ|exc,thea(207)iclyde bytheopenatekaGiaanes berbd by apnare fenton odontal‘wavefunctions. Thus,forexample, iftheneutron-proton potentialhastheform | VO) =Vil) +6OuVlr) theBorn Apposination eds 1 ite Mu=TfaveVORwhereEanepeethentlandalpinstofheneonpoten — : D.Scattering ofIdentical Particles 4 ‘When twoidentical paruicles scatter, thereisnowayofdistinguishing «Feaesccumats purteoeghtnpctindicdenoeietinaeefmsfame,ceoanerationdensesenate pri ba BRAT omens dies 402 Quantum Physics scatters through 8theother goesintheditectionx—6(Fig.24-7).Classically, t00,thecrosssectionforscateringisaffectedbytheidentityofpeticls,since thenumberofcountsatacetaincouncerwillbethesumofche counts due to theewopatticles, Thus a(t) =(6) +a(n—0) (24-90) Inquantum mechanics thete isnowayofdistinguishing theewofinal states,so thatthetwoamplitade: f()andflr—0)caninterfere. Thus thecross section forthescattering oftwoidentical spin zeto (boson) particles, forexample, evparticles, is de yO +fle—9 aio ‘This differs from theclassical result bycheinterference term de ag7FOV +fe—OF+(POfe—9)+fOfe—9](24-92) anditleads toanenhancement at7/2, forexample, de )?(4),=4A(2) (24-98) | ‘compared totheresult thar would beobtained wichout interference: de i ale()..-21G) aoe Whenthescatteringoftwospin1/2panticesisconsidered, forxumple,proton‘pocon scarering orelectoa-clectton scattering, then theamplitude shouldreflectchebasicancispmmetcy oftheotalwavefunctionundertheinterchange‘oftheewoparticles.Iftheewoparticlesaeinaspinsingletscae,thenthespatialswave function issymmetric, and des :FE 0+fe (2495) che twopatticles areinaspin cxiplet state, then thespetial wave function is sntisymmetic, and AH=176)-fie-9)* (2498 a=8)-fir-9) ) Inthescatcering ofewo unpolarized protons, allspin states ateequally likely,andthustheprobabilcy offindingtheewoprotonsinatiplestateistheetimes Collision Theory 403 1slarge a8finding them inasinglet state, sothat Bde 3d tide aaa taae a =H/O~flr—01+f +fer—OI - =OLE +ifr O1* FO fe —8)+FOfe] (2497) bPForprocon-proton scatteringaswellasfor«—ascatcering,thebasicamplirude 1EB fis thesumof«nuclearcamiftheencrgiesatenottoolow)andaCoulomb D>term.Whethertheidenticalpaticiesbebosonsotfermions,thereissymmetry uFunder theinterchange @—> —8.Symmetry considerations alsoplayaeleinthesatering ofpariles by2©gyallatice,Ifweignorespin,sothatwedonothavecoworrywhetherthetlectton does ordoes notfipitsspin ("up"” —+""down” orvicevetsa), then at tow energies, chesceting amplitude 6)isindependent ofangle (S-wave scattering), andthesolution oftheSchrédinger equation by#single atom FNocased atthe lattice point a;hastheasymproticfoem Aira ‘Hr) OM +froa) (24-98) Now F Ble—ay]=A(t—2eyta)? 4 ara)wie(.- 228) 5 Shr bia (24.99) and since Ais avector ofmagnitudefanditpointsinthedicetion,thepoine ‘ofobservation, iristhefinalmomentum k’.Ifwedivide outthephase factor|e ®°*chewavefunctionhastheasymptotic form vearazewnre® so(Z) tio 40chatchesatering amplitude is fO=fe*™ saw k (24-101) “Thetotalampliudeisthesumofalindividualscaringamplicudes whenwehave &situation inwhich wecannot tellwhich atom inthecrystal didthe scattering, This isindeed checaseforelastic low-energy scattering when recoilisnotobservedandspinsarenotmeasured. Thus,forthecferentprocesswehave .lzcol (24102) $04 Quantum Physics Ifwehave«simplecubicaayoflacepoints,suchshat ae=a(nads +tyty+mts) —NS tasttnt, SN (24-103) (spacingsreincegralmultiplesofainalldizections) thea pees SOE eee We use Seema geeaey.ot = ee oa ee pene pw ne er =SREP euro coobaitheresale |desitatNN-+9)sia?(N43)sata+8) |an Steal siatey/? Tinta? A108) where ay=4,—2a,—(vp=integer), etc. (24-106)‘Wecanmakethegeneralization exhibitedabove,sinceachangea+a—27,withvaninteger,doesnotchange(24-105).Theexpression (24-105)isnotveryTransparent However wheaNislg,echohefactorbecomeseysong) peakedwhenanaeaatonfa,wing sin? Nw7a—4eN(wu) (24-107) 4formulaeasilyderivedfrom(22-36)byasimplechangeofvatiabes,weget de + (2")* yea"[fl*(xy(2N)(ea—29)(24-108) Nowthesotnumberofatomsi(28)andhencethecrosssectionpratomis fo ys sly—y 2wy a(e4) (24.109) ‘Tsthedecaleossectionisverysmal,excepinthetectiongivenby vone2s (sen E Collision Theory 408 Bi.whereicistronglypeaked.TheconditionsaboveatecalledtheBraggconditions,Baadcheintegersvemy,»aecalledtheMiloindieoftheBragplane.B‘Therelationsjustderivedcanbegeneralized tomorecomplicated crystals,fpThey areused tostudy arystal structure, using neuttoas oFX-rays asincident Egatcles, otusingaknown crystal tostudyX-rays thatarcemicted inatomic Pysastions involving energetic photons. fg GM Problems PBIB, 1.Stowchacforacentsalpotential V(e)=(7),thematrixelement My. | Bee24-77) may bewriten intheForen 1ash f°4 Mana1dev)sir FNotechatthisisanevenfunction ofA,thatis,afunction of - AP=(py poi/At : 2,Consider apotential ofthe form Vi) =Vor Calculate, using theBorn Approximation, thedifrental cross section de/d as#functionoftheceacer-ofsmass scateringangle6.Compareyourcesulwith thedifferential cross section foraYukawa potential a Von=va [Alreadydonein(2485)-(24-85)]. Tomakethecomparison,adjusttheparame: Etersintheewocases sothatchetwodifferential cross sections andtheir slopes Fate thesume intheforward direction atA=0.ftmight beconvenient to Lick some definite numerical values forVo,Vx,2,and&codepice thisgraphi Fally.Canyougiveaqualitativeargumentexplainingthelagedifferenceberween thepredictions forlarge momentum ennsfers? j 3.Consider thepotential j VO)=Poe Iftherangeparameteris«=1.2fm=1.2X10-¥emandVp=100MeVin F)magnitude, what isthetotal cose section forproton-procon scattering at100 MeV centerof-mass energy, calculaced inBora approximation? 406 Quantum Physics (Note. Thiscalculation involves «numerical integration. Icisuseful tousethe relation Hat =(pr—pd?=20 —cos) We (dQ=2xd(cos)=pie) anddotheintegration between A*=0andA*=4pt/R?.) Giveyouranswer in nillitarns (1mb=10 em 4.Suppose thescareting amplitude forneutron-procon scaceing isgiven bbytheform £00)=EF(A+ Bap ox)& where andfyaretheiniial andfinal spinstates ofthe neutton-proton system, “Thepossible states are beexPxP gaxPxP PAP PAPxPx? xPa? xx? xa? Use drsby=APM +20a +oe) where atten (63 eeniegre) ” 2 oo 7a “Nie sntenprusinineines= (2)aairr=($x-(2)tocaleulate all16seattezing amplitudes, Make atableofyourresults andalso tabulate che cross sections. 5.Ifany oneofthespinstares(e.g.,initial proton, orinitialneutron, etc.) isnotmeasured, thecross section isthe sumovertheunmessuced spinstates, Suppose boththeinital andfinalprocon spinsarenotmeasured. Write dows ‘expressions fothecrosssectionsforthefinalneutron“up”andthefinalneutron “down,” given thattheinitial neutron stateis“up.” What isthepolarization P, defined by etaat pectoet attel where ¢7isthecrosssection withthefinalneutron upands0on.6.UsetheablecomputedinProblem4cocalculatethecrosssectionsfortriplet»eipleeandsinglet»singletseating,respectively Showtheplet—> Collision Theory 407 i singlec seatering vanishes. Check your results byobserving thatsince (inunits | ar ; Hr+foy=8 fone has op: dy=288-3 : =3 when acting onsinglee state ‘Notethattheamplitudeisindependent ofma30thatmsmustbethesumeinthejn andGalspinstates. There aredhreestates iathetiple, allcontributing Janequal amount tothecross section, andonlyonetothesingler crosssection, (Gat toalain tnlies achs& 1 ying 9 1ge FPA —PAF +Boo-bx)HeOPM —LPP)ve v2 theamplitudes areadded forthefour tenms before squaring. Can youexplain why?) 9.Tecan beshown hatthesolucion oftheJ=0Schrédinger equation forthepocential, )=2So :VO" ONGere iy “| which bebaves aspenpcocically likeeis wie ABE $1) +ABOpete BOEHD+BOWDY ad Ger+DREA “Thesolution thatbehaves asymptotically like¢*™isF(—&,).Thustheregular solution, which vanishes atr=0,is F a)=(F(60) F(A) —F(-49) Fb} Usehisinformation toobsain thescatering amplicade F(#) =[5(8) —1/2 Discuss thesolution forvarious limiting cases. References Scattering theory isdiscussed inallofthetextbooks listed artheendofthis volume. Inaddition there exist aaumber ofadvanced treatises that are devored tothissubject alone,Mistaccesiblecosunsatthisleveli 408 Quantum Physics N.F.MottandH.$.W.Massey,TheTheoryofAtomicCollisions(ThirdEdition)‘Oxford, The Clarendon Press (1965). More formal is L$.RodbergandR.M.ThalerIntroduction fosheQuantureTheoryofScaring,‘Academic Press, New York, 1967. More advanced ate ML. Goldberger andK.M.Watson, Colliwn Theory, Joho Wiley 8Sons, 1965. R.Newton, Seattering Theory ofWaves andParticles, McGraw-Hill Co.,1966. | | | |chapter25 |TheAbsorption ofRadiation inMatter ©The processthatistheinverseofradiativedecayofatoms,namelythe Pps ofphotons necompanied bytheexcntion ofwns, aaalsotakeplc.~BBRForphotonenergiesexceeding theionization energyoftheatom,theelectronSipexcited cothecontinuum, This iscalled thepheteeletrie ect, andisanim- portant mechaism inthe absorption ofradiation inate, P ‘According cotheGolden Role(22-55), thetransition rateforthe process a ++ (atom) —(atomy’ +¢ 54) Fis given by int A—Be-e) :na[EIRalt(io—By— a atfefaya(225-Bs-#2) fears [nts(Se) iata(te 25 : 20[fyMeye =faoie|Mal (52) Fn theabove expression, mistheelectron mass, thedele function sepesenss . encigy conservation, ys temagnitude ofthe binding eneegy oftheelection fim theatom, andinthelastline, .isevaluated acchevanishing oftheargument ofthe det faction. “Thematin element isgiven by + (262Y Fa : ECE)" fenio owasvae 3) ‘Thevector potential isnotmalized, asinChaprer 22,toonephoton inthe volume V,and¥i(r), ¥y(r) atethewave functions fortheelectron intheinitial . 409 410 Quancum Physics andfinalstates. Ifweconsider ahydrogenlike atom, andassume thattheelectron isin cheground stace, wehave w=3,(2)er ry “Thefinal-state wavefunctionshouldbetakentobeasolutionofthe Schrddinger ‘equation withaCoulomb potential withE>0,Wedidnocdiscuss thesesola- ‘ionswhen westudied thehydrogen atom. Theycanbewriten inclosed form‘buttheyarequitecomplicated, asistheintegralin(25-3).Ifthephotonenergyismuch larger thantheionization energy, thentheresidual interaction ofthe futgoing electton withtheionchaticleaves behind becomes lessimportant,andwemayapproximate ¥(r)byaplanewave.Sinceweassumethatweonly hveoneatomfaourvolume,wewllhiveonlyoneelectroninthevolume,and hence the normalization issuch chat Lwie) =See as.WO=Fp @s-) ‘Thefactor Vthatappeats inthephase space [Vd®p/(2xh)"] corresponds tothe ‘ame nortralization, thatis,thecwofactors arent independent, Thesquare of thematrix element issomewhat simplied sincedhefinalstateisaneigensate ‘ofmomentum, sothat Fe pei) =epAflAi) (25-6)Hencethesquareofthemauixelementisam(2)2H!11(ZYOpp vault=(2) 52(2)wa x|fiponiespain (7) ‘Wewillevaluatetheintegrallater,Athispoinewenotethatheateagainhasa 1/¥bbehavioeducothefactthacwearedealingwichasinglephotoninchevolume V,Instead, wewillconsider thecross section forthephotoelectriceffec.Tohaveaflaxofonephotonpesquatecentimeterwemusthaveadensity Ofphotons 1/¢percubic centimeter (sothat cylinder ofunitaresbaseand length ccomtesponding toatimeinterval of1seccontain onephoton), dati, ‘wemustmultiply thetatebyV/e.Weget,combining (25-2) and(25-7) the ifferential cross section eee 2(y=*t2y pot BhGahAwe)eeVag)OPO \ xAYfereen-nrm +Bile @)) “TheAbsorptionofRadiationinMatter401 Tathisexpression 4 isthesolid angle itowhichp.points.Theintegraloverall clectron directions yields theroa cros section» forthephotoelectric effect. i Ifthetarget atoms atedistibured withadensityofNatomspetcubiccentimeter, BEthen inaslaboftargetmaterialofareaAandthicknessdx,thereareNAdxtarget i ‘atoms. Each atom has across section@fotthereactionunderconsideration, so PE thatthe coral efecive arenpresented tothebeam isNAdx. Ifthere ate BE incident particles inthebombarding beam, thenthenumber ofparticles thatimeractinthethicknessdofthetargetisgivenby . a inceracting particles _erst section incident panicles~ cool area Bs ctacis, Be dn Néode|} oe nde (259) FE 7 A i ‘Theminus signindicates thacparticles areremoved fromthebeam.Integration ives fs)=mye 5.10) 4‘whereneistheumberofincidentparticlesandn(x)isthenumberofparticles' eftimthebearn aftertraversing athickness xofthetaxget. Thequantity \= 1/Nahas thedimensions ofalength,andiscalledthemeanfreepath.Onesome- times speais ofthe meanficepathforthe photoelectric ec, forpaiproduc. © tion, andsoon,even though what iseased isthecross section 3 ‘Togetan ideaofthemagnitudes ofmean freepaths, aotethat N=Nio/Abs‘wheteNp=6.02X10*isAvogadro's Number,pisthedensityingramspet© cubic centimeter and Aistheatomic weight. Cross sections formolecular Collisions canbeestimated from theproperties ofgases, andthey turn outto have magnitudes ofcheonder oft0- cn, consistene with thefactchatatomicdlimensionsaeoftheorderof10-*cm:Isthisareasonableguessfrchephoto-tlectc coss section? Weshall soon extnine theteasons fotwhy itisnot © Tathemeantime wewrite themean feepathincentimeters, inamatecal of density pandatomic weight 4,with thecos section expressed inwaits of Sort cnt called Bera thus, y-t-4_ 1 :No p602 X10%© 416 -418 5.1i »(bans) on) H Bythesamecoken,nuclear csssections readcobeoftheorderof10cmt 3 (earn), puice physics cow secon weofthe ore: of0-" ntCaliban), Sing : Alert motu fre rections, sad evea down co10% co foethe ence? ae | scaringveosowene 412 Quantum Physis “Toevaluate thecross section in(25-8) weneed towork outtheintegral faeAIODl (25:12) 1weusetheintegealevaluatedin(2487),which,witalightchangeofnot.tion reads on te BrOt (25-13 fore ass oo) ‘wecas,bydifferentiating withrespect toyobsain Htcar : forecnaee @s.u4) ‘50thatwecanfinallycalculatethecrosssection.Aftersomejudiciouscombining offacrors, weendupwith 4. +(22) (eey 3ay>8(2)(me)Geaay Gas) where (ik-pd_(er-Pd a=fh i Since theelectron andphoton energies arerelated by famBo+BE (25.16 ‘weseethatforenergies quite abitabove thebinding energies, faa p2/2m. Hence fa Une) ne©PM seLip,pore L(Y 2!odybetpe wtf (t2)5,-6,wife (ia) =P(1tied) a7) fornontelavstic elecrons, ps<mcWehaveusedthenotation py=Mikfor for reve elearos on should sly usecheDine equson todescrib se roca, Bec ter then hePorodecc ectseeeimporastwhen=?MeN ‘TheAbsorptionofRaioninMater413 1 thephoton momentum, andasusual the~denotes heunitvector. Thus Aza(4)(epot [ones2(1-25.-8)f @s.0) Ifwechoose thephoton direction todefine thezis, andchewophotonpolariaiondirections«°°pointinthexandptetons, espe, Ben, wing Te B.=(sin#cos6,sinsin4,cos6) (25-19) BE wehae(9-40) sia6costand(pe£8)=sat#sat9sache BY gverageofthenumerator overthetwopolarization directions (wearecalculatingBe phoroaec cross section withunpolarized photons) i 2 (Bes)=F(sin® sin®6+sin?Bcost9)=}sint# (25-20) Falso@ BeBe =ono @ay sothu, wing p//2m =E,weget ms “[er+Z(—cosDi Lael E norigh elements, thecondition thatweimposed eae, he>Bawhich is Fequialotto E>}me"(Za)* (25-23) issatisfied overareasonably widerangeofenergies. Ifweinsert(25-23)intothe Cross section, wefindthathedenoraator simplifies, andweget Snavira (ZYa z(:~eeeJ (25-24) Fer usdiscuss ssios axpect ofthis formula (Q)Fase, thevague guess that since atomic sizes tend tobeoftheorderof1enthecrosssectionsshouldbeofexder10"cm?iswrong!Tetuehacthefacor as ofthat engaivde, bateismulled by(1/137) which s 414 Quantum Physics dimensionless, buthardly negligible! Weshould trytounderstand how one could be30eng inert tohave Some guidance what onemust bectf Shout inmabing eines fe igooe thelastangula actor, which wewil Sinus later weSeethatwemay, wth thehep of E=jmvd write chefactor infront as avtatne' 22)"=seurew(£) -2(2ye(Z) wa» “Thisisamovewseflfo,1shows,fstofalthepresenceofsinglefactwhich shoul alveys bepresent when asingle photon isetd oraborbed ‘Thecoupling ofthe wecorpoenia toacharges proportional othe care‘andthesquareofthis willlead tothea.The factor (49/2)? isabetter measure of theactofheatom tan nce weteConideting abyogee atom of Charge 2Whee ronuins letter high power ofthe aio ofthe bial” seloety oftheclecwon iatestom tothevelocity oftheoutgoing fee Teton cistheratio (aZ¢/y,) [rather chan just(¢/v.), which isalso dimensionless}. hueappean, bese themux elameat involves theovetap betwen thefeelectronwavefunctionandtheboundelectronwavefunction,thatis,thesquareturrixcomentnreltedotheproblyneameasurement ofthemomentaofthebound electron yields p,.Thefunctional dependence flaZ</1.), inthis tase theighth power” eanat begused atonfneal quiliante grounds Forexample, iftheelectron wave function wereGaussian [¥x(r) ««~”“], the fallof with increasing velocity would bemuch faster than theeighth power. “Theron why2garsitacomakeisthattheomentumdstrbutionof theclecton 8foalized inaregion ofspend ht we Zam 25.26 apaltRjLas (25-26) ‘andforpa>>Zameoneisfatoutiathetailofthemomentum distribution. This,in bycheenceraity selon, depends ontheSal dstbution oftherefenciontoddependssemiielyamthesenpriontheangleTmomentam. This dow make pooro-deiaegasion iaauceat pli 2ey veel Thre tor snthepase pce sohatthmai lone quel ge on cin poo a. ‘TheAbsorption ofRadiation inMarcer 415 (2)Theangular distribution ofde/d8isgivenby F@=—ae (5.27) ©(0=G0) cosa ™ We note, ist ofall, that thecross section vanishes inthe forward ditection. “This isaconsequence ofthefactthatphotons aretransversely polarized. The smattix element isproportional top. ¢,andwhen p.ispanel tothephoton f—-momentum, cisfaccor vanishes. The factor inthedenominator has, because of thefourth power, astrong influence ontheangular distribution, When #4/¢ 4‘approaches unity,thisbecomes verydramatic, butevenformoderate »,/¢there 4ssignificant peaking inthenearforward direction, where thedenominator isatitssmallest.Thiscorresponds totheminimumvalueofthemomeneum transferbeeween photon endelectron (p;—pt3 Moredetailedcalculations needtobedonetocovertherelativisticregion.“Theformulederivedaboveworkswellintheregionofitsvaldity.*Arverylow A‘energies,amoteaccuratewavefunctionfortheoutgoingelectzonmustbeused.Sach awave function will reflece the Coulomb interaction between the nucleus andtheelecron.Therewill,ofcours,benophoroeffect belowthethresholdforinning theleas-bound electron fromanoutershell,Astheenergy increases § above thechreshold, electrons from deeper shels willbephocoproduced. Ifone om plotstheintegrated crosssection, orpreferably themassabsorption coefciont*©Nef,a8functionofthephotonwaveleagth,onefindsthedacashownonVy aiesolesRelgecorapntr oeetoofthen=Tce 3 axons; theL-edges coteespond tothevarious electrons inthe =2scates. The {edges occurat thebinding energies ofthevarious electrons. Moseley's empirical law states chaechey arelocated at @-«,)* a=36S ey (25-28) .whereem,the"screeningconstants,” areapproximately givenbye5=28+1.Thisfoxmula isjustwhac weexpect forthe nsotbitals, andehescreening isthe effect ofall the others electrons. ¥ Atrelacvistic energies thecrosssection dropslessprecipitously, withan(Bim)behaviorinsceadof(E/m)-*”butbythetimeenetgiesof0.5MeVarereached, thephotoelectric efect ceases tobeofanyimportance asfarasthe bsotption ofradiation isconcerned. Intheenergy tegion of0.5-5 MeV, say, icistheCompton Bffs thatisthe dominane absorptive effect. Hee freeelectrons scatter photons. Atlowfrequencies theeffect canbe “tecalculating absorcionofration,theesethatwedsnedostbeulipted |bysinetherearewoeectsinhegroundsteexcepinydioger. "This sequal toNye Awhere Neis Avogadioseunber tdisthe atomic weighs. 416 Quancum Physics x0 2s _9 Tre 2109} Ps rr rT) aK Fig.25-1. Massabsorption coeficieat Nefforplatinum asafunction ofphoton, widens ndestood dhssially; electromagnetic rdiation impinging ontheelectionscclentesit,andtheradiationemiedbytheaccleaedchargeisthescaredtndation. Thecssially cciated Thomson coss section is se (ey(2) om tnquantum mechanics, theseatterag amplitude (matix element) mustbero- posional to,sincetwophotons ateinvolved, Sincetheperturbation iathe ‘Hamiltonian is Lpnen+eae (2530) ‘when bothterms iatheexpansion of(22-11) atekept, weseethanan#contri- ution tothescattering amplitude cancome from twosoutces.(thefistsourceisaistotdercontribution fromtheerm#AY)/2me.(i)thesecond source is2second-order persbation term fom the coupling, epA(ed)/me. Since webave0¢developed thesecond order perturbation formalism, wewilestice ourselves costating theresults.(a)Acthreshold,withthegaugethatwehavebeenusing,V-A(e4)=0, thewhole amplitude comes fromtheterminvolving #A%(r,))/2m¢. (b)Thematrix clement insecond order hastheform —¥(flew:A/me\n)(niep-8/meli) ‘ = oe (2531) “TheAbsorptionofRadiationinMauer417 where che “sum over intermediate states“ alsoimplies integration over althemomen,when“n”includescontiauumsateIsnotenoughtoincludeiarenmedite onelectionsatescoxrrpooding totheaequence nbereonte | ‘andtheintetmediate statescontaining anelectronandtwophotons, correspond: Eeingtotheprocess at nomty tlowty RE eoumns oureatiisnecessary toinclude thepostibilty ofthe “vital” creation(ofanelecton-positron pairbytheincidentphoton,followedbytheanaiilationofthepositonbytheincidentelecaron,withtheemissionofthefalphoton,as Gand eheprocess atnratututatOoyty 5 The calculation leads tothe Klin frmala eV it+xfai+s) 1oom(LYEAF9 Dogargop »(3){=eee aan ] 1 Le +jlo+25)—a| thes xno (25-32) ‘whichisinexcellentagreement withexpetiment. Atlowfrequencies thisbecomes on8(SYa-m (253) andathigh frequencies («> 1)thisreads ot(SYEoog2e+ 0530) “Thus theCompton cots section, 10,dps offahigh enegies. AtenergiesaboveafewMEV,thedominantabsorpeiveprocesssparredo | Tein rematlable acthat photon ahighenough energies, Rs>2m } can “materialize” intoanelectron andapositon (Fig. 25-2). Thelattercanbe q “Thefactthar(25-31)lookslikeanoff-diagonal versionofthesecond-order energy sti,eoumenocet 418 Quancom Physic Y ay “BeGl wiab&eH\Gh "0 Not Va ;Mt y, : \ Ye\ \ (Aoy AN le\ S/ \,oe x T=ICr) Fig.25-2. Tora sbiorption cocicient forloudandaluminam as«function of negy, inunits oftheelectron restenegy (O31 MeV). Thephotoelectric crossSetonforAlinegligibleomthescaledepictedhee. properly called an“antielecron'; itasthesamemass25theelectron andche Stine spn, butitscharge andmagnetic moment havethesame valve with “opposite signasthose fortheelecteon, andthenonrelativistic coupling withthe clectromagnetic fieldisobtained bythereplacement ofpbyp—eA(rd)/e.‘Suchmateialzaion cinonlyoccurinthepresenceofahidpaticle,anucleus, foeexample, since energy andmomencim conservation cannot hold forthe process yore Toseethiswithoutgoingthrough4longkinematicalcalculation, considertheinverseprocess¢-+«7intheceterofmassframe.Theelectronandposittoneve equal and opposite moment, 0that theGaal sate hascoergy 2(mtt +pie)? andmomentum0.AphotonofenergyEmustcarrymomentum B/c.Ifthereisaaucleus present, itcanabsorb momentum andenergy (fora ‘massive nucleus thiswillbeverysmall, p#/2M), sotharitbecomes possible to talance energy andmomentum.“Thecalelacionof ‘yh mucleus >¢-+ +mucous isbeyond thescope ofthisbook. Thetheory ofquantum clectrodyaamics hat TheAbsocption ofRadiation iaMazer 419 isused inthese calculations also shows thac wecantransfer pastcles from one: sideofcheequationtotheother,providedwechangethetransferredparticlesco itheirantiparticles. Thusonepredictschat 1a nucleus +¢*—> nucleus +e+ 1 shouldalsooccur,withamatixelementverycloselyrelatedtothatofpispo-duction. This isinagreement withexperiment, andthe lstprocess isreeponsible : 3 focusmic rayshowers. . ‘Anincident -yrayofvery high energy (itmay come from thedecay #°—+2y, withthe #*produced whenaprimarycosmicrayproconhitsanucleus atthetopoftheatmosphere) willmake apair, witheach member canying oughly halftheoriginal energy. Each member cinproduce aphoton, asindi ‘aedabove,andtheendproductscanmaizeFurtherphotonsaadpairs.Showers F coming from extremely high enetgy events occurring atthetopoftheatmos-fF pherecancoverareasofseveralsquaremiles!LessspectacularshowersincountersFate used toidentify photons ofelectrons. Anincident paticle tacischarged, i butmuch heavier, willbedeflected les,andwiltherefore radiate less. Detailed calculations show hae energy lostinmaterial through these pg processes follows thelaw Se Bhs) =Bye (25.35) |) wherethe“radiation length” isgivenby : (nt/h®) A - 4 36) ae tAZ*a*Nop log(183/24) (336) Ug ie whereNo=6.02X10"isAvogadro's number, mistheelectron mass,AistheEF atomicweight,Zisthechargeofchenucleus,andpisthedensityofthematerial 4 ingrams petcubic centimeter. The“pai production length” isgiven by ze Ime=2k (5:37) ae ‘TheformulaisnotgoodforverylowZ.TypicalvaluesofLare Ait 330m -AL 97em j Pb 0.53.«m i Bremutrahlung isthedominant energy Jossmechanism forelectoas athigh ‘energies. Atlowenergies ionization dominates, Lack ofspace keeps usfrom discussing thisessencially classical effect. 1This process elled Bremrablang, andcambeundestod classi, «chug tected itheCoulomb eld ofthe sucess neceleated, ana hence date, : 420 Quancum Physics Problems 1.Calculate thecross section fortheprocess y+ deuteron +N+ P “Theprocedure isthesameaschatfrthephotelectcele.Inthecaleultion ofthemattx ceiment, thefinal sate wave function isagain 1yin)=oemWe=Jp where pistheproton moinentum. Atlowenergies thewavelength oftheraia- Tons much ager thanthe"si"ofthedeuteron,that"=1,Tocaeulate fore vee observe thatMrEn)Mr wtMS yn=o where thereduced mass Mp/2 wasused, Now useintegration byparts toshow that o-fayes[rwMeley,-ao+] =fencnen (fhMi) 5.Mefacine fe (-FM) aBefore vey. leading co = tn fare gen —fore veon Sincetheinteisoalyoverherangeofthepotential,whichiseryshore,we ‘eanreplace «oa therightsideby1.Another useofcheScheédinger uation leads to =ferer veovenBofarose with Eathedeuteron binding copy =—225 MeV. Forthecalculation of thisine cake yaya Meet gywetoe rn =o <n ropely nonmalized. Forwhat energies would youexpect thephoton waveKengvobemuchlargerthanthefangeofthepotetialreS12fin?(The TheAbsorption ofRadiation inMattee 424 justification forignoring theincegral inside thepotential range (¢<14)liesin : that ro&t/awhereat=MpEs/f"). 2.Theprinciple ofdeuiled balance relates thematrix elements forthe reactions Ata>Bt+o and B+tboAta WW |3 thus 3 Lao =D) Mal* ‘where thesumisoverbothinitial andfinalspinstates. Taking intoaccount that inthecalculation ofarteotcross section oneaverages overtheinital spinstatesandsuisoverthefinalspinsates,showthatfortheaes M+ V+ 1)ae_Int V+ 1)Ax POGpdE) dQ, ppd) dO ! where Jo,Ja,JuJoatechespins oftheparticles, psandpsarethecenter ofmass momenta ofpatticles6anda(Iand11mustcakeplaceatthesametotalenergy), =EyandEyatethecorresponding encrgies oftheparticles, andty,d,arethe solid angles inwhich bandaareobserved. Usethisresult toexpress thecross section fortheradiative capture process N+P>D4+y7 interms ofthecross section calculated inproblem 1,Note thatthefactor | 4 @I-+ 1)forphotons is2sincethereateonlytwopolatization states, andalso thespin ofthe deuteron is1. 3.The cross section forthe reaction EDP +P hasbeen measured forincident x*laboratory kinetic energy of24MeV, and found tobeequal to3.0X10-* emt. (@)Acwhatlaboratoryenergywouldonebeabletocaryoutatestof \ detailed balance bymeasuring thecross section for | P+Por+D(Thepionmassismet=140MeV;Met=940MeV;MoS22). }(b)Giventhatthespinofthexis0,whatisthepredicted crosssection for chis reaction? 4.What istheradiation length ialiquid Xenon, forwhich Z=54, A= 131,and p= 309gmcnr? 5.Suppose theelectron were bound rothenucleus by«squarewellpo tential,Caleulatecheenergydependence ofthecrosssectionforthephotoelectric 422 Quancum Physics effect. Assume thatthephoton energy ismuch larger thanthebinding energy of theelectron,andthatthepotentialhasashortrange.(Hint.Seeproblem1.) References ‘The mechanisms responsible fortheabsorption ofradiation inmatter ate discussed inmost ofthemodem physics textbooks listed atheendofChapter 1. Foraverycomplete discussion oftheexperimental rechniques used iathe measurement ofthevarious effects, eealso IE,Segre, Nuclei andParticle, W.A.Benjamin, Inc.,New York, 1964 ||chapter26 Elementary Particles andTheir Symmetries | |es Inthischapter wediscuss anumber ofropicstelated cochefundamental> inraions ofelemenary pacer Altough iscape scene EE cualiative thantheothers, because whatever theory existsinvolves conceptsbea tooadvanced tobediscussed inaquantitative way,wewillseethetquantum:conceptsareessentialintheanalysisofavarietyofcomplexphenomen. A.Electrons and Positrons InChapter 25wementioned thatinthepresence ofasecond body (to conserve momentum) aphoton canmatetalize intoanelectron endapositron. ‘Thepositron canveryappropriately becalled theantpanielecocespondiog tothe dens tsenea toecrepe forthe sigote ee che tiher dearomagoctic popes’ fog" oppouhe sagnetc dipole meses)Whenaneecucepositonparsproduedmsbublechantesysary: characteristic pattem caused bythefactthatthetwoparticles bend oppositely a inamagneticfieldisapparent(Fig.26-1).Antipartides are«necessaryconse. < quence ofrelativistic quantum mechanics, andthepositron waspredicted in ‘SambyDine woyeu before Andgnon dacoed epee ‘Weliveinaworldbuiltofprotons,neutrons,andelectrons,Positronsare tare,sincetheyhavewobeproduced wiltheexpenare oftneaflac Den!{1MeV) conesponig tothe production osfssnten)oadcsdey havebeensued onlunde specal experimen iferesuccy Neate, uation dhondjmoms ssi sade ts nena elie ee Tomagnstic neato’ dootdningeaty crepe geaha ene charge bewcen cleo solpositon Hist Badcastes foes ean \Ae \ereict) are Vofe (©) it ios , / / - i Woy) | fephNees (oh | Fig.26-1. Bubble chamber picruze oftheconvesion ofray intoclecton- potion paininthepresence ofmater (hydrogen), Inonecasetheparis produced [ithe coulomb field ofan elec, intheother ase iisprodaced inthe coulomb feldof«proton. Toeracks curve inthemagne fedofthebubble chamber, sith therer pails bending les. (Coumesy oftheLawrence Berkeley Labor tony, Universe ofCalifornia) 424 HlementaryParicles 425 thegeneral notion oftheinwriance ofthelawsofphysics under particle: antiparticle conjugation, inwhich notonlyelectrons arereplaced bypositrons (Gndviceversa) butprotons byantiprotons, neutrons byaatineutrons, andso foth, With thisgeneralization,wecanconceiveofantimatterconsistingofanti. ‘ace, with posittons bound «othem byattctive Coulomb forces. Aconse-‘quenceoftheinvarianceprincipleischaallehephysicalcbserables arethesame, 0that anobserver could nottellwhether heandhisenvironment wete made ofmater orantimatter. Whether thelawsofphysics obeythisconjectured invariance lawmust besetded byexperiment. Ienowappeacs thatthe lawsof thestrong (aucleat) andelectromagnetic interactions do,butthatthe lawsof theweak iaterctions donot.Theexisene ofautipatcies follows fiomquite general lawsofrelativistic quantum mechanics,‘: Letusconcentate onpositronsforthetimebeingandaskwhathappenswhentheyareproduced inthevicinity ofelectrons. Sooner oflatertheywill GBcollidewithanelectron,andiatheinverseoftheproductionproces,theyvillannihilate eachothe, withtheliberon ofthetotalenergy consisting ofthe kinetic energies andthesumofcherestmasses 2m. Thisenctgy willbeliberatedintheformofadation.Toconserveenergyandmomentum, «thitdody (eg,anucleus) mustbepresent, ofatleastwophotons mastbepro- Fie _—_tuced, a8waspointed outcate. Aascinacing possibility isthatthepositron loses energy byionization, thuis,long-range collisions witelectrons, sothat slowing down rather thanannihilation infight occurs, Theposition willgenenilybecptutedintoanorbitaboutanelectron,Thepositonandelectronaac eachother andform anatom thatwecallpstronium. Theatom isde- scribed,infastapproximacion, bythesameequationaswasusedfochehydro- F gen stom, except that thereduced mass isx=an/2, $0thatthe ionization ‘energyis6.8eV,the“radius”is1Ainsteadof0.5Aandsoon.Posicronium does exist and aloftspropertiesateinacordwithexpecations.? Theground stateofpositronium isan!=0cate,andofcheewopossiblesates, and*S, theformer lieslower. Thisiswhacchehypetine splitting (Chapter 17)would indicace, buthesituation bereiscomplicated bythefactthatthe possibiltyof annihilationreallymakesthisproblemdifferentfemthatofthehydrogen atom, andthehyperfine interaction ispray cancelled bypurely relativistic effects. Once positronium isinanState, thewavefunctions ofthe ewoparticles ‘ovelap significantly, andannihilation"becomes likely. The°Sstatecandecay into two photons, andtheuansition rate canbeestimated asfollows. Foreachphotoaemited,chemarrxelementwillhavean¢init,sochatthesquateofthematrix element willinvolve ¢,of,equivalently, a.Theannihilation atemust ssobeproportional rotheprobability thatthe cwoparticles overlap, Areson "Thisincndes sucheste efecsatthespots coupling, hypeine severe, ndanefeciveimaginaryarintspotentialdetotepssoftaabationins fophon 426 Quantum Physics ablemeasureofthisis|¥40)|*, where #0)isthehydrogenlike wavefunction of theground sate,evaluted at20separation beeween theparticles. Thevalueof ‘thisis(1/1)(1/ae)*, whereao=K/yca.Tomakethedimensions comeoutright,wwemustmultiplybya(length)?(time)?madeupoff,¢,andg,butwithoutdayfactossof.Theestimateisthus aya (Yaywe ree ()a (a)os(m2 1 @e() es) ‘The constant»isnotdetermined bythisdimensional argument. Thetatethusis R=0269 X10°sect (262) CCompatison withtheexperimental rteof0.8X10!sec! shows thatthe Constant 9iapptoxinatsly 3.0.Aproper evaluation ofthisconstant relly fegires telaivstie quancum mechanics, since werequire thematrix clement forthesnaihiaion ofaparticle andanantipartice, andthisconcept doesnot enter into thenonrelativistic ScrBdinger equation. PPosizonium inthe*S,sate canalsoannihilate, butmust dosowith theemissionofducephotons,Tounderstandthis,wemostconsiderthepropertiesofposivoniam undercharge conjugation, this, undertheinterchange ¢©‘Weobserve (cf.Fig.26-2)thatcharge conjugation canbeaccomplished by(1)an inversion, accompanied by2)«spinexchange. Asingle state, with S=0,is(odd,andawipletstatewith$=Lisevenunderthelerexchange,sothattheeffect ofspinexchange is(1), Theeffect ofspace reflection isusually (CA)! thats, evenangular momentum states areevenunder refection, andso ‘on,burthere isanadditional factor of(—1) thatarises from chefactthatanti-pattiesofspin1/2(also3/2,3/2...)haveparityoppositetothatofpartcles.*‘Thisimplieschatwecanwritefortheelectofchargeconjugation c= (-1)8" (26-3) “Thusheground sate'ievenunder charge conjugation, andthestajst above it,5,&odd. Similarly, "2,isoddunder charge Conjugation, while 2preseven, and50forth. Nowtheelectromagnetic Geldisddunder cOnjugh- tion, Frexample, theequation vB =ep (6-4) Ths follows rom uke geal popeies ofwelaivitic guano mechanic, aodhacaptives consequence hefromtheoneenone ove. Forexpe22)MELEEctooftnewprotonspodaceipoonismsnoaePeSRATESoeraty peentcalarso echhee‘ bore outbyexpernes Elemencary Panicle, 427 b ee i f / fein foe |/ Fig.26.2. Equvalace ofcharge conjugation and(aversion) %(apn exchange) forelcton-poston syste. [can onlybeinvatant underchargeconjugstion ifchanges signwhenthe charge density does, Ineffect, under charge conjugation,’ e—>—e.This leads to “a seriesofselection rules,which statethatposironitm inasatewithagiven F Sand can only decay intoaneven numberofphotonsif5+Fiseven,andinco © anodd numberofphotonsif$+1sodd.Thusthe,statecanonlydecayinto {dn oddaumber ofphotons. Since each addiionl photon seduces thenceofdecaybyafactorofa,atleast,thesmallestallowednumberofphotonsisfavored B.Baryons, Antibaryons, andMesons . Inthelastsection weexirapolated thenotion ofcharge conjugation to 4 particles other than electrons andonthatbasis made «conjecture about the g cxistence ofaniprotons andantineutroas, These particles have actualy been found toexist, aehough icidfcok tomake them, since ictakesacenterof amassenergyofatleast2Mye!1880MeVromakeapiratrese.Theantipeoton thasbeen found tohave charge —1,mass equal tothatoftheproton, magnetic moment equal andopposite rothatoftheproton, andother properties expected fom theory. Since cheneutron isneva, Onemight askwhac distinguishes the ancineutron from theneutton, The answer isthat there appears toexist & *Undercarsconngucionp—(i)A->p+oe)A.Wedono:changehesignof thenambr/cboe intendageeinofAyoegetBandBeenectot‘hugecoujgnonThsmesosthepolaaionvcoreuasformedincaaegeeeSince tna ae Jovlee thenue ofquunan evlng te oreba thee oeSrableumsteaysundecheausaformaon 428 Quantum Physics quancam number likecharge, called Baryon Number, Ne,which isconserved, and which hasvalue +1(bydefinition) fornucleons (neutrons andprotons) and—1 forantinucleons. “The notion ofaconservedbaryonnumberarsesfromtheempiricalobser- vation thatthereaction inwhich e~andPannihilate does notseem tooccur. If the reaction + Psu os,equivalent, Pact +“seu ‘couldoccur,thenmatterwouldnotbestable.Wecantriviallyselimitonthelifetime ofthe proton;theexistenceofveryoldrocksshowsthaticislongerthan 45X10?years. Actually onecandobetter. Either oftheabove reactions ‘occurring in4scintillator would givetisetoradiation thatcould bedetected. Byproperly shielding alarge scintillator toeliminate pulses caused byexternal sources such ascosmic rays, andlooking for“spontaneous” pulses, onecan secalimit onhow often theabove reactions take place ia,say10%atoms, “Theabsence ofsuch sponcaneous pulses over acercain period ofobserva- tion hasbeen translated intoalowerlimitontheprotoalifetimeequalto2X10" eats! Ieisfaircosaytheetheproton isstable. ‘Actually thetexction C+PONtY doesoccurinnucle,sothatitisnotquiterighecosaychatthenumberofprovons isconserved. Thecomect statement hereisthatthenumber ofprotons plusthe ‘number ofneuttons isconserved, justlikecharge. This number iswhet wecall theBaryon Number. Weposculate thatunder charge conjugation thisquancurn‘umber,justlikecharge,changessign.Thustheantiproton andtheantineutronhave Ny=—1.Electrons andpositrons have Nz=0,andthisistheformal ‘explanation fortheabsence ofthereactions @-+P—+“stuff,” ands0on.Baryonsandantibaryons canannihilateinto“stuf”thatmusthaveNy=0,but sayhavecharge1,0,—1intheprocessesP+N,P+P(orN+N),and B-+N.Whatevertheannihilationproductsare,theymusthave@totalenergy ofaleast 1880 MeV and thisisaverydistinctivefearoreofnucleonancinucleon anitilation Alchough iisconsistent wich thesymmetry laws toexpect P+ B (ete pairs) +photons tooccur,itturnsoutthatmostofthetimetheannihilation productsarex-ansons alsocalled pions. These areparticles fistpredictedbyYukawain1935coexplain theshort-range nuclear forces inamanner analogous tothelong-range electo- Blementay Panis 429 L ——_magnetic forces. Thepions were viewed ascounterparts ofphotons.‘ Pions ate‘notpartofoureverydayexperience, sincetheyhaveshortlifetimes,10-*secforthexand10° secforthe2°,They allhave restmass oftheordet mc?140MeVandNp=0.Thex”istheantiparticleofthex+;theirmassesanddecay patterns arethesame. Pions areresponsible foratleast part ofthenuclear forces, andassuchmust berather strongly coupled tonucleons. Thus theanalog ofet/he (221/137) isanumber gi/hec &15.Hence theeulethatthesmallest ‘number ofphotons inareaction ismost probable does nothold forpions. In the annihilation P+B—pions BR&chenumbercanvaryquite4lot,consistentwiththeconstrineprovidedby‘energy conservation, The average number appears 10bearound 5,when the nucleon andantinucleon areacrest.Pions havebeen found tohave spin0and negative party, andthusatecalled freuducalar particles. Siace theyatespinless, they cannot have anyelectrical moments, foexample, magnetic dipole moment. “Thus theonly difference berween ax*anda.x~is itscharge, andthe#°isits ‘own antipaticle likethephoton. | C.Isotopic SpinConservation ‘Theneutron andtheproton atereally verymuch alike. They differ in(a)theircharge,(b)cheirmagneticmoments,and(c)theirmasses,thelsttoabout‘onepartinathousend.Thenuclearforces,ontheotherhand,donotappeatto distinguish beeween neutrons andprotons. Thus thebinding energies ofmizeor‘nucle,thatis,nuclethattransformintoeachotherwhenneutronsarereplacedbyprotons andviceversa, arealmost equal, withthe discrepancy explainable interms ofthedifferenceintheCoulombenergies.Ifchencutton-proton massdifference alsowete anelectromagnetic effect (and itssizeisconsistent with this possibility), then one could blame allthedifferences between neutrons and protons onelectromagnetisin. 4 Heisenberg, andCondon made thebold proposal thatchenature ofthe ‘nucleons mustbesuchthatificwerepossibleco"turnoff”cheircouplingtothe clectromagnetc field, thatis,theelectric charges, thenthere would benowayofdistinguishing betweenprotonandnevtroa,andthatthesetwoparticlesshouldreally beviewed astwosubsates ofasingle eatty, theaucleon, This notion_grewoutoftherealizationthatanelectronwithspin“up”andanelectronwithspin“down” inamagnetic fieldaresillchesume electron, even though the energy isdifferent. Inthiscase themagnetic feld canbecummed off,andthe “symmetry breaking” bemade codisappear. Fornucleons, thesymmetry. “See Special Topics section 3on“The Yukawa Theory” 430, Quantum Physics breakingelecrromagaetic interactioncannotrellybeturedoffexperimentally,botthis noburtier toimagining «world without elecromagnetism. Thepro. posal wasthettheanalogy with spin should bequite exact, tha iythat the(roenandtheesronshldBe"upand“down”statesof“issosin1/2cnttycalledtheaceon.Theaucleonis221=1/2sateandtheprotonandnevtton aceeigenstates ofI,with eigenvalue +1/2 and —1/2, respectively. ‘Theantiproton andtheantineutron aioform dovbles, butnow itis the sntiproton chehas1,=—1/2, andtheantinewcon J,=+1/2, Nuclea forcesfrenow“aucleon-aucleon forces,”andtheequalityofP-P,NP,andN-Nforces canreadily beunderstood ifoneassumes thie theaucleon-nacleon potential conserves theotal angular momentum inisotopic spinspace, thatis, thanapeinseme“Two nucleons, each baving I=1/2canform anI=1triplet andan 1=osingle. Thesats, incomplete analogy tothespin triplet andsingle, have the form pp aipleeVi(PN-+NP)singleeyi(PN=NP)(26-5) NN “Thenotation PandNhere isthe isotopic spinanalog ofspin “up” andspin “down” spinors x4 BythePauli Exclusion Principle, theP-P scates arelimited tocrlly satsymmertc states 'S,Pag. 'Dy «spin conservation demands thatche N-P system intheI= 1state alsoobeys chesame aymmecy. Thus theP-P andN-P forces inthese sates willbeequal (and equal cotheN-N force), but theforces inthe %,¥P,*Daza, «tas ofthe N-P system canbediferent, since these comtespond to1=0.That theforces arediferent isshown bythe facts deuteron without «comesponding P-PandN-N bound sate exists. Thissituationisanalogoustotheexistenceofspia-dependent potenialthatgive@ diferentforce foetheeiplet and singlet spin sates, even though thetotal sngular momentum iconserved. Ifwe introduce operatorsI(=laf)obeying the“angular momeatum commutation relations Unk] =i, Copel) 266) wwecanconstruct thewhole isoropic angular momentum formalism inexact ‘analogy with theordinary angular momentuen formalism, except thatthere is90analogoforbitalangulatmomentum connectedwithmotioninspace.spinConservation implies thatintheabsence ofeleccomagnetic interactions, the Hamilonian hasthepropery that ual =0 eon Hlementary Pericles 431 Furthermore, eigenstates ofIand1,wiheigenvaluesIl+1)and—1< I,<Tan beusedasbasis states. Theexistence ofthtee pions, x*,«°,andx, allwithspin0andpariey ~1endallofalmost chesame mass, fsinverywell withtheexistence ofchesymmetry; wecansaythatpionform anI=1triple with the1,=1,0,~1states represented byx*,2°,anda ‘Themere existence ofthese -spin multiplets maybeviewed asevidence fortheunderlying symmeny. Wecia,however, point coseveral other moce {direct manifestations ofthesymmetry ofthestrmg interactions thatareseenin the nuclear forces {@)Nuclei consisting ofZprotons andA-Z neutrons willhave yaZr4-2) 2 e =Z-A4p (268) B= sothatanyoneofitsstatesbelongs toamultiple oftotalJatleaseaslargeFf a5|Z—4/2],Onemighthopetofindevidenceforothermembersofthese| ——_-mukiplets inneighboring nucle, andsuch evidence hasindeed been found.|‘The/-spinpartnersofagivensetoflevelsarecalledanalogstatesandhavebeen theobject ofintensive study bynuclear physicists, Figure 26-3shows «par- ticularly clean example ofmultiplets. "0,F,and"*Ne have proton/aeutronBF —_—_aswokers(6,10),(0.9),and(10,8),respectively. Forthefistandthirdcheground |states could belong totheseme é-spin triplet; thespectrum shows thr'Fhasa ‘ground statethathasspin-paity 1,sotaiteannor belong withtheOFground F stares of"Ne and 0, but that thete isanexcited OFstate that could bethe 1,=Omemberofthecriplet.Furthermore, theseiremarkablecorrespondence between awhole sequence ofenergy levels inthethree nuclei, indicating chat ‘hey areallparts ofanI=1multiplet.Thefigueeshowsthatcheyarenot degenerate inmass, butchartheydiffer byseveral MeV. Thisistobeexpected j because theCoulomb repulsion berween theprotons doesnotrespect #-spin symmetry, Theenergy differences canbeaccounted forquantitatively inthis way. (b)Fspia multiplets alsoappear inexcited states ofnucleons. Ifoneex-aminespionsandnucleonsemergingfromahigh-energy collisionofapionot :proto withatarget proton, onecandetermine their momenta andenergies. Ifaparticular nucleon andpionweretobedecay products ofasingle entity, then that entity would have tohave I= 3/2of1/2, since theaddition of I=1/2andI=1canonlyleadtosuchstates* Furthermore, ifthedecayingstatewereatrestandofmassM,thenthepionandnucleonwouldhaveequalaadopposite momenta, andthesamoftheir energies would equal Mc.More‘generally,if(E,p)denotethenucleonenergyandmomentum and(¢q)denote “eis, ofcourse, assumed thatapi iconserved insucha“decay” oftheexited stare. Mote sot this ater! 432 Quantum Physics 8 cy " r—— sn cSa (se ce te A * Gan 4 #188) = r—— os 7—-r (438) — ‘ =—-* se) o—— Te (eas) = 2081 +$bcS es : —01108 2b r—_—usm ol Feige e040 ¥eijdy00 Fen geet Fig.26-3.ThelevelschemesfornucleiwithA=18TheI=1levelsfor"O G.=1), YF(Z,=0),and Ne (I,=1)show aremarkable correspondence.TheI=0levelsforFarealsoskerchedin.(DatatakenfromF.Ajzenberg-Selove,[Nurdeer Physics A190, 1(1972).) thepion energy andmomentum, then telativity tells usthacthemore genetal relation is +o @t ae =are (26) ‘Thus bymeasuring energies andmomenta, andstudying such combinations of energies andmomenta forpion-aucleon pairs, itispossible tolook forsuchdecayingstates.Figure26-4showssomeexamplesofsuchmassspectra.Oneofthemost common states found isanI=3/2state with Me =1236 MeV. ‘One knows thatitis 1=3/2, since themass peak occurs inthePrsystem, Benen Pasider 488 which as =32.eas, ofcouse, beeneecked thtthepakocca inheUiherseesThemullssullySsefp336)Icerenee Scns tatadeadstyofthesnuscaneionketone edoe clonindecathepnandetyte3/—Teseehenbas ‘widthofabout 120MeV. Thisisthe“narural linewidth,” anditindicates thattelisineoftheT=3/2aes ; i wo =aya ~Dox16x1sTOPXWHse (26-10) Nowonder theAcannot bedetected asaparticle leaving atackinabubbleGunter! é m= 1 |x |Feo 3 |° « pairs from reaction «~+P—+wte-N. (6)Plocofevents (#*P) asafunction of 434 Quaccam Physics x0 gis 5 i 510 3 Loe °ie Tao 30 709FP mas ier » Fig. 26.4 coatinued ‘Theconservation ofspin allows uscomake predictions about the telative decay rates of Pst Neal Net “Teprocedure iscompletely analogous toourdiscussion ofspinandintessicy rulesatcheendofChaptee22.Theinitialsaveof1=3/2andy=1/2maybe wwiteen intems of T=Land f= 1/2wave fonctions asfollows ry ju Yurus©fea°P+EeN (26.11) VTP ENS vete Yay fepesents the. Thus theprobability offadingaPy®stateis2/3 andthatoffinding Ne*stateis1/3,These predictions ateborne outbyan analysis ofthedata “Espin conservation basbeentested inmany teactions, andsere isa0ques sionascoitscorreciness, subject rosmall electromagnetic corrections. Thefact that these comections atesosmull, forexample, thc cheneutron: proson ass sifference iss0smull, made stpossible toidentify thesymmetry. When sym retry breaking islarge, thisbecomes much more diffcule. ] +’ \ Wer . VW SY fr é‘Ven | theK*undergoesthedecayK++”+(CourtesyofLantenceBekeletake 436 Quancam Physics D.Strangeness Inthe late1940s itappeaced asifchebasicingredients ofatheory ofthe stronginteraction, thosetesponsible forthemuclar forces, wereestablished. ‘Theforcesactedberwecenthenucleondoublets,andthe“glue”givingrisecochese forces wasthepiontriplet, aspredicted byYukawa, There remained thein superablezechnical problem ofmaking reliable calculations (because thepotential isstrong, pertabation theory canner beused), buttheexpectation Wasthehis problem would, sooner oFater,besolved, Itwastherefore veryexciting when round 1950cosmic rayexperiments, andlaer,thefrsthigh-energy acccleators, indicated theexistence ofanewsetofparticles, Thefistparticle sodiscovered‘wastheA°:itwasfoundthatwheaacloudchamberwasexposedtocosmicays, fcertain umber ofV-shaped tracks were seen. When theexperiment was repeated inastrong magnetic field,itwasfound thatthe tacks bentinoppose directions; checucvature ofthetracks inthemagnetic fild determined dheit ‘moments,andtherangedeterminedtheitenergies.Fromths,theapplicationof, (266) allowed thedetermination ofthemass oftheparticle thatdecayed into thePand#7:itwasgivenbyMyc!=1115MeV.Theapexofthe¥,matking thepointofdecay,appearedacertaindistancefromthepointwheretheproduc:tioninteraction occurred. From thisthelifetime oftheA*could bedetermined, andwasfound tobe2.5X10-¥ sec.From thenumber of A'sseen in&given rnomberofphotographs itwasdetermined thattheywereproducedwithacrosssection ofthe orderof10-#em.Inadditiontothedecaymode(seeFig.26-5). BoPte thedecay mode MONE ‘waslaterestablished. Another pairofparticles werealsofounds these wetethesigmas, withAse=1190MeV,lfeimesrs"=0.8X10°"sec,r==15X10- sec,and dominant decay modes etePe Nat Nee ‘Atabout thesame timeother ‘racks appeared chatworefinally interpreted as ‘caused bynewparticles withNp=0.Therdecaymodes were(seefrexaraple, Big.265). 3 Elementaty Paricles 437 4 peor F ee } ty g eo & Xrete Inspiteoftheeiserent foalstates, thesewereautibuted :osingle partces: because imeachcasechemasses cameoutmyc?=494MeVandthe Mfetimes alsoclusteredaroundthevaluesrai=1.2X10-'secandrye=08X10-500,TheseKmesonsweralsoproducedwthrossectionsoftheonerof10-7cn Thediscovery ofthesepcticlessndthetproperties causedaciesshe data,takenatfacevalue,werenotcompatible anditkeseriously pled taut «quantum mechunis could notdescribe thebehavior ofthese particles. Toace this,letusdescribe theatic clement forthedecayA?—»Pebythenusber G.According totheGolden Rule,thedecayxeisgivenby 2oyHUMEaetapdb , BOS ate~5OGeiye esr) i With reusivitic kinematics (pistheceoter ofmasciomentum) Bm[Me +PED [ime +pan ‘dp/deevaluated utE=Mtcanbecalelared, andweget,writing @in |dimensionless form as @aahe (26.13) the rate A(me) (be)(Bay(Be nesmG)(&)Ce)(ore oe) Thedimensionless numberAitcheanalogofthefinesraccureconstantaPorting innumbers wegetforthedecay tee R06 AX10%sect (2615) 458 Quanum Physics feo which weGndtae8&0:7X10-". Thisnumber iverymuch smaller thanthefinestructure constant andsuggests thatcheinteraction responsible forthedeny mmuch weaker thantheslecuongnetiintracton. Onthefb and ifthe Ais produced inaconjectted rection suchss rtPoe te conecanusetheA°Pr~couplingtoestimatethecrosssection.EstimatesoftheFreaeuedincur disarm ofthephoroclecc efecsuggest tahecon TEjas mane proporcona 8.Teages ae”involving themasses of theparticles involved chusgives, withtheabove enn(ZY15x10a (2616) “Thisdiffersfromtheexperimental valuebyfactorof10"andnominorchangesinthe etnaer casoeuse husappeas thatthe Asproduced by200g imerscon addecays through awenk iteration “Ihewayontoftederma wassuggested byPais,whoproposed that theproduction process necessarily involves another oneofthenewparticles, 30 havesccoat hvoling ps ofthe newparle could proceed only, eres rections involving onlyoneofthen woul hav©goslowly. Tus, thecoojcreed acon, rtPowee should noeake place, buthat et PSea focexample,could.ThesuggestionofAsiaPrdutonttned ove1beceefeesoondeterminedthattheproductionof+Awastnays“Accompanied bytheproduction ofaK."ThelimitationsofthePaisproposalweresecnnensiltnetherprice,namedtheCascade(2),masdiscovered,PeInninyedencehomedctodecaywithfeteof7X10°scacon ies Boe tr butitdma decay acon cothemode EONtE resmass wasfound 0beMae =1321MeV. Ifthe Z"hlonged wit he3% thewheeedecay mode has4paiofthe"new" pls, andshould go reply the “worl” hentheesty BoNte soul gopity, insend ofwot a Elementary Paris 439 fe mom op 4 | a :its"|= —Oe°wow] owe) baryon sone ¥ Fig,266.Hadron specu asknownin1955benethediscovery ofother {paces peeiced bystangeness theory. Theenergies areonlyapprotiaate, td4)thenewparticlespedicedbychestrangenesstheoryaegivenbydhedatedlevee * Orderwasbrought intothejumbled situation byGell-Mann andby5)Nishijima,whoindependently proposedtheextensionofthenotionofspintoJ Ghenewpues tndinodued sevgrants mania heentiPs speccrumofbaryons(theamefornucleonsandthenewparticlestheultimately § endupasnucleons) andmesons was,by1953, believed tohavetheformshown; inFig.26-6.Irisleartharthe3—A°masdiferencejstoolargetoputchem{¢—__intoan spintiplet.Thus,intheabsenceofequalmasspartes,theAhadvoif beanI=0sae. Since thepeoduction crosssection inthe teaction iC HPO ER vslargeenough tobeviewed asastrong (tather thanelecromagnetic) process,i-spinshouldbeconserved.Thelfsidehas1=3/2or1/2,andthustheK®‘mustbelong tooneofthesemultipless. Theabsence ofenyobserved K**or A,needed tomakeupaquact, showed thatthe K°would avetobeputof a01=1/2double, withI,=—1/2equaltotheI,ofthePxsystem.TheKe vwasundoubtedly the=+1/2partneroftheK°,Thelifedimesofthesetwoparticles could beverydiferent, since theweak interactions need notconserve‘spinanymorechantheclectcmagnetic onesdo.TheK-isundoubwedly thesncipatcieoftheK+,andisJ,=1/2partnersdenotedbyH°.NowethactheRe‘tnnoc beidentical withthe K°sincetheyhavedffeencealues of1.Howisit : possibie for#neutral system nortogointoitselfunder charge conjugation? ‘Wesawthatthe atineuton difred fromtheneutton, because theyfeed in 1 thevalueofbaryon number Np.What ®thequantum number thetcstinguisheshecween the K°andthe K°? 440 Quancurs Physics “Toanswer thisquestion, letusconsider therelation between J,andthe clectrical charge. Forthenucleon system wehave 1 Q=h+) and forpions o-k “These«wocases,andthatoftheaninucleon, canbecombinedin No onus “This formula, however, does notwork forthenewparticles. Ifwemodifythe formula bytheintroduction ofanewquantumnumberScalledthestrangeness, sothat Na, S . gabe Ste (2617) wefindthatfornucleonsandpions=0,butfortheA°wemusthaveS=—1.‘Thus theK*,anditspartner, cheK*,must have S=-+1,andheace theK-and the mustbave5~~ftassuethcgconugconiodunghe signofS,a8suggestedby(26-17).TisisthereasonthattheK®andtheK®arediferent. How this manifests itself wewill seelate. Continuing with ourextmination ofthebaryon spectrum, weseethatin theabsence of**and/or 2, itseems natural toassign T=1tothe2's. Thishowever pradict theexistence ofa2°,ofmass close to1190 MeV. Why wasthedecey Borie (analogous tothe A*decay) never seen? Gell-Mann pointed outthatthe electro- magnetic decay Posty did nor involveachangeinstrangeness,incontrastcotheformerdecaymode. ‘Wich chepostulate thatadecay inwhich strangeness changed byoneun, last = (26.18) should go“weakly” witheypical lifetime oftheorder of10-*sec,while srange- nessconserving reactions should gostoagly, orelecromagaetically, itwas possible copredict chatthe2°—>A”+decay should beverypid (7~10-* sec),leaving noopporeunicy frtheweak decay totakeplace. Acareful study of 1°decays showed thattheyfrequently originated inareaction inwhich some momentum andenergy wasmissing. Aneximination ofthemissing momentum 4 Flemenacy Paricles 444 | andenergy showed chatthemass ofchemissing particle wasconsistent with0, 4ndthatthephoton andA°were decay products ofsomething with mass 1192 MeV, confirming theexistece ofthe2°.How dothese notions, thatis,con- "servation of$inthestrong andelectromagnetic interactions, and|S]=1in SF heweak ones, work forthe2?Theweak decay te Boe te 7suggestschathestrangeness ofthe3~mustbe0ot~2.Theformerassignment4) isincompatible withthe absence of * andhence wemustwke $=—2.Thisimplies thattheZ~hasI,=—1/2, and % imtheabsence ofmultiply charged 5'sweasign toie11/2.Thispredicts a partner, the", which should havemass arouad 1320MeV anddecay according 8 Boxee Jthedecays2+2+watenorpossiblebecauseofthemassesoftheparticles ©involved). The2°waslookedforandfound!Theexplanation fortheabsenceof , BoNte 5: liesinthatitischacactrized by[4S]=2,which ispresumably doubly weak,2characterized pethapsbya8ofmagnitude 10-*.iE “Thenotionofstrangeness conservation inthestrongandclectromagneticis‘teractions, and[AS]=1intheweakones,haspassedeverytest.Thedasi# ——Gaacion ofparticles byé-spin andstrangeness, otequivalently byhyprcharge Y i defined by te Y=Ne+S (2619) i isgiven inthefollowing cable. | Baryons Mesons “Anuibaryons: yor Newt a= 0 Ne=—t 1 1 RN KR FB i ° 1 rs re ERP oo ” ? x j 1 _ at BE ee RE 442 Quantum Physics Subsequent determination ofthespins andpatties ofthese particles showed that allthepatticles inthefirstcolumn were 1/24 andthose inthe second column were O-.Ifwearelooking forpatteras, whete isthemissing I=0,¥=0pseudoscalar meson? E.Unitary Symmetry ‘The search forafamilial relation among chebaryons andamong the mesons wasactively pursued iathelate1950s. Finally in1961 Gell-Mann, and independently Ne'eman, discovered ageneralization of#-spin, bearing the | " | Y8 » 1x « hb x x «x | a x x 2 x | a, ETT, | 2 x Kx n bx we ge mx a x oe x Oa, Fig.26.7. Some SUC) representations, Thenumberof cronses ateach sitetepre- sents themultiplicity. Thus the27consis ofaY=2,1=Ostate,Y=1,1=3/2 tnd1/2sates, Y=0,1 =2,1,0tates, and50on. Elementary Paricles 443 technical name ofSU(3), Since group theory, thetoolmostwidely usedinthe search, isbeyond thescope ofthisbook, wecanonlygiveaveryqualitativeideaofhowthefamilialelationlooks.1aSU(3),statescomeinsupermalkiplets.Each supermuleplet willconsist ofanumberofstatesthatcanbelabeledby ©ispinaswellashypercharge. Figure 26-7shows theseveral supermuliples,the octet,8,thedecuplet, 10,ndthe27.Figures 26-8and26-9showhowthe 1/2°baryons andtheO~mesons fitintotheoctetpattern. Themissing 1=0, ¥=0pseudoscalar meson wasfound iatheexamination ofx“#7x° masses in bubble-chamber pictures, Astudy of (me =(Ex+Bot EY~Ap +pt pot showed astrongpeakingatmy*=580MeV,andananalysisofthedistribution‘oftheenergiesandmomentaamong,thethreepartiesshowedchatthedecay {anninpie!dcguatunnoninooradsceMesbuareeeke BEcomelations inwtx'e-,sty,showedthatthei-spinhadtobeaero.Sincethe FRdecaywasclearlynotweek,thestrangeness hadtobe20 aE Ifthe SU() symmetry wasindeedabadlybroken(badly,comparedtothe: onlyslightly broken i-spin symmetry) symmetry, thenitshould alsoberelevantqtotheexcitedstaresofthenucleon. TheI=3/2,¥=1(1236) resonances Hieoveded parmnes, In1960asecofAYxresonanceswetediscovered,The/-spinwas cleatly1andthehypercharge was0,anddeasiledrestsshowedthatthespinand Se_pasicywere3/24,s0thattheseso-called2*(1385)weremostlikeypartnersofthe‘4(1236) inasupecmultpler, Thesimplest possible sssignments werethe10aad the27.Gell-Mannandothersconjecturedthatthe10-wastheappropriatechoice. ‘Thediscovery ia1962 ofaZxresonance ofmass 2531MeV, withI=1/2 y x : 1 \\ \ \ j| a ae Ne ent Noro Naot ; 4 Ne fext 1 a Fig.26-8. Octe: pattem forspin1/2baryons. Thediagonal linesatelinesof constant charge Q * 4446 Quantum Physies y x S | r Pees Q © F rT ‘ Fig.26-9. Occet pattern formesons. Included istheT=0,Y=0particle 4°discovered in1961. (a0Zrresonance,forexample)sronglysupportedthisassignment. Whatwasstillmissingwasthelastmemberofthe decuplet, anI=0,¥=—2negativelycharged particle.WhatwouldicJooklike?Herethepatcernofmassesoftheother members ofthedecupet, (1236), *(1385), and=*(1531) suggeseed equal spacing increasing linaly with |¥|. Ifthiswere cobemaintained, the missing partice, called theO-,would havecohavemassiathevicinity ofMae=1675MeV.ThiswouldgivetheSTauniquesignature;iesmassistoolowtodecay stoongly into2+K,andhenceitmustundergo|S]=1decayto A°KotZr.Itsproduction wouldalsohave@specialsignature,sincethelowestpossible Y-value foraninital stateisY=0forK~+Pcollisions.Thus,co makeaY=—24F,twoK'swouldhavetobeproduced.‘Amassive search forsuch aparticle wasundertaken, andin1964 the first9picture waspublished (Fig.26-10). Theproduction process was EPO Ta and thedecay @rte Looe Loa,ss pteac Hlementery Parces 445 “Thefacecatia thex”decay Bort, both+-tays produced pairswasveryfortuitous andmakes thepictureatextbook ‘exampleofwhattheoreticians dreamof,butexperimentalists seldomsce.The sassofthe9wasfoundtolieremarkably closetothepredcred vale,at |1672MeV, Thespinandparity have notyetbeen measured, since todatechere ‘exist only282~pictures, butthere isaodoube inanybody's mind ofwhat the outcomewilbeTheSUG)symmecry, justlikespinconservation, makespredictions aboucdecayratesamong particles inthesimesopermaltple. Thusthereare |predictions relating At —Pf xt,B¢*— 4°+xt,andEEthatareingoodagreementwithexperiment,giventhefactthatthesymmetryisbroken, Thegenetally accepted viewisthatSU(3) isindeed anundedlying sym- meiryofthestronginteractions.Themechanismbywhichicisbrokenisnot |yetwellvadrtoos, hough soa pater (ach theea pig flem ese devi \FrettainPEt \PV/|VBP AAS +2 P boAE Ba \{ jeager Dae pore ayy NA] ery eye BBL ta 3) fe Th TF . it Fig. 26-10. [email protected] ofthe@isbelow theZ+Kmass, sothatasrangenes violating (weak) decay isinvolved, Whats sen inhebubblechamber pictures thesequence 58" fe Bt aS ohn Dy), Ae Po Pron Barnes aal,Ps. Ree.Lets, 12,204WB64), courtesy ofBroothuven National Labortocy and Be.NP. Samos 446 Quantarn Physics ‘thedecuplet) follow from some simple postulates about thenature ofthe symmetry breaking. Allofthe many resonances thathave been found canbefitted inco supermuliplets, although frequently there arestillundiscovered partners. Jeisaremarkable fact that allofthemesonic resonances (and there arenow 1°,1%,2and(probably) O*octets) andallofthebaryonic resonances fitinto‘octetsformesons,andoctersordecupletsforbaryons.Thisabsenceofhighersupermultiples, forexample, the27,cannot beunderstood onthebasis of SU(3) alone. Itdoes follow from 2simple composite model ofelemencary particles, called thequark mode! fistproposed byGell-Mann andbyZw, F,The Quark Model ‘Thequestionofwhich particlesareelementaty hasbeenapressingquestion inthisfundamental field,andwiththerecognition thatthe"elementary" protonandneutronhadsixpartnes,itbecameclearthatifthereweresomeelementarybuilding blocks, chere would most probably befewer than eight. What might these building blocks belike? Wehavespin1/2particles aswellasspin0,1, 3/2, «particles. These canbemade outofspin 1/2building blocks, butnot‘outofspin0buildingblocks;similarlyweneedani-spindoublet,atleast,0‘make upispin 0,1/2, 1,and3/2states. Inaddition, weneed atleast onemote particle dffecing inhypercharge from thedoublet, sothatvarious Ystates can beconstructed. SU(3) happens tohave, asitssimplest noncivil states, athree- particle representation anditsanciparticlerepresentacion. These representations snared “quarks” canbeused tobuild upother SU(3) representations,useike angularmomentum 1/2canbeusedtobuildupangularmomencurs staresofJdiferent from 1/2, The rules tuts outtobe se5- 841 3@3= 643 30303=10+8+8+1 sochatitisplausible coassume thatmesons atemade of“quarks” and “anti-‘quatks,”andbaryonsandtheirexcitedstatesaremadeoutofcreequarkseach,Inthiswaytheabsence ofhigher representations canbeunderstood. Jnorder ‘0mainteia the formula y geht thefollowing quantum numbers ateassigned tothequatks, which arelabeled with lowercase leters related totheir spin concent, p,»,andX Hlementary Patces 447 Particle Np B ¥ 2 ? i a i a i *i 4 +4 3 4 3 a+o ° i a x+ ° ° F + * a + 4 4 a ? +4 i + = =i Toconstruct thecomposite wave functions foraquatk-anciquatk system, we stattwiththehighest ¥andhighest Istates; loweting fycanbedonebysuces sively converting p> »or 3»p.Togetthestates ofonelower uni ofY,convertapahoran9toaX,Forexample, KY=() Hence = (i) Togetthex*stare,convertXintheK*,Thsyields xt=(pn) F andchen, succesielyt Lee vit q r=) ‘TheK-andK°aejuttheentipattclesoftheKdouble,sothat 4B=0) =o ‘There remains theI=0,Y=0,n°state. Itcanbeobtained byconvertingp—>hintheK+,Whatonegetsispart9°andpartx°,butsincewealreadyknow‘whatpart”is,the4°canbefound byinsisting thatitbeorthogonal cothex”. “The choice turns out tobe Lak—a 7=7gOM=Ph-wi) -__£Thesiou sigeinthe =isatechnical subey. eft, thesaipinorto (it &-3 : 448 Quantum Physics “There remains afal possibilty Nom75(ob+mw+OH) ‘arthogonal coboth#°andn°.Thisis,infact,the1inthedecomposition 3X3= 8+1,anditdoescorrespond toapacicle chathasbeenidentified, the9/(958), ‘which alsohasspinandpatty 0 ‘Thebound seates with L=0,thatis,the'Sstates, arethepseudoscalar ‘mesons (recall thattheparity ofanaatiparticle hasanadditional minus sign). Onecanimagine L=1states, forexample, "P;(i, 1-)bound states, andalso *Px4 sates (21,1°,0°),andsoon.Many ofthese have been found. Thequark model, being mote specific thn SU(3) yields more predictions, comelating clecays ofparticles thathave different spins "Theehree-quatk wave functions canbeworked outjustlikechequatk-anciquarkwavefunctions.ThehighestY,Jystateisthe(ppp)state,whichcanbeidentifed with theA+. Itspartners areagain obtained bysuccessive conversion of p= a=(pp) 1 ar =yaen+bmp+mm) a=Felontape+mp) a =(one) ‘The2*+isobtained from (ppp) bychanging pro. Thus 1 x= op5G+ P+MP) woe ebNEpe EH=Sag(NEPhPhe+Now+Mp) 1 zt=ata+he+a 7h heha) Successively 1 zea a »yEM = +NA+ mY) 1 =Lan ten+Ni) B= eM HA EM) aod =m Blementary Paces 449 ‘Thefour multiplets difer inchenumber of\quark. Ifiisassumed thatthe symmeny breaking completely resides inthe facttactheNqui issome 150 MeV more massive thanthe(p.x) double themass partetn ofthe decapler can beundetscood. When thisargument isapplied tothemeson octet, itdoes not wotk aswell with a150 MeV mass difference. However, che relations i=2my+binding 2 1my=2(2m)+1(am)+binding | Fam) +5 s lead co 1 sme=1Ging +me) 256 (26:20) {which isknowa asthe Gell-Mano Okubo mass formula. Tewotks toabout a 2 10percent accuracy, butisverygoodiftherelation iswritten forthesquares of © themasses. Incidentally, cheseformula willwotk forother octets, andfor x thebaryons itwillhive theform j L 1 5(ove+ms)=1Gm+ms) 063 metim)=1Om+ e621) | Teiquite accurate andissometimes usedtoestimate where partners ofincom. ; plete octets eight belocated Thequark model lends tomany other predictions. Forexample, iftis assumed that athigh enegpies allquarks andanciquatks interac identically, leading t0equal cos sections, then itfollows chatinPPcollisions there are nine possible interactions andinxPcollisions there aesx,sothat PP) 3 oP) 3 ‘sey" 2622) Surprisingly, thissimpleminded counting wodks verywell, both inthisinstance andin*many more. Icisanurgent andasyetunsolved problem inparticle physics tounderstand whyquarks, which ust beverymassive iftheyexsc at all,since otherwise theywould have beeen seen, actinsuch &simple additive manner, Other questions remain. Why dothrce quarks bine, butnottwo? More deailed considerations show thatquatks, even though assumed ¢ohave ; spin1/2,actasifeheydidnotobeyFesmicDinac statistics, Whyisthisso? We do not kaow. 450 Quantum Physics G.Parity Nonconservation Inadditioncochetronginteractionsandtheelectromagneticinteractions, thereexist,iaoarureweakinteractions.Theywetefistdiscoveredinbetadecay, that isthe teaction NoPte +3 andrelated ceactions such espostton decay PoONtete andthecapeare reaction O+PIN4e with thelastewo occurring only ianuclei. What was observed was4nuclear decay oftheform AZ AZ+D te ‘Theelecttons didnotcome outwithafixed energy, astheywould havecoifeis wasatwo-body decay, although themaximum electron energy matched that available foratwo-body decay. Faced withthechoice ofgiving upenergy con: servation otproposing anew partici, Pauli in1931 postulated that thete exista ‘cutralparticle emitted inthereaction with theelectton. The properties of thenewparticle, named theneuriny, were thefollowing: 1,Charge conservation required that itbeelectrically neuer 2,Theequality ofthemaximum clectton energy totheavailable energy requited thatthe neutrino mass beverytiny; itisnowbelieved tobezero. 3.Studies ofchespins ofthe initial andfinal nuclei required theneutrino tobeafermion. Itisnowknowa tohavespin1/2. 4.Theneutrino wasnotfound when icwasfestpostulated. Thereason isthaticinteractsveryweaklywithmacter.Thecrosssectionforneuttinoabsorp:tioncould becalculated withadetailed theory proposed byFermi in1932, anditwasshowntobe10-cm?atlowenecgics.Thus,inspiteofisesotericnature,theexistenceoftheneutrinowasacceptedbymostphysicists,anditwasfaallyidentifiedin1954.Nowadaysneutrinoscomingfromthedecayofhighenergypions areused tostudy high-energy neutrino-nucleus collisions “TheFermitheoryofbetadecayexplainedacassofweakdecays,including those involving&newparticle,themuon(a),whichisforallpracticalpurposes anelectronofmassmyc!=105MeV,whichwasdiscoveredinthe1940s.Iecouldalsoexplin inprinciple, ifnotin detail, decays such as *)wo ,(3)+ 7 Elementary Particles 451 sincethese could occur hough theeps reoP4+N pans(S)ey ” withtheN-Nannihilating. Withthediscovery ofthestrange particles, weak sorte and : Kotte Kotte ‘Theaterdecays anced much awentio. Astheexpeinenial datbeg to point othe fcthatthe Kmeson hadspin patsdox noes Thedese Kot Fito ewospinless particles implied thartheorbital angular momentum alsohadFsovanish Sincebthpionsweeofnegaivepry,sheimplcaonwastatheKhadpositive parity. Oncheotherhand,adecailed studyoftheenergy disttibu- tions ihedecay Kase suggestedverysongchatalheeponswerinstaterelivecoeachothe,tsmight havebeenguested fomthesll amoun ofLneteeoegy sve, tothethreepions inthedecay. Thismeant thattheparity hadtobe(—1)3, thai of,since there werehe negative pty pares thefalse ‘hese conclusions wereinconsistent wichthewelleababed principe oftheF——inmaance oftheweofpisunderspaceection ‘in1956 LeandYang, inaveyimporsnt peer, sed hequestion, ‘Housdowerealyknowthatpartysconereed intheweakinteractions? TherewasnoF.—coubeabour thewlcty ofpentyconservation inthecectomageety ee {actions Paty conservation inplis some scecton rls, ndthet ested toahighdepres ofaccuney, Thisdgreeof sccm bohighenough, how er, 16s tothing about thecontrton afpay ashe wel neacconevel.Inthedirectstudyofweakinteractions, there‘arealsosomeselection tales forexample, sheKshould notbeabletodecay intoSveandSot he coer, whatissce rocheck putnonconervatio istoamine ealObserable tatallows ustodistinguish beeneen Owwend aedoseeete at Acre inairr Thequestion ianediaey ages, Would no wateofss slecton, moving wth momentum, beelected int sate with nomearun 452 Quancum Physics =p,andwould aotthisdistinguish beewecn thetwowodds? Theanswer isthacthiswouldnot,providedchatbothstaresareequallyprobable,sothatifwesce tneleczon with momentum ~pwearenotforced totheconclasion that we | lveinthe“mitor” word. The existenceofeliptcalorbsinplanetaryphysics isnotevidence against theinvariance ofthelawsofgravitation under rotations, unless some ellipical orbits areprefered coothers. Thus todistinguish berween fourworld andthe“mirror” world more subtlety isneeded. Suppose wehadaone-dimensional potential chat violated patty con- servation, chais,Suppose itwasofthe form V6) =Vena) +Fouls) (2623) andsupposethatVese(x)isverymuchsmallerthanVigea(2).Ifw(x)areche | eigenvalues of iy=E+Veal) (26.24) then thelowest order energy change duetothepresence ofVoaa(s) is at,=[7deae)Vaal)0) (2625) ‘NowtheHamiltonian Hyiseveninx,andhence,asdiscussed inChapter4,theeigenfunctions 1,(x)canbechosenaseigenstates ofthe parity operator, that is, they areither even orodd Consequently, Asvanishes, thati,ameasurement ‘oftheenergy cannot beusedcodistinguish between aworld anda“micor” swocid, Thesecond-order energy shift willnorvanish, butsince iisquadentic inVoaa(2), itscontribution willbethesame inthetwoworlds. ‘Theargument cinbegeneralized‘Wecansimiladyarguechatadetermination ofadecayratecannotdistinguish berween thetwoworlds. Ifparity isaotconserved, itispossible for ‘thematrix element forsome trnstion tohave theform M=Mone +Mas (2626) BytheGolden Rule, thedecay rtehasthefrm R=a3)Mera+Matt}(E) (2627) Tathe “mimor” wot this takes theform R-S|Mam-muloe) (26-28) Blemencary Panicles 453, (Onthefaceofit,itlooks asifthedecay atesatediferent, Wemust, however,becatefulindescribingthedifferenceberweenthetwoterms,Fitstofll,Mosmustbeascalar;itcannotbeavector,sincethiswouldsingleoutadivection in space andulcimetely imply lackofangular momentum conservation, Thus if there aremomenta pyintheprocess, thea Mera cinbe&function ofvacious products pips Iftherearespinvectors present, theniccanalsodepend on Se), andon(pi-S,), burnotonpr’, since thelsti«pseudoscalar quan- ‘icy;whereas momentum changes signoder aniaverson, cheangular momen- ‘umndoesnot(€grXpdoesnot,andhence thespins Cannot). Ontheotherfhand,Mocamustbelineainxpseudoscalar quaatity.Thusiftherearemorethen threeindependent momentainthefinalstateofadecay,apossiblepseudoscalar ispreps Xps.Intwoorthtee body decays, thepseudoscalars mest beofthe form S-pwhere Sisoneofthespinvectors andpisoneofthemomenta. Hence, iffordefiniteness weassume that M= A+ BS-p (26.29) then Z|Maen4Moual?=DlA)?+DIB*Sp)? £2 (AB +A°B)S-p (2630) Inadecay ratechespinstaes areusually surnmed over, thais,nomeasurements involving thecorrelation ofthespinandmomentum atemade. lnthatcasethe Jasescum vanishes, and therates arethesume fortheworld and the “mior” world. Icisonlyifthepresence ofacorelationsuchasS-pismeasured,thatis, ifapeadsscalar quantityismewsared,tatpaitynoaconservation canbedetected.LecandYang then suggested several experiments involving theweak interactions inwhich such cortelations could bemeasured. Within afewmonths q ‘oftheappearance oftheipaperanumberof experiments showed thatparitywas indeed volted intheweak interactions. What does thisdotothecherished notionthatthelawsofnatureshouldbeinvatiznt underinvetsion, tht,sotospeak,it shouldnotbepossiblecoinstructanextragalactic beingonhowtomakearight- jhanded screw? Itnowappears hactheweak interactions notonlyviolate puticy conservation, butarealsonotinvatianc under charge conjugation, Theyare insarians under coibined charge conjugation andinserson, CP.This, from aroutine ‘acrempt todetermine theparity oftheKmeson fromitsdecay grew (a)the ‘experimental verification thatpaity isnotconserved intheweak interactions, (b)subsequent clarification ofmany aspects oftheweak interactions, not possible before partynonconservation hadbeenobserved, and(c)thediscovery ‘thatpacure shows more imagination thanphysicists, asinsupplanting Cand P byCPinvariance. . q 454 Quantum Physics H.TheK°—K*System ‘Asourfinaltopiewediscusstheimplications oftheconsequences ofthestrangeness theory thattheK°isaotidentical toitsantiparticle, cheK°,since these implications make useofsimple quancum cheory andarequite staring. ‘Asnoted before, what distinguishes theK°from theK°isthe strangeness, and inaproduction process ieisclear which ofthese isproduced. Thus inthe reaction rt PoNte wweknow thataK*isproduced; iathereaction K+ PONtR ‘wealso know that itis theK®thatisproduced.Giventhattheparticlesae pseudoscalar,wefind that P|K*) =—|R*) cP|R) =—|K°) “ThusbothK°andK°maybeviewedaslinearsuperpositions ofCPeigenstates;ifwe weite 1 pe Re SKK) R=(-h+k) ; Vite Bee Ck+m) (6st) itfollows that CPIKs)=[KiCPLR)==|Ks) (26:32)SinceCPisconservedintheweakinteractions, andthextx~systemwith2er0angulac momentum iseven under CP,inthedecays Rote Rowe itisreally onlyKi,thatisdecaying. BothKiandKcandecay intosome ofthe ‘other modes, forexample, wry Ingencal, bothKandKocKyandKare equivalen basissates inatwo-sitespace. Thestrong interaction production process actsasapolarizer, producing a particular particle K°,sty,ofequivalently, aparticular coherent mixture of and K;(coherent, inthesense thatchephase relationship isfixed). After 10"! . Hlemenary Pics 455 secseangenessuologemeasanything,Theweakdecryintor#¥~2c28antnalyer, and picks OutKGll Man andPaisin1953pointed outthatthe remaining Kcompose persis, an,sine xan decry noche chanel available totheKy,presumably hasalonger lifetime andshould belooked for, with oneofthe akemative dey moder. TheKywalooked forandfound. IFaad2lifetime ofabout5X10-*seccompared wich0.8X10-1secfortheKi. Other ineesing elects emerge. PasandPiccion noted thatone stars wth4K°beam,chenafter10°seconeisletwith1/+/2Ka,thatis,abeamofthe form4(K°+K°).If,beforetheKydecaycantakeplace,matteisinterposed,then, because ofthedifferent strong interactions oftheK°andtheK°com. ponent orempl, REPO EP Ry poRae 4 K+ POKt+N R+Porte R+NOREN Par theparticularphaserelationisdestroyed, andoneaolongerhasapureKebeam. Hence, 2puswilagin beser,sncethemitre cowwliavane some ‘Thus under theidealized conditions that alltheK°areabsorbed, and theK”inclseatedinthefrmection,whaeegefomteinerposedabofmaterial is$K°.Thephenomenon, known asregeneration, hasbeen observed. andstudied indetail. (SeeFig.26-11.) Theverification oftheprediction ofGel.MaonandPaisasstigsupportooureliinthevalidityofquantamechanics astheproperframework forthedescription ofsubatomic phenomena. ¥ Thisis«goodnoteonwhichtoend.Thereader, having mastered the cai htwehavepresented, steady godeeper intthesao ant mechanic iawhich more sopiniated mathomtial ooe neneesty. Sachsey wlrnghimoreotheroaofkaowledge,betinthenes |fonofdhesrctueofelementarypriestenergiesofblionsofwattheproperties ofmatter at10? K°,otthenature ofnuclear matter onthesurface andInsideanewtonsat,Whereverheosechoseogo,thetwilbeencimaandsurprises. icheers aenerstor Ze Fig. 26-11. Schema ding ofregent epesinet 456 Quantum Physics Problems 1.A.10 MeV positron collides with «hydrogen atom acrest.Write down theenergy andmomentum conservation relations, Taking intoaccount eefactthactheprotonmassisMyc?=940MeV,whatwlbethecoergyoftheemittedahoton from thereaction «+ HP +3? 2.What isthethreshold energy forthe production ofanantproton in che reaction PH PaP+P+P+P ‘One oftheinitial protons isatrest. 3,He?(PPN) andHP(PNN) atelikely candidates foranspindoublee (@here isnoeineutron nucleus). Consider thereactions *°+He®P+D<<raw Show thaté<pin conservation predicts that a(He’) ote) o(H) Hint,Writeouttheinitialstare(whyisitanispineigenstate?) intermsofthe (analog of¥,) andHe, H®(eoalog ofxy,x-in spin1/2wavefunctions), aswasdone in(26-11). 4,Which ofthefollowing reactions canproceed strongly, which weakly, aodwhich notatall,andwhy? P+ PoP +A+ P+ PP+ Ae+xt +P Ee wht P+ Kt PLES +R +e PePaw teen Korte te Kotte 5.Consider «beam ofpions impinging on2proton target. What isthe threshold forK®production? What isthe threshold forK~production? (ting, Sere inthecenter ofmas Fame.) 6,Calelae cheparameter introduced inSection Dofthischapter chat characerizes thedecays (@)2» N-+ wtand(b)K*—rat+#°,Theratesare 06X10 sec“ and 1.7X107see~ respectively Blementary Particles 457 " 7.Apatticle isscen todecay weakly (lifetime oftheorder of10! sec) 4sfollows: Xow bat ‘Whatcanyousayabouttheparticleonthebasisofthisinformation? Consider(2)limits onitsmass, (b)éspin, and()spin,parity. tacowhich SUG) super- multiplet, oftheones discussed inchischapter, could thisparticle fir?Whacwouldbethesignificance oftheobservation ofsuchaparcicle? References Plementary particle physics isdiscussed archeadvanced undergraduate and fiseyear graduate level in DL, Perkins, Intaduction toHighEnergy Physic, Addison-Wesley Publishing ©o,, 1972 See also E,Segre, Nacle andPartcls, W.A.Benjamin, Inc,,NewYork, 1964 ‘Anelementary, nonmathematical discussion maybefound inthefascinating litle volume K.W.Ford, TheWorld ofElementary Particles, Bhisdelt Publishing Company, New York, 1968, Recent developments areusually written upinareadable form inScewifc Americas, special topics 1,Relativistic Kinematics 2.TheEquivalence Principle 3.TheWentzel-Kramers-Brillouin Approximation 4.lifetimes, Line Widths, andResonances 4 5.TheYukawa Theory 9 |special topic 1 Relativistic Kinematics Inthissectionwesummarizesomeformulasthatareusefulinsimplifying theeffects ofrativce cuasformations fiom oneefaeace fame toanother ‘Appel aplication aes inscaring: theory deals withthecee ofmss fame, experiment withthe laboratory frame, andthe reslts ofthe twomust be compared. Thesimplifying technique tobeused isbased ontwofess from thetheory ofspecial ela: (a)Thescalarproductoftwofour-vectors, A,=(de,A)andBy= (Bs, B),defined by A-B=A,B,=(AcBy—A-B) (ST1-1) isivan under Lovee easformations (b)Theenergy andmoma of«pati sansform aafourvector Ebe(Ex) (ria) Fhe square of“length isgiven inemsoftheres assofthe pariie ee P=neEHptmate (sT13) ¥ Ingeneral, acollision betwoen twopesticles, leadingcoowoparticles inche * finalsae, forexample, Alps) +Blips) >Clpc) +Apo) "‘willbecharacterized byjustewonumbers,Thereasonisthat thereare4X4=16 ©Gerencomponents offourmoments;theseaerestedbyfourmacscond’tions (ST1-3), andfour energy andmomentum consereation cond‘tons, furthermore iavatiance under uatsation andwader torauom implies thatsi sor coodinas, thecere oftas mementam, dheenenttion ofeheseater ingplane inspace andthechoee ofaxes imcheeplane areinelewon ° £= (bat pa)? (pe+po)? (ST1-4) 461 462 Quantum Physics ‘withthe second termfollowing fromfour-momentum conservation, and 4=(te—pa?=(popa? (sr13) Another possible choice is 1=(bo—pa?=(pe~pa® rie, “These threeatenotindependent,sincechereadercaneasilyconvincehimselfchat Pas+Pax=Pon+Pomtheenergy-momentum conservation lawimplies sttdwmmt +matt +mkt +mp (ST.1L-7) “Theinvariants have thefollowing significance. :Consider thecenter ofmass frame, inwhich pat pr=0 (st1-8) There 1=(Bhat Paa)®—(pat pi? =(@+By =e ors) thatis,itis,withinthefaccorof,chequaeofthetotalcenterofmassenergy.Wefollow custom inlabeling thecenter ofmasscoordinates withanascetisk. ‘The significance ofFissomewhat clearer inthespecial (butverycom son) casethatpaticles AandC,andBandDate chesame, as,forexample, in the reactions rHPortP and atesyte Intha case, inthecenter ofmass fame, Ph= ph p= —Pe By+p=Ee+Pb (srL210) and mame mem a (stat) imply thar (wie +niece +DEA+mbet =(pee mit+ (wet mb Relevistic Kinematics 463 tha is, 3 Ey= ES B=ED (ST1-12) Then Ea SV ee +=g-por= (2-B)—pipie ==(pi—pot (ST 1-13) thatis,icisminus thesquare ofthemomentum eansfr inthecenter ofmass fame Notethat#isrelatedtothecenterofmassscatteringangle.Theaboveyields =—pit=pe+2pPE=—pit—pe+2/p%|[pelcos#* (st1-14) The laboratory frame ischaracterized bypp=0 Pre=(rae, 0) (ris) Thus =atpot=paltpat+papa =mgd +mkO +mE! (ST1-16) and #=(po—po? =m'+mabe—Imp =Qa pol® =mde +me —DEERE +Apal-pel =mad+mec—BMEct+2\p4"||poX|cos6”(ST1-17) ; This, withthe helpof “3 Eat+mact=Bol+Ent (st1-18) fad theinnarianeofsands,chatthefactthaandthavethesamevaluesinthecenter ofmassandthelaboratory fumes (oranyotherframes) allows usto compute thetelation berween center ofmassscateing angle andlaboratoryE_scacteting angle,indbetweentheenergiesintherwofares.Thetransformation properties ofdifferential cross sections, de/d (cos6) ateobtained from chestatement, which canbeestablished when oneformulates, scaceting theory relacvistially, that dis aninvariant, Hence de 4 (sr119) i 4664 Quanturs Physics isinvariant and, toexhibit thecross section inaform inwhich thetransfocma- tions from onefame oanother atemost eaily doe, itis bestcowee ita8 fonction sandf,Wewillsotcothishet. ABaGoal peripheral coment we notethatheexpesion forthe les(2xh)* isnocrelativiscicll invatint. However themanifest invaiance of f.fave=) =fevfaraure—pt—mt) -fpg =tf__#p Wet me 2)@as nyt (ST1-20) shows that fEs= (sr1.20) isinvariant. Matix clerents intlativistic theories always have cheparticles soamalized notaccording to yenWw butaccording to yew ViVE" sothatthe necessary factors emerge from thesquae ofthematrix element. |special topic2 TheEquivalence Principle Acconing cotheslativy pencil, thelis ofphysics mast besuch as comate tunposble todstingush between twoneil aes that ce an cach other nly inthieone moving with #constant velouy tlie tothe Grher. Common experience sugges snob eqslences #7 motposte toanguish bysimple mechanical bseraoes whether asytem tia 2 hier pavitacea! felotwhether itisiagy feepion oeabet Constant accenon afthspeopine magaaie andduseeon Tate ga, eon cl heequlence ely hols tse soul ough sohtthe | fale ra.Baseinpeopoued tahisequlenc esfandamentl pinesif ‘ofnature,andchataiflawsofphysicsbeinaccordwithit, Theaceprne of hasme faseahing comequentes, Bt of we uae thathefom acleation amast bepede byt fe Fan th eed by Pom rea) ‘heequivalent gaviaional ld may beproduced byaus «dnc R tay, poviel theprveional oce F=G™R (sr22)c isade equal omabyanapproptne choice ofMandargs) Thetis, how: tre 00 eson whytheetal as ofthe bjt ich appa inthe tnion (OF21)should equal begrvinooal mas atcee hegene na pote ena 7 Mn ;va)=-69" (23) charges dontotheColom’ potent andoncould nagion, focomaple, dt hes 466 Quantum Physics the“masses” that enter into (ST2-3) atedifferet from theinertial masses thatappearin(ST2-1).Thisisnotpossibleiftheequivalence principlereallyholds.‘Suppose, forexample, thatarather extreme siceation were tohold: electtons are ‘notsubject togravitational forces. Inthaccasecheinertial mass ofanatom is (approximately) M(A.Z) =AM +Zmwhere Mandwacechenucleon and leceron masses, respectively. Ontheother hand, thegravitational mass that eaters into (ST2:3) isjustAM, tothesame approximation. Thus, inside asatelliteamassofleadmightbefoating,whereasamassofmaterialforwhichthe 2/A ratio isdifferent would fall,This suggests anexperimental testofthe ‘equivalence principle; according totheprinciple, allmaterials should behave inthesamewayunderacombination ofgravitational andcentrifugal focces.Thistest, showing theequivalence beeween inertial andgravitational mass, hasbeen ‘attied ovr,andestablished toanaccuracy ofonepattin10!inrecent expeti- ‘meats byRoll, Krotkow, andDicke," butwasknown tobetruetoanaccuracy ofafewpartsin10?fromeatlyexperiments ofEotvbs(1890,1922).Theprin-ciple oftheexperiment involves suspending cwo equal masses ofdifferent ‘material (gold andaluminum) from arorsion balance. Any difference inthe acceleration ofthe cwo masses toward the sun asthe earth moves initsorbit ‘would result inadefection, which infact, as notobserved. ‘Another consequence wasmentioned inChapter 2and inChapter 22 {Section B).Aphoton ofenergy Fhasgeaviatiopal mass E/¢*: toestablish this, ‘we consideranatominanexcicedstate,withmassMataheightxabovesome referencelevel.TheworkdonetolifittothatheightisM¥gx.Whentheatom decaystothegroundstate,ofmassM,aphotonofenergyMMe!—Me!is emieed, ‘Suppose that thephoton isabsorbed atthereference level. Theenergy absorbed is(E+E,)where F,isthegravitational energy chatitacquired iathe fall,Ifthe atom intheground statealsodrops tothereference level, thetoral -work done bythegravitational fieldisMgxonthearom.Iftheabsorbedenergy isusedroexcitetheatombacktothestateofmassM*,weaebucktotheoriginalsicuation, with M*acthereference level, provided Mtge =E+ Mex (st24) Thus E Ey=(MP 2) =oe (St25) Consider now aphoton ataheight x.Leitsfrequency there bev, sochat E=b (st 26) +For adenied desertion oftheexperiment, sceR.H.Dicke, The Theacal Sasiconee ofBxprimetal Relat, Gordon and. Breach, Science Pubes, New York se TheEquivalence Principle 467 ‘The photon energy stthereference level is +e m4 8) sr29) ‘This must equal ho’,whete »isthefequency ofthephocon messuted atthe reference level. Thus theration oa) (eras) implies tatthefrequency ofthephoton ismised snd a£ (st29) “Thepeniod, which isreciprocal to»,isthuschanged according to aT_ ag ATL 10)7 © (ST2-10) ‘This prediction wascoafimed in2terrestrial experiment done with theMéss- 3 bauer effect (Chapter 22,section B).Theshiftismore dramatic ifwecompare thefrequency ofanatomic emission lineonthesurface ofmassive sarwiththefrequencyofthelineonearth.There ba a_GM (sr2-11)i +R Forthesua,whosémassis1.99X108gi,andwhoseradiusi6.96X10cm, theshift issmal, Ax/v =2.12 x10-% This asbeen measuted forthe sodium lineiasunlight (Dicke, loc.cic)andobservation agrees withtheory to5%.‘Theshifeiscalledthegravitational redshiftsincetheearthis“up”relativetothe 4 Another prediction follows byanalogy withelectrostatics, Justlike@ charge isdeflected byaCoulomb feld,witha defection anglegiven by ower wp 3S (sr242) where Zeand¢arechetwocharges, themassofthemoving charge, visis *asymprotic veloiey, and isthe impace parameter, sowillthephoton bede.flectedbyalangemus,forexample,thesun.Sincecheforcelawisthesume,we 468 Quantum Physics justmake thesubstitution Ze—+GMm and —>¢.The impact paramere isroughlytheradiusofthe sun, s0that fortwo stars, or wa ct ST 2:o (243) “The numerical value ofthis is0.83". The actual measured value ofthis, observed bylooking atstarsneathetimofthe sunduring atotal eclipse ijustowice thisvalue, Theexplanation ofthisdiscepancy liesintheEinstein Theory of Gravitation, which isbeyond thescope ofthis book Teshould besttessed that Planck’s constant does notenter (ST2-8). The resule isapurely classical one,andcanbederived without theustofthecon- nection E=he,which isaconvenience, butnotnecessary? *-The orice paper ofFiat iveryreadable, andisspine josation io‘hePrincipofRelat,cllecionaxgpaperspubisbyDoverPsblaton, ne -topic3 The Wentzel-Kramers-Brillouin Approximation ‘Thisapposition method isparclacy weil when onisling with soy vying potent Hea what tismeas wilbecome cas ene aus tose teequason Ue), 2m ;AS + Beveoly) <0 (st34) f Hx) =Rex) (ST3-2) Then Pe PR Mm igAS BY) waren,a[SEsBS a(S)|(33) sothahediferentequationsplitsatoevo,bytakingtheeaandimaicayparcof(ST 3-1)after(ST3-3)hasbeensubsticuted. Theimaginary partgives ak as Raette Ee? (ST3-4) £(bef saps)«0 . 6 se srs) sco 470 oan Pye ‘Therealpartreads PR 1 (SY, aml—VO $ekaSY+BEOlamo which, when (ST3-5)issubstituted, becomes PRO, mB VOeet Ree (13.6) ‘Athi point wemake theapproximation hat Lae G1 1 (ayreser ela) sro sothattheequation becomes e et 2m[E—W(x)] (ST3-8) Tas c a£4 yarf+ =Vine (139) and bene 560)=[4VaniVe (sr310) ‘Thecontin othe aly cambeasad intoaement abou the vatiation of(2). Itwillbesatisted ifV(x) vaties slowly inawavelength, which varies from point copoint, butwhich forslowly varying V()isdefined by i L3 x=4 srsay =a)~Fanl—Poe ory Ache pins whee E-Va)=0 (sr3.12) specl venmentis requie, Bese inheapproxima ST3-8 (9)apes {Die single Thisano besods mene atthe apponinason (S37) tot beporthere Thepei points eae trig prea here the cel pice woud ttnarounds camonlyove whee VES)20. Thewayoftancing seas natwing pot ieootech berronedhaeThefastest weavesleontoeeotheingpoint fwhere E> V(x), say),oftheform Ma) =RefaERATIVE (rsa) 7 ‘TheWentzel-Kramers-Brillouia Approximation. am andasolution cotherightofcheturing point fwhere E<V(x)], andwhatwe needis«formula chainterpolates between chem. Inthevicinity ofthe turningPoin onecanapproximace -/(m/K)[E—V(x)byastuightlineovetsmall interval,andsolve theScarédinger equation exactly. Since itisasecond-ordet equation, therearetwoadjustable constants, oneofwhich isfixedbySting the solution to(ST3-13) andtheother byfcting itso Vs)=ReASi 6NRPOTT=B (3.14 thesolution tothe right ofthe curing point! Theabove solution decreases in amplitude asxincreases, Thetotalattenuation atthe nextturning point, when ED Vis) again is Yeu) fsVesoppy feVeRO Ei,(ST3-13) whichisjusethesquarerootofthetransmission probability thatwefoundin Chaprer5, “Formone del, sealos ayofchemore advanced books omqutstuon mestaricsforexpt,J.L-Powelland,Catenans,QuantumMesania,Addon:WesleyPublishing Co.(196i): LLSci, Quatem Mechs, McGraw-Hill Book Ca[1968 |specialtopic4 Lifetimes, Line Widths, and Resonances Inthisscion wewldiscus ighly improved tesinent ofeastonites,whichwilinatehowtheespnestialeybaiecomessowThTotedathena! sophnton usumedoftenadewilaketneteesmen somewhat esclepae tha spose ‘Tosinplify therolem ssmuta posible, weconsid aaor with jusowlevel thegioud Sate withenety 0,andring onl se, wth tncigy Tewotes aecoupled tothedecuomgt el wich wel | Ske ever sodatnopolaiuson ves apron WealelyconethesubseafdienesofHacomingoftheGeelsateforth Hob=Bb, (ST4.1) 4 ‘andoftheground state +onephoton, $(k), forwhieh Heal) =ek)$k) (r42) todlimit ourselves tothese inanexpansion ofan abe Fanci, Thiis cenainly justified when thecoupling between thewostates, ¢yand4()through theprea Jnsoal, sindewongnaiccourling st tenieokeres ‘hewos thee, photon sates egg Nove ee (ti@(k))=0 (ST4.3)f——_evenen heissuchdeeents(A)andHatehesame,Thesesorthogonal betse onehas photo inictnd theeter doc nottodbocce "Te slut oftheequation Eoihao (He+VW) (ST4-4) ° ‘This was first derived byWeisskopf andWigner(1950). os 474 Quantum Poysics maybewien intexms ofthe compete set Ho)=aee4Fanon anyeM (8143) ‘When thisisinserted into (ST4-4), da un aNa Bey +ifi4SEP0900)4fidee) Hs) «ged =Balt) fPhe(k)(kp)eM0h) Fane votfakitesoO"vou) results.Ifwecakethescalaproductwithdy,weget deEwaininivien) +[aie#-™(eV)000) Since V,acting onastate, issupposed tochange thephoton number byone, GaiVigs) =©.Wich thenocion (hy —B= Rutt) (Vie) =ate or46) theequation becomes a.[us Ma) (sr47) If-we take thescalar product with @(q), andagain usephoton counting toset(Ha)160k)=0,weget,afterlitlemanipulation, sing&normalizationthat is) =1k -@) (ras) theequation AAD —9p*Mra) (r49) ‘Since b(k,0) =Oftheexcived state isoccupied at1=0,«solutionofthisequa- tien i bas)=nrfaeHFa0) (sr410) Lines, Line Widths, andReconanect 475 Wenow insce thisinto(ST47)toget HOFfeeioagieeme faor(srt)a : Nextmultiply bothsidesoftheequation by«~™andintegrate overtimefeom 0%0 ©.On the left hand side art Padie [raedhe me =lealt)] o™als frac fadyAErnf” dean =af.Bema 1 (sr42) where wehive used (0) =1 (ST4-13) (Ontheright side wehave =gefemanc aseeofartoe Now, ascanbeseenfromFigute ST4-1,theineegral loverthefirstocant inthe #4plane, caoalsobewritten as f*abate)doeff*aeny Fig,ST4.1.Theineprationoverthefstoctantcanbedoneeithetbybolding1fed incgpmcng long theeral stip, andtheasuming thesips Rome PS tof=a,orby firsttaking theintegal along theborzontl stipfom f= eoFo&andthensummingoveallthehorizoetlipsfom?=tO?=o 476 Quantum Physics andbere, telasintegal canbedone, sothat- 1 egygeneCmrfaerar=—Efanimany[arane afm MME [arsine rasa--FefeesSarat SrA) Wecansolveforfrdial)#™toget - weIaeMtafmTm)?(oF415) LS i “Thereader familias withthe theory ofLaplace tinsfrmations wilrecognize the hove assuch, Theinversion oftheLaplace tansformation oftheabove form needs some discussion, which canbefound inthemore advanced lierature. We willargue asfollows, Although wedonotkaow howtoexact a)fomthe hove, wecnexamine therelation inthelist chat 2+0,1wemake the “Ansete aya" (ras) weget, a 1Ftd") 7fyMOEee keer ‘which implies inthe limic +0chat ceim-fonlace =knifoul ied (sr.417) We can write thisas7fg,MOO! foLim=je/auie +) _EPpq,LUO?wll)~ iarene 1 a ; +Ffeximao| sore (sr418) Inevahating therelprtofwemake useoftherelation d Lim <= tet) (orsHnoaoe7to) ” Lifts, Line Widths, andResonances 477 0hafaly, weget a IMO) Le mot-@[™ob)=[aeiman|fot) (ST 4.20) Whea thisisexponeatiated, wefindthatthecoefficient of¢yiny()is en em (sr421) where Faid2faiV|oe))|*afe(e)—B] Y= stpans[oxioirienot ww(STH) ‘Thus theprobability offinding $(¢)inthestaegyafteratime#is,cotheextent thatoutsolution isapproximately conet, |a)|* = (ST4-23) here isthe decay atecalculated inperturbation theory. Furthermore, the coxcillatory behavior of(0)ischaracterized bytheenergy Eofthestare¢, ‘hifed lythexcnd orderperterbaton energy sii,ascomparison with(16-10) shows. The only difference isthattheintermediate states summed over here F foam+continuum, andthusthelimiting process shown in(ST“18 mest be usedtodefinecheincegalwhencheenergydenominttr canvanish 4‘Anotherquantityofinteresistheprobabilitythatthesace4)cadupin 4thestate$(k)at#=©,Thisisgivenby|B(k,«)|?,where,accordingto 5GrAi0)andGT46),wehave eawe)=2geen[attee He Ha)=ryfa Mk) 1 _ 5 Faw\ nawate)=29 where _do=fiMac)| a=A fiaB=dk’) Thus M*(k) Wk, 9) = 0) an veo (rae) ick)—EBfoFoayyti 478° Quantum Physics andtheabsolute square ofthis, eeLt Videe018=ay any +Gray Sra) yields theLoceataian shape forthe linewidth, thai,thephocon energy is cencered about the(shifted) energy oftheexcited level, with thewitlth described byfry/2. Theenessy shiftissmall, andusually ignored “Thesume form appeats inthe sattering problem. Consider chesateringofa“photon’ofmomentumK,bytheatominthegroundsate.Thestateofthesystem isagain described byequations (ST4-1,ST4-2,ST47,andST49) fexecpe thaiitally, which heremeans at#=~, chestateisspecifically given aso(k.), sothat Hq)=tq—k) ate—@ (ST4.26) ence theintegration of(ST49)gives Hai)=Hak +Eamri@ f'araroorean ‘Thequantityofinterestistheamplitude foratransitionintoafinalstateiawhich thephotonhasmomentumkyat=++,that,iis (Uk)|v©))=ky,+©) =Blky=ke)=;Mk)fdad!)(oy=wk) (sr428) using theprevious equation. ‘Substituting (ST4-27) into(ST4-7)yields theequation ep MO)—ayJkL)|nd(r)(ST4-29) “Thismaybeinegeted, taking incoaccount thata(—>)=0,cogive ay=MOaelatid -$farinaos f'aenerf"ade)"(ST450) Nowthenegaf"de%snotwalldefi.Thesaadardproceiowriteitintheform LiecimesLineWidths,andRomances 479 : eatin tigfar4 tim gran ‘The use ofaconvergence factor,whichisthenallowedtovanishinawell- defined was, issomewhat similar tothe texment ofthe Coulomb pocenial as thelimiting caeofascreened Coclomb potential iaChapter 24.Next, ascia beseen fom Figure ST42, Fig.ST42. “The incegal in($1430) caneither bewritten os “sun” ofthevervesep,asiatheestingao,Sumofthehorizonsstp,sin(ST432).Th sume lnechange ofonce wed inBq,(ST433) excep httheveiSlee =tisshiedwo+=. [ieee [arse =araenefare =f agen eer sotha MOE 8fyMNF grgyngesncn (y=MD FFpgLMOOIEEYagen - Ooty ok) J el q (ST 4-32) F—_Aecording co(ST428) thequantity ofincerest fornonferwatd scattering (sothat thefests canbeignored) is 480° Quantum Physics *aytwDf gga [lao files+ie),|ae= Mae)|*fyeufalae!) [HOH fy ae [eee . PMO) yeFo +i iLacgoltf°sale"ifod?fared fefenREL”ara[7arene Pa sr 4s) ‘where inthelaseterm weagain rewrote theintegral withthehelpofFigure ST42.Thesintegration ciaagain bedone with thehelp oftheconvergence factot trick, sothat (ST4-33) now reads at detweMU)lay—0) fi ae Files+fe) +hfe“w(k)fwoe L 1«aa atl whichisintheformofanequetionfortheunknown.Thismaybesolvedtogive©eyfe,BEMODBoy—0) fas id) x——_ 1___—1fm[td*(ST4.34) a] ao) Fe Hence, inthenonforwatd diction Hy) =—Z-M*(k;)-29M(ks)Bay—on) 1 - imol* actinfree .=2May —fi)Mlk) M04) ey—B=faega+iefemale ~ :x) (or435) ‘Theamplitude peaks strongly whentheincident (andfoal)energy «s(4) is seardheencrgy oftheexced sateoftheatom, shifted toE+EB,asin(ST Tap). This jsies checomments made atthe ndofChapter 18 |specialtopic5 TheYukawa Theory “The else aucear physics experiments such asthesatering of parcicles bylight auc, andhesady of«deny lictimes showed thatthe nuclear forces were such that theradius ofthenucleus (with Anucleons) was R= nA (ST 5-1) with ~1.1. 10-¥ cm.Thus thenuclear deny (ncleons/unit volume) | Nas constae independent of4.Withlongeangefetes,suchathelecuo- static forces, there would beA(A —1)/2 “bonds” andonewould expect che + density toincrease with4sincecheKineticenergyonlyincreaseslinearlywithA. Theconstancyofthenucleardensitythusstrong]suggestedshotnagefore.‘These forces, eading tobinding enegis measured inMev. rather than electionFelts, badeobesguicanlysonge thanelectonngoeticTroe; Insewch fora mechanism that would give 1stosuch forces, Yokawa ia 1955 drew ontheinsights gained fom the successes ofquantum elec. Synamics andproposed hismeen try ofecear fr. Atte lve ofthis book, = aly2qualittive description ofthattheorycanbegiven.Theintetoction be- ten evocharged panes atrstorverylowvloces) canbedevrbed ia terms ofan “action ata-disanee” Coulomb inteaction. Amote acute description involves thefed concep; checharged puils areSources of,and intent with elecwomnagntic fields (E38) andthus they incrct with each ochee through theintcemediary ofthefield. Inadescription thatisaccurate onthe Gquntum level theelecuemagnetic feld iquansined, and thequanta, the Photons, aethecuits ofthefi Two charges canineract bythe fllowing Inednian. Gige “I” emits «phoron, We know from eacyy-momearum Conservation thathiscannot bea reaphoton, ofequivalently feaprocess titer thephoton hasanewegy that does actcomspoed toitsmomenta +Sethe dcasion ofaneing inChart : “SSurtssooetar fate nguge sadsai tbe ake tery pal, ienould bemouy otek ef pton sac tying tod ofhl eh 46: 482 Quantum Physics (ie, £=po),of,thephoton isreal,butenergy issorconserved inthereation® ataty ‘When thatphoton isabsorbed bycharge "2"theimbalance inenergy canbe corrected, since that process ytase alsocannot conserve energy. Thequestion is,whyshould such an“exchange” ‘ofphotons giverisecoanateraction ofrepulsion between changes? Theanswetreallyischatourvisualdescription oftheexchangeisjustaninterpretation ofche second-otder perturbation energy shift duetoaperturbing potential H).The foumula (16-16), describing theenergy shiftfromtheinteracion-free energy of wocharges ¢,andes,reads “Thesum isoverall intermediate stetes thatcanbeobtained from H,acting on thestate [¢,). Oucverbal description corresponds totheintermediate stare in which eis initsintial state, and«has,through theaction of#4,emitted a photon, sothat|»)=|¢’y.s) here. Theenergy oftheintermedia state isthe recoil energy ofthe«~/(pe)* +(ma), plusthephoton energy pe.Thesum ‘over intermediate states corresponds coanintegration ove allpossible photon ‘momenta consistent with momentum conservation, that is,anintegration over alldiections. Since thecharge "2"could betheonethatemits thephoton, onemustalsocalculatethecontribution fromtheemissionofthephoronby¢sanditsabsorption bya.Wewillnordothecalculation showing thattheCoulomb potential emerges, because, infact,electrodynamics iseather subtle. Thecaleu- lation will bedone formesons. ‘What Yukawa suggested inhisextremely sigoificanc paper isthatthere existamesonGeld,thati,afieldthatisdifferenfromtheelectromagnetic fed,whosequanta,themesonhaveaistemass,whichiscoupled0protonsand‘neutronsinamanneranalogoustochecouplingofphosonstochargedparcicles‘Theexchange ofthese quanta willthengiveriser0aninteraction becween nu- cleons. Inthisway, iacerctions asdiferentaselectromagnetism andthenuclear forces would atlease shareacommonuniversalmechanism *Lerusgothrougha +Weassume momentum consrvtion inboth points ofview. They tun oattobe completly equivalent‘Theidenthasallaceracionsproceedthroughan“exchangeofquanbasganed such wideacceptance thateventheweik intetacons atebelieved bymany people 1be tmedinted bya"werk intermediate verortezon” whose popecties aededuced fom what is Keown about theweak interaction. Such parce basootbeen discovered, butts site ontatent toassure thaitveytasiv,whichwool!explainwhyicesmotDeeneeta ving accelerators TheYukawa Theory 483 verysimple calculation assuming catthemeson field issaat, andeatwecan ‘rite thesimple Hamiltoiaa forthe aucleon interaction a=F—pote) (15.3) ‘Themeton fedwilbewritten inanalogy tothe vector potential in(22-27) a8 oes)=&)Pre absorption) arehNan +(2)oer? (emission) (ST5-4) wv ‘Weatenotinapositiontojustifyallchefactors.Theappeatunce of«inthenotmalization factor hasthesame source asinthephoton problem; itcomesfromthefactthattheenergyofamesonquaatumisfu,Themomentum ofthemeson isRl,andsince themeson has&mass», wenow have theeoetgy- ‘momentum relation (hea)? =(Filke)* +(pct)? (ST5-5) Letusnow calculate thevarious terms inthesecond.ocder enetgy shife (ST5-2). ‘When avcleon "I"emits chemeson, wehave aR0RV an s+mesonleplay=«(E)"mmeorsg Noreference ismade toaucleon "2",since itisunaffected byHydusing the emission by"I."The absorption by"2"leaves "1"unalered,andwhatenteris (leone+meson)=674)" (srs) “The enegy denomiator i Bara ~Bait =(Bins+Bae+fs)=(8s,+Be)ho (S138) B Evdiffers fromEysincenucleon “1”recoils upontheemission ofthemeson, However, therecoil energy is(fk)#/2Miy andchsisgenerally small inthenon. relativistic approximation, sincethenucleon massssolage: Hence theenergy ‘denominator justaThus theenergy shifts given by 20 pen men ab2BEEgang ten “The sum isoverall meson momentum sete, andasalways, cismetas an integration over thephase space _ [vee va: =m-fie fe (st5.10) 484 Quantum Physics Hence aregt owe ae V(Qn)?vfory ef mown ae)Sine Gel : aein-nd--fas ST5a 40) MGTG (roa) [Note thatwedidaottke intoaccount momentum conservation, butintegrated ‘over allmomenta forthe meson. The reason isthat, ineffect, wetreat the nucleons asinfinitely massive (they donotrecoil andarealways atr,andrs, respectively) andthisanymeson momentum isallowed intheinermediate sate, Remember thac thissavery crude calculation! ‘Theintegralcanbedone[itisinfactthethree-dimensional Fouriertrans- form of(24-87), andicyields for 2aayeeincnin ap-- eee ae de [esl ‘This should bedoubled, because thequcleon "2"could bedoing theemitting. “Thos theenetgy change duetothemeson field is nina aE=-¢——_ ST5-12)onal g ‘Theenergydependsontheseparation, x,anddropsofffasefor>fi/ur.Therange istherefore aot (rs.3) ra Given hat 421.4 X10° cm, weobtain aFe x5 x108 “a 14X10 = ox ay Tax 109 16x 10 150 Mev (5.14) Ifthemesons atenotscala, burpacudoscalr, then #coupling likechat shown in($153) does notconserve pay, since thekinetic energy iseven and thepoteatal isoddunder inversion, Onemst therefore make 2scalar oxof pouedosclar meson field, os deviatvr, There ae,ontelcion, rwalter ‘The Yokawa Theory 405 tives; oneiscohave mesons always emitted inpais, with thecoupling 2°) (srsas) 2forehepairwise emission ofphotons divecothetxm (/2mc)A4; theother isto contract anaxial vector (anaxial vector isike magnetic field), anddot ‘eine thenucleon spia operitor, sothatthe coupling would be h 5woe ST5.16 Figg©Hed) (sr5.16) ‘This would allow single emission ofmesons. Both couplings could, ofcourse, be present, Ifchemesons azevector mesons, thati,essetialy “heavy phorons,” then thecoupling could be Pe) (st5.17) Tall cases, however, therange issllfu. “The mesons predicted byYukawa were finaly found in1947. Thelong range partofthe nscleon-nucleon fore isductopimesons, whose mass was found tobe140MeV! They were found tobepseudoscalar, and, likephotons, theycanbeemitted incolisions ottransitions, Thecoupling (ST5-16) explains4greadealaboutpion-nucleon scaring(¢heanalogofcheComponeet)inthelowenergycegion,andYukawa’sideaisfandsmental coaltheunderstandingwwehave about thesong intracions. Indetail, much more bashappened There atealso vector mesons (spin-patty 1-)andspin 2mesons, enmany others. They canallbeexchanged, andemitted, and,sincethecoupling tothe "© nucleons isstrong, theycanbeexchanged notjustonce, batmany times, The Calculation aebeyond present-day mathematics, anditisacurious factthatherulerforceproblem,whichstartedallthis,isllesswellunderstoodthan,forccamplehigh-energy scareting,Fromourpointofview,iisveryimportant0note thateven inthis newrealm ofshore distences andstrong force, thee is20 reason tosuppose cha quincum mechanics isnoxthecorrect way rodescribe : «ThefateBaisathee0makefiments. appendices A.TheFourier Integral and Delta Functions B.Operators 1 ae |appendixA TheFourier Integral and Delta Functions Consider function le) aiseid, wth peso 2,sotat fle) =fle+21) (At) Such fapion maybeexpanded ia«Fourie Secs inthe iva (—L,), and | thee haste form fo)=dvcos™ 4Basin (a2) 5 ‘Wemnayrewrite theserics intheform f=Zseon (a3) which cep, sae ;<(or cosmL(eri4griety - if1g komik gins i|sin =35(ert ey “Thecoefficients maybedetermined withthehelpoftheorthonormality relation, 1 f* cxfimei 1 man Tous 499 490° Quancun Physics Letusnow rewrite (A-3) byintroducing An,thedifference between two successive integers. Since thiisunity, wehave fo)=Za L jooFm Y =EZamen ao Letuschange thenotation bywitog 7 é (A-7) and reas as We also write Le AW)2°95 as) Hence (A-6) becomes Aw f==Vaab (A-10) ifwenowletL—»«,then&approaches acontinuousvariable,sinceAkbecomes 7iniesialy sal. IfwerealltheRiemanadenonof2intel,wesce chatintheline(Ao) mpibewienintheform a aPOTS oe.pe vel Pak rent ‘Thecoeficieat A(E) isgiven by A)=van fas comet Fade 1= ihe ,)“Tefdefls)€ (aay) Equations A-11 andA.12 define theFourier integral transformations. Ifwe inser thesecond equation intothefstweget feRPae” apne (nas) “The Fourier Inegral andDelta Functions 491 Suppose now thatweinterchange, without question, theorder ofintegrations. Wethen get . wo . #0)=ffy)[zfiaa] ay Forthistobetrue,thequantitya(x—y)defined by Pewpa fl”aor eyo bfa (A) andcalledtheDirseDeltajunctionmustbeaverypeculiarkindoffunction;it must vasish when x57,adTemusttendtoTafinityinanappeopriatewaywhen x—y=0,since therange ofintegration isinfinitesimally smal, Icistherefore notafunction intheusual mathematical sense, butiiseather «“generalized fonction’ or«“dstbution."* Itdoesnoehaveanymening byself,Butit ‘anbedefined provided italways appeatsinthefoam” a [Atue9 withtheFunction 3)sulficendy smooth inthernge ofvalues thatcheargu- ‘ment ofthedelta function tikes. Wewilltakethatforgranted andmanipulatethedelsfunctionbyitself,withtheunderstanding thatattheendalltheelationsthatwewritedownoalyoccurundertheintegralsign.“Thefollowing properties ofthe dela function canbedemonstrated: @ Has)=ua5) (A126) ”Thiscanbeseentofollowfrom £0)=]5/0)He») (aa) Afwewrite x=afandy=a,thenthisreads q Kes)=|efsafler)alae—01 Ontheoxher hand, fet)~fanftenHe-9 - which implies orresult. 1ThetheoryofdsbutonswasdevelopedbydematerascianLausent Shea ‘Aninoue weument ey befound inM} Ligh inrdaicn earn neha ‘nd Grated Pectin, Canbioge Usiverity Pes (938) 492 Quantum Physics (ii)Arelation thatfollows from (A-16) is 1=#8)=—[x-2)(x+2) (A Bet)=Tle0)+Bert2] (a8) “This follows from thefactshacheargument ofthedelta function vanishes at x= asadx=—a,Thostherearetwocontibucons: B(x? —a*)=(x—2)(x+a) 1 1 =he-+ae +) Tera Otae? 1 = ee0)+ee+a) FaqWe)+aea) ; Mote gently, onecinshow that _yphens) UO=ETadleon as) where thexate therots offs) itheinterval ofintegration Inaddition totherepresentation (A-15) ofthedelta function, there areotherrepresentations thatmayproveuseful.Wediscussseveralofthem.(4)Consider theform (A-15), which wewrite intheform. 800)=2imJanc (420)ad, ‘Theintegralcanbedone,andwegec —ite Xe)=timLA Be ie shin SE (a21) (b)Consider thefunction A(x,4) defined by. Aua)=0 0xcna 1 -b -acnce taBoonacxc (an) <0 4#<x Then a(x)=Lim(xa) (A-23) ‘TheFourie legal andDeen Funeons 493 Icisclear thatanintegral ofaproductofA(x)andafunctionf(x)thetis smooth near cheorigin wilpick outhevalue aeeorigin Limfdsfle)8G8)=f)Limfdkd(x) =fo) (0Bythesame tokeo, aaypeaked fonction, nomalied toonit an under willapproach adele faction inchelini thatchewidth ofthe pen {goes to210, Wewillave itothereaderfoshowthathefollowingaesepe- sentationsofthedeltafunction Lie A(x)=Limyeta (A-24) and (x)=Limveo (A-23) (@)Wewilhave occasion codealwith orumormal elwomias which we denote bythegene symbol P(x). These have thepropery tat f#PaCe)Pale)ts)=Ban (426) where 1s) maybeunity oFsome simple fncion, called theweight function Feefunction thattybeexpanded inaseisoftheseorthogonalpaloma, yeaa we fod=Eaa (A27) {Ifwemultiplybothsidesbyw(x)Pa(x) andintegrateoverx,wefindthat ane|9(0)Pa) (428) {Ifweinsertthisinto(A-27)andpreparedrodealwith“generalized functions,” i ‘wefreely interchange sum and integral, weget fo)=EPa)f0)£0)Pa toe =foro(=Pa(x)we(y)x) (A-29) “Thas wegesillnother representtion ofthe dla Fanon. Examples ofthe . Rag) teLegenve polynomals, Hermite polynomials, and laguese poly Doms, allofwhich make ‘heir appeaance imquantum mechani problems, 494 Quantum Physics Sine theda fenton nays appt pied by+smooth fneion under eg sig, weangie meng tt eines, Forexample *weyae)=[et +il [te £0=feLipo -[2 ve = -faeFO2falas --(2)deJeug (A-30) andoon Thedels fncin ianexemel fl tol adshestant wll tocountr iinevery patofteri pis Theimegal ofde function is eine (aan =x—a) hich isthe standard notson frhsdiscontinuous fnction. Convey, thederivativeoftheso-calledstpfictionistheDiracdetafunction: geo =o ws |appendixB Operators Inthisappendix wediscuss some topics related tolinea operators. The secofadmissible wave packets aresquie itegrable functions. Since ¥69) =aah(o) +BAG) 6) issquare integrable, if¥x(x) andYs(x) aresquare integrable andafarearbitrarycomplexnumbers,wesaythatchey'sformalinearpace.AnoperatorAoathis space is«mapping Me) =#9) (32) where (si alsosquare integrable, Among allcheoperators there isasubset Called linear operators, which have thepropery that Aagx) =aAyx) (B3) whereaisanarbitrarycomplexconstant,and Abs) +Bhs) =abs) +BAB). ow withabeing complex nuinbers. Afurther subset istheRermitian operators for Which theexpectation alue forall admissible ys), (ae=fare) ave) s) isreal. First weprove thatforalladmissible yxandy» fiFG) AdA(a)de=fiLAG)" ale) de eo) holds. “Thecealicy of(4)implies hae [aves aves=fetary 60 en 495 496 Quantum Physics Now substitute for(s) V2) =Hi) +abs) 8) “This implies chat fosENNAlda+Me)=|ded+m4)(tb+dv) (B9) Using hexmiciciy, thais, faeviaten |devcani® i=12 (B10) weobrain w[iia2[vias=focanieteforme ox Since isanarbitrary complex number, chetelatons forthecoeficient ofband forthe coeficient ofM*must sepasately hold. Thus festan=fairy (BA) ‘Thenext result thatwewish topeove ischaeigenfunctions of«hermitian ‘operator conresponding 10different eigenvalues areorthogonal. Consider thetwo equations Abs) =ashe) snd Laptayl* =abs) 613) Note thatayisrealsince theeigenvalues ofahermitian operator aereal. Takethescalarproductoftherstequitionwith¥3andthesecondequationwithJ.Thos feotsantey =afetae [aster vile)=afe oy Subwacting, wegee (as—as)[HeoAls)d=[eian=facao =0 (B-15) Ths, fay oa, wehave J50wo)dem0 (16, Operas 497 IfwedefinechehetmitianconjugateoftheoperatorAbyA’,sothat fbx(ay)*nn=|devices (B47) then fora hermitian operator AnA B18) Wecanprove that (4B) =Bat was) Todos0,wemoze that fiV(AB) =f(ABE:)*te f(Bn(AH) [eave fiWBA (B20) Ageneralization ofthisis (ABC...2)=2h...Bat wa “Thos,aproductofewohermitianoperatorsisonlyhermicaniftheewoopetators commute: (48) =BLA! =BA=AB+[B.A] (2) ‘Another resul isthatforanyoperator 4,thefollowing Ata 4A-A) 62) aa will behermitien, [Next weprove the“uncertainty relations." Wedefine (a4) =(a)=a=(A=(A) en Le . u=a- ay v= B- () 2) 498 Quincum Physics and consider o= tay (B26) Theo 10)=favre o 2) With AandBhemnitian, soareVand ¥.Wemaythusrewrite: 109=[scuy+aver(or+ave =[aor wy+fawnoo safauuprrn-Woww) =farwsersaurny =aaymany+afavetury>0 =(aa +NB) + AB) (B28) “Thecisimamwillocutwhen 2B) +HAB)) =0 (62) Substituting thesolution =—4) . X=43aay 30) ico 10), weget ABI |a8) 4"~“cast*xan*° that is, (ay (any >4(1A) (sn Iacideoally, theminimum value occurs when issuch thatUYandVYare proportional toeach other, Fortheaseofthe operatorsand,thismeanschat AM), .7de+BO)=O (B-32) whose solution i vx) =Cerern (B-33) Operon499 2ground stateeigenfunction oftheharmonic oscillator, Iisimportant tonote that cheuncerainty relation (aay(AB)>4(iLAB)))* (639 vasdetived without anyuseofwave concepts ofthereciprocity betweenawave formanditsfourier transform, Theresults depends entirely ontheoperator Ptopertes oftheobservables amd Weconclude theappendix bylisting some properties ofcommutators. @ (4a =—[6,4] (835) ro) (A.B) =(4B) —(Bayt =Ba —AB =(BA 636) (il)16AandBarehermitian, s0is4,8), Thisfollows directly from the prceing opi. (48,0 =ABC -caB =ABC ~ACB +ACB CAB =ABA +1408 x (6)Temay besbowa term bytem chat ABeA=B+(AB)++(af4.B)] +x[4lataall] +... 638) “This isknownatheBaker-Hausdorf emia,andsofsomeutleyinmanipuls- tionsofoperators (vi) eis ensilyexablished chat [4{B,c]] +[B{c.4]] +[¢f4.5]] =0 (B39) “This iscalled theJacobi identityAmoteextensivediscussionofoperatorsandthelinearspacesthattheyaredefined onmaybefoundinJ.D.Jackson, Mathematics forQuantumMachnis, WC.A.Banja, fac,NewYork(i982) |references" XG. Baym, Lect onQuatam Mechani, W.A.Benjani, Inc, New York, 196 This isaveryappealing book, withustheright mixate offormalism, intuitive gumeas, andapplications. Itshould beconsidered anadvanced book, Acesible 1thestudent who hascoveted thematerial inthis book. D.B.Beard andG.B.Bead, Quamtum Mechanics with Application, Allyn sad Bacon, Inc, Boston, 1970. ‘Asaumber ofqutatum mechanics copics aredeveloped stound particular appli ‘cations. Byitself, thebook isprobably notvery useful forself-study, but thevariety ofapplications make itauseful reference book. Thedemands ‘made bythebook ateprobably comparable tothose ofthisbook. xR. Becket, Flecromagnetic Fields andInkeractons, Blaisdell Publishing Co.,New York, 1964 “hiss notely» texsbook onquanta mechanics, butabout halfofVolume2 isdevore tothedeerpion ofproperties ofmater with thehelp of {quanto mechanics, Cerin appliations aediscussed, andthestudent‘ayfindthisausefulcollateralreference,onthelevelofthepresentbook. HLA. Bethe and R.W.Jackiw,InermediateQuantumMechanics(secondedition), ‘W.A.Benjnin, Inc, New York, 1968, “This book contain detailed dscassios ofelcletional methods applicable to thetheory ofatomic structure, multiplet splitings, thephoroletic tffec, and atomic cllsons. Mech ofthe material ft tobe found in toyother eeubook. The book isthus anadvanced tex, a8well 45an exhaustive reference book.HA.BetheandE,ESalpeter,QuamMachanisofOmendTuothen“Atoms, Spanget Vetng, 1957 ‘This reprint oftheauthors’ article intheHandbuch derPhysik isanelaborate, ‘dead, defi westment oftheproblem athand. Itisabook about atoms andaotabout quantum mechanics,andthelevelishigh,Ieisan reel vfetence book\¢_D.Bohm,QuantumTheory,PrenticeHal,Inc,195:‘The book isdiscursively written, onalevel comparable cothepresent book. heauthor peysmich ttevion cocheprinciples ofquant theory, and {ives anexclen dstsson ofthe qurmuar theory ofthemeasurement frocess, There atefewapplications eadnotmany peoblews.Manyhookshavebeewenshouquemmechanic.havesdfomome| ethene bes, loca stfe,andpobiblsuesonethesThetneee Soaks orquam cheney ied sor 502 Quantum Physics YS. Borowite, Fundamental ofQuantum Mechanic, W.A.Benjamin, Inc.,New York, 1967. This isawell-wricen book, about halfofwhich isdevoted tothetheory of ‘waves andtoclasical mechanics, Thelevel iscomparable tothatofthe present book, E,U.Condon andG.H.Shorey, TheTheory ofAtomic Sper, Cambridge University Press, Cambridge, 1959. Thisisaverydecaled reference book onallaspects ofatomic spect, although icdoes notmake useoftherecent techniques thatdepend ongroup theory. Thebook isvery advanced, andchus itsshortcomings inche techaical developments atenotimportant foranybody butthespecialist. Tes very useful forthe student. A.S,Davydov, Quantum Mechanics, Addison-Wesley Publishing Co.Inc, 196. ‘This isanadvanced, comprehensive textbook. Thebook isalicle weak on fundamentals, butceeats many physical systems. There areexcellent discussions ofrelativistic equations, group theory, second quantization, ‘andsome aspects ofsolid-state theory 4-R-H.DickeandJ.P.Witke,ntaduton QuantumMechanic,Addison-WesleyPublishingCo,Inc.,1960. enjoyed thisbook verymuch. Itisonalevelcomparable tothepresent book, anddiscusses afew copics, nocably quantum statistics, thatarenotcreared haere, Theproblems areexcellent. AP. A.M.Dirac, ThePrinciples ofQuantum Mechanics (fourth edition), Oxford, Gharendon Press, 1958. ‘Thisisasuperb book byoneofthemajor creators ofquantum mechanics. The ‘student who hasstudied chematerial inthisbook will have notrouble swith Dita ifheisaallserious about mastering quanturn mechanics, he should sooner otacergothrough Dirac’s book. RP.Feynman andA.R.Hibbs, Quantum Mechanics andPathIntegral, McGraw- Hill Book Co,, 1965. In1948, R.P.Feynman proposedadiffereatformulationofquantummechanics. Tnthisbook theequivalence ofthisformulation tothestandatd theory isdemonstrated, andthe“pathincegral"expressionforthegeneralamplitudeisexploitedinanumberofcalculations. Theselectionofmaterialisveryinteresting, andthepoint ofviewisdifferent fromtheonedeveloped by theauthor. Thus thissomewha: more advanced book presents anexcelleat complement (0thisbook. __R.P. Feynman, RB.Leighton, andM.Sands,TheFema LectotPhy Val.3,Quantum Mechanis, Addison-Wesley Publishing Co,Inc., 1965 Inthisintroduction toquantum mechanics, Feynman abandons thepath integral andappeoaches chesubject fromchepoint ofviewofstatevectors,‘Alargenumberoffascinating examplesarediscussedwiththeminimum References 503, ‘offormal apparatus. Asuperb complementary book, whose onlyshort: coming istheabsence ofproblems. K, Gottfried, Quantum Mechanis, Vol. 1,Fundamentals, W.A.Benjmain, Inc., 1966. This is«veryadvanced book, distinguished bythecarewith which thevarioustopicsarediscussed. Thetreatmentofthemeasurement processandofinvariance principles isexcellent. The student who hasmastered the ‘material inthisbook should beable toread Gotetied’s book, providedhehasacquiredthenecessarymathematical equipment. W.Heisenberg, ThePhysical Principles ofthe Quantum Thesry, Dover Publications, Tic, 1930. ‘Thisreprintofsome1930lecturesgivenbyHeisenberg onthephysicasig-nificanceofthe quantum theory stillmakes good reading. Thediscussion oftheuncermainey relations isparticularly useful. FA. Kaempffer, Concepts inQuentum Mechanics, Academic Press, 1965, This isnotacextbook inanysense. Avariety oftopics arediscussed, The . selection oftopics isimaginative, andthediscussion informative. The level issomewhat above thaeofthepresent book H.A, Kramers, Quantum Mechanic, Interscicnce Publishers, Ine., 1957. Thisbookbyoneofthefoundersofthesubjectisaitsestinthediscussionof spin and the introduction torelativistic quantum cheory, both rather advanced subjects. The student who iscomfortable with quantum me- chanics willfndbrowsing through thisbook enjoyable andrewarding. LD. Landau andEM. Lifthice, Quantum Mechanics (Nowelasvitie Thor) (Second edition), Addison-Wesley Publishing Co., Inc., 196. ‘ThebookbyLandauandLifshitzisoneofaseriesofsuperbBookscoveringalloftheoretical physics. Itishard tothink ofthsasatextbook foranybut themost sophisticated students. Any student, however, once hereaches the advanced level, will find much that isuseful inthis book, There isanassumedmathematical facilityonthepartofthestudent %FMandl, Quantem Mechanics, Butterworths Scientiic Publications, London, 1957. This book contains a gooddiscussion ofthe foundation ofwave mechaaics atlevelsha c O n p a r a B l ewiththepressorbook ‘A.Messiah, Quantum Mechanic (in2volumes), John Wiley andSons, ne., 1968. ‘This book gives «complete coverage ofquantum theory from eherreatment of ‘one-dimensional potentials through che quancization ofthe electro- magnetic fieldandtherelativisticwaveequationofDirac.eisanadvanced book, anditassumes amathematical sophistication that fewfistyear . ‘graduate students possess. Itisanextremely worthwhile book. 504 Quantum Physics YE. Merebacher, Quantum Mechanic (second edition) John Wiley andSons, Inc., 1970. ‘Together wichthebookbySchif,thisschestandardfrstyeargraduatetextbook, aaddeservedly so.Thecomplese range ofconcepts andphenomena is tweated with economy andtaste. The book should beavailable tothe student who hasgone through themaceral inthisbook. J.D.Patk,Intradacton totheQuantumThevry,McGrawHillCo.,1964“This aeactive book iswrittenonthesamelevelasthepresentbook.Amongche topics discussed byPark, andabsent inthisvolume, isthesubject of quantum statistics, which istreated with clasicy. W.Pauli, DieAlgemcinen Prinzipion derWellennechanib, Handbuch derPhys, Vol. 5/1, Springer Verag, 1958. ‘Theadvanced student who reads German willfind inthisreprint ofa1930 article byPauli aconcise definitive discussion ofquantum mechanics. ‘There arenoapplications, butallofthe imporrane maces atethere Jil. Powell andB,Crasemann, Quantum Mechania, Addison-Wesley Publishing o,, Tne, 1961 ‘Theseeeagthofhisbookisinthepainstaking workingoutofelofthemathe-‘matical details ofwave mechanics andmatrix mechanics. Probably allof ‘themathemarical aspects ofthese subjects thathavebeen bypassed inour book canbefound here. There isagood discussion oftheWKB approxi-mationaadofthegeneralpropertiesofsecond order differential equations ‘There atereatively fewapplications, andthere aremore exercises then problems 1M.E.Rose, Elementary Theory ofAngular Momentum, John Wiley &Sons, Inc, 1937. ‘Anadvancedtreatmentofangularmomentum andthemanyepplications ¢o‘atomic and nuclear physics odJ.J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley, Inc., 1967.‘AnexcellentBookatleveljustaboveMerzbacher andSchiffFortheadvancedstudent. ALD. 8.Saxon, Elementary Quantem Mechanis, Holden-Day, SanFrancisco, 1968, ‘This book isonthesame leve! asthepresent one, anditisauseful reference,sincetheselectionoftopicsisjustalitledifferent,asitheemphasisandthechoice ofapplications. Thebook contains anexcellent setofproblems. LL. Schif, Quantum Mechanic (Chird edition), McGraw-Hill Book Co, 1968, “This isoneofthestandard firstyeargraduate textbooks. Itispethaps alitte to0 ‘compact, andthusmost suitable forthe well-pepared student. Thelevel‘ofmachematical sophistication assumedisabovethatofthereaderofthepresent book. D.cerHaar, Selected Problems inQuantum Mechanic, Academic Press, Inc., New York, 1964. Refecences 505 Both thestudent andtheteacher willfindthis averyuseful volume. Most oftheproblemsareonthelevelofthisbook.Thereareafewotherbooksworthliseingaspareofthegeneralreferencelist.Thesearebooksthaedealindeel with some ofthetopics chataretouched onillustatively incis volume BL. Cohen, Conepts ofNuclear Physic, McGraw-Hill Book Co., 1972 (for ‘nuclear physics) .Kinel, Inireducton toSolid State Physic (fourth edition), John Wiley &Sons, Inc, New York, 1971 (forsolid-state physics). E,Segre, Nuclei andParts, W.A.Benjamin, Inc., New York, 1964, (for ‘experimental aspects ofnuclear andparticle physic). D.H,Perkins, Introduction toHigh Energy Physia, Addison-Welsley Publishing 0, 1972 M,Abmowitz sndI.A.Seegun (Bd), Handbook ofMathematical Functons, ‘National Bureau ofSeandards, 1964 |W.Magnus andP.Obethettinger, FormulasandTheoremsfortheSpecialPunctons ofMathematical Physics, Chelsea Publishing Co,, New York, 1949, A.Erdelyi, W.Magnus, F.Obethectinges, andF.G.Tricomi (E¢s.), HigherTrade Pacts(TheBaeanPrjec), Volumes1,2,tod3 McGraw-Hill Book Co., New York, 1953 1.S.Gradsheeya andI.M.Ryzhik, Tables ofInegral, Serer andProducts, Aca- demic Press, New York, 1965, physical constants" No(Avogadro's number) =6,022169(40) X10*mole ¢(velocity oflight) =2.9979250(10) x10!cmsec"? «(elecon charge) =4.905250(21) x10" ese =1.6021917(70) x10-* coulomb 1Mev =1,6021917(70) X10-4erg f(Planck constant/2x) =6.582183(22) X10-* MeV sec =1.0545919(80) X10°" ergsec a(Bne structure constane#/f) =1/137:03602(21) &(Boltzmann constant) =1,380622(59) X10-ergK~ a.(eecton mass) =9:109558(54) X10" gin =0.511041 (16) MeV/c? im,(proton mass) ‘=938,2592(52) MeV/c? im =1836.109(1:) ann (1/12 Xmos) =931.4812(52) MeV/et a=(h/mcen) =0.52917715(81) X10-*cm R(=mta/2) =15.605826(45) eV=1Rydberg G(gravitational constant) =6.6752(31) X10-*em®gr sec* pte Bob mgoeton) = =o.7a438i(t8) x10-¥ MeV gause! *compet bySayJ.Brody,spreeinKierofMadrePhi,4,2, +eae Theigus inprices corapond tetheSal wacenany fear nda Spaton) iotheht igi ofhen amber s07 index Avvogin, ofae lamats, 409 ‘forCalon eaten, 39 eater 4 Tinton,400 absoptnyorm pein aprons, 314 Mainofnaarmens,43 Bownpebebaiynepeatin, 4 ‘tapedtialangusWotentom, BoveEitansta,148 He Boson 8fospi23 oundteriapoteslwet0,185 adatuse tere (poten), 361 Boxpotential 6,151, 1018?‘hannoBoomeee222 Brgxd, 1,100,408 ‘heey89 Bevgnfort,92 ‘namtn, 167 Bremsiirg419 mutaeons159,232 Baldingpip,303 ‘Sadlon Bohr ato 2jSheraton138 Catesofmasmation,145,156Sefer 15 Ciabishypetm,200 ‘Seaspole169,171 satmlquaeofman120 smaccen209 motionofdeeninmetel217Ansoptn, 11 cioofsarrly231 ‘Matacon3fandpotns 428lheretsemeyea403 ‘atberyon27 Goldend7 ‘atpres23 Golsrosioigofeslines,365 ‘ectntedprods,438 ellsoneon.3 ‘Neier, 29 omataon tts499satgecae frpandeysisiat penodicui307 {Srdnglrneat,159,171abtlonsaten 398 completttcommingober,relaninasins,37,360,314 HoConplatewtfegos, 65,114, Bandsrc 108 fanBepenton65,471 Compafe,10Buyenousber 2a Titaorm,417Betancton,235 enatsorpdorofadn,416 ‘phe182 compton ating (2 dec 150 Cenzrcionofsgtmomenta,158 Bk oo eatin, 1 Stementiny 1 ery deny 31S Stay 6.30 . se etc2 otpete.47 Batrstmiemotei confitothemeson,67,121 Botove136-39 Conespondeate pani,9 Bakr eauependence pinch, 9 icon movon inmga el, 216Bolumcen378 intenofdpokradon,38 SaturnpoSttydkuibton, 6,322nomalaton afectopon,44 Scedcamker 3 chpstandiflrental30.—_Bermanmantna, 397 ‘Sintinrpann 36 so st Index forblackaise,385 coordinates, 215foridenticalparticles,402 clasicalmotion,217‘optical theorem,384 onrespondencelimit,218 Photclecti effect,413 ‘lectromagneti energydensity,344 Telatiitevariance,463 ‘Elomentaryparties,423, intermsofboundstateenecey,394missivepower,1 fntermsofphase shits, 383. nergy operator,seeHamiltonian ‘ou, 385 Energy site, fstordet, 256 totalelastic, 383 Tplium ground sate, 286 Clindscal coordinates, 215 Alii enced states, 289,second order, 257,477 ‘Davision-Geimesdiftration experiment, 13 Botvos experinients,seFineequivalence DeBrogliewavelengthrelation,13 principe Debye frequency, 371 quilt conditions ineavity, 974Depenerecy, 70,375 Equipartition ofemcigy4{orcentral potentials, 176 Equibalence principle, seBinstein{orCoulombpotential,198 Exchangeeffectinhelium,289 lift inatoms, 302 Exchange operator, 147 Degenorate eigenfunctions,118 Expansionpostulate,62,112 Degenerate perturbation theory, 258 Expectiton values, 48‘Stateffect,263 Torrinhydrogen,206inmattelanguage,265 Exponentialdecayrite,365,477 Delta function potential, 77,93 easy’ ofstates, seePhase space ermiDiras statistics, 148DetailedBalanceprinciple,375. entenergy,87,152,163, Dicke-withke cage, 21,170 Permions, 148 Differential operatorsforangularmo-FeynmanHellmanntheorem(robles), ‘mentom, 168 27 Differential settingcrosssection,380,Finestructureconstant,17 382,399Flux,47,70,183 DiracDatafunction,68,491 Flaxconservationinradialequation,184 Discnotation,113, Fourrectors,461, Dispersion, 119 Fourier integral, 27,490 Doppler broadening, 367 Fourier series, 489Doubleslitexperiment20,35 FowlerNordheimformula,87alongPetitlaofspecificheats,6 iceparticles,67 Sehrodinger equation, 45 Effective range formula, 394 radi equation, 181Ehrenfestthorer ofcasiimit,122‘igenfanctions, 39 GaugeinvarianceofSchodingerequation, ‘orthogonality, 496 218 Eigeevale, 39 Gauge transformations, 210 Eigenvaloe equation, 38 Gedankenexperment, 20forinfinite box,60 (GollMann-Ne'eman unitary symmetry, 42forLa,169 GelMann-Nishiima srangenestheory,439fnmatesform,231 GelbMann-Okubo massformals,443 insiein equivalence principle, 37,466 GallMann-Pas predictionofKy,454 Binsinmodeloftoe,371 GellMann-Zweig quarkmode,45 Einsteinphotoelectriceffectformula,7GoldenRulefortransitionrat,350 Electricdipolemoment,260,338 Gravitationaldeflectionoflight,467 Electricdipoleapproximation, 352 Gravitational frequencysil,37,373,467‘electionrules,354 Groundstate,61,130transition ratefor2P—» 15,357, Group veloc, 31 Becrie quadrupoletrantions, 354 Gyromagnetie ratioforelectron, 235 Electroninconstantmagneticed, ‘Schrodinger equation icylindtie ——-Hamfonian, $3. coordinates, 215 Index St forator, 299 Lagrange maltipice invasiational principle,oreleetoninextemalld,211,342 300, Harmonie oulator,11,127 ‘Laguetepolynomials, 199 ‘operator methods, 127 ‘Lambda particle discovery, 436 ‘lgenfunctons, 133 {armor frequency, 213‘matesform,28 Las,376arte self-consistent method, 300 {ee-Yang onpatty nonconservaton, 451 Heisenberg uncertainty telations, 33,36, Legends polynomials 175,175,206120 Levinsontheorem,393‘Heisenberg picture, 135 Lifetime, 368, 473eisenbergexplanation offeromagoetism, Linewidth366,478291 olsonbroadening, 366Heitl-London method formolecles,329 Doppler broadening 367 etiom atom, 283, 303, ecoll hit, 368 ‘exchange ineracton, 289, Linear operators, $8,114, 495 Sstorder evel shifts, 286 Lorentz force, 211, 223ingluenceofspin,286,289 Lorenz invariants, 461 Hermitianconjugieoperator,115 Lorentzianlineshape,366,478 Hermitian operators, 82,118,495 ‘Hund’ Rules, 291,308 Magic numbers, 188 Hybaid orbitals, 336 Magnetic dipole moment, 235 Hydrogen atom, Bobr model, 12 ofpin, 235, radial equation, 195, Magnetic dipole transitions, 355felativstceffects,271, Magneticfaxquantization, 220,secteur,197 ‘Massabsorptioncneticen,$15 Spin-orbit coupling, 274 ‘Mass formula forbaryonsandmesons,49, Hydrogen molecule, 316, 323, 327 Matrices, 229,Hiyperchage, 441 Matexproducts,228,‘Hypeetine structure, 27 Matrix representation ofoperators, 229,230 Maxwell'sequations,209 Identicalparticies,146,250 Meanfeepath,411scattering, 401 Meisner effect, 221Inducedsbsorptionandemission,373 Mesontheory6fnuclearforces,seeYukawa Inclatic collisions, 384 Miller indices ofBragg plans, 408 Infitebox,reeParticleinbox. Molecules,313 Intensityofspectrallinesinmolecules,321clasifeation ofstates, 324 i Intensity ofdipole radiation, 358 ‘electronic ences, 316 Intensity andspin, 358 ‘orbitals, 328,332 Inteal conversion (problem), 362 ‘onedimcnsioaal model, 93Interpretation ofwavefunction, 46 ‘pecticheats,322 Interpretationofexpansioncoeficients, 64,structure,327 113,typesofmotion,314 Invarance under discrete displacements, 89 Momentum opersior, 49,142 “doplacements, 143 hecniticity, 52 particle-andpstile conjugation, 425 tigenfunctions, 67 foutions, 157 Momentum conservation, 142, 143 seflections,seParity ‘ofphoton,10 Isotopic spin, $29 Space wave function, $0 ‘conservation, 434 Mossbauer effect, 368 smoltplets, 431 N-pucticle eystem, 141 KK system, 454 Hamiltonian, 141 regeneration, 455 Neutrino, 450Kitehhoflawsofthermalradiation,1,2Nestron-proton scattering,395Konig Penney model, 98 ‘potential spidependence, 396 512 Index Normalization ofmomentum cen: atresnance, 389Tanetons,68 foxsquarewall,189,389 Normalizationofeigenfunctions, 112 atdeal,389, Noesconnectionwithegy,62 Phasftexpansionofcateringamplitude, Nicleon sotopc sin, 430 335, 308 Phasespac,348 ! Obsrvbies,119 focmanyparticlesat,350 Operators31,58 Phaseofweuncon,46 Termitan,$2,495 Phooons, 372 Aiea,58,495, Photons,10 singandlowering,130,171 Photonomentum,10stots emi poems, 127. ahr aneon, 40,5 Optica bee. otdlimgzation of deuteron, 55 Ottis, 291,328 Phowelects ele, 8Orthoncrmaliy condition, 61,112,496 angulardependence41s Oritoheim, 392 Gros tin, 413. Overap intra, 318 ‘ney dependence, 614 sautycoment,410, Pairproduction,418 ons(phmesons), 251,28,485 showers419 ancconstant5,16 Pavedelectronsandbonding,331 hankradiationformula,5 Faispropos ofsociated production, 438" Yesaon, 373, FrisPiecion regeneration experiment, 455 Planewaveenpressed inspherical harmo,Prahebum, 292 19 Paranagretc resonance, 237 Potarabty, 262Paty,65,67 Polarizationofphoton,45 nconeratoninwea itectons, med one357 q "50,Population inversion, 377 otpon,251 Postionoperator,133 selection ales 354 ‘genes andthe interpretation ofwavePaseval's tore, 50 function, 133Palwavesatrigampitude,381,34Pogtron417,23Pateiabos,60 Pestonia,425 Se gy anlation rat,426 i Inthedimensions,162, chargeconjugation,426 aulexcusionprinciple,145,150 Potentialbare,64 | Puliprinciple, effectonmolecular spectral Potential string andphaseshit,154, |intesis,321 189,380,369, 390andisotope pin,430 Potential eatienng iBornapproximation,sndtwospinorstates,250,251 397,401 Pllspinmatics,233 Potentialsep,75 Penetration ofwave fetins, 78 Potential wel, 73 Period wav ftctions, 52,99, 348 ound sates, 80 Feld pant 98 ddparty bound sateconditions, 83 fPeriodicfable,307 Prestonofpi,238 Permanent dipole moment, 260 Probability onsetion, 67 Perturbation theory, calculation ofnuclene Probability interpeation ofwarefancion, ‘ee om meson exchange, 483 38 Feraraton theory, convergence, 266 Probably imterpretation ofexpansion codegenerate,258, citicients,644Expansionpostulate Testonder, 256 Propagation ofwavepacket 30 second order sit, 257,477Sccondordermatielement,417 uiansication ofangularmomenta,15,17, time dependent, 341 2 timeindopendsi, 255 Quunzaion ofeectromagpetie Sd, 342 ihapsiftforradialslain,186 ancamofraion,7 Intex 513 (Quantum electradymamicfomofvectorSettingmatricnonedimension(pote), potntal348 106 uaodel446 Settering,independence, 395masonofmesons,447 Sehvodingrequation32 Soman oftaryons, 48 ieparle 43 smerny 403 ‘indica oon, 215 iilconan,43 Ratiequation,161,176,178 {oNpair142 ‘eurandrepoarsttions,180 paidpotential,53Symproti lation, 128 ‘pantion ofcentr ofmassmotion, 155‘Shionforhytossn,198 ‘pationofangucoordinatey,16 otofslutons,301-205 Separationoftsdependence,31 Radiationofetme341 ineiments138 odiaone419, tinedependent, 57 Radiativeanions, mateselement, 351 tnendependen 57 BP1Srte366 Sehraingrpete,135 Rakcgoperators191,171 Schwartnua(prcbem),123, KamsserTowneffect,80 ScreenedColomponents39 Rangeofmicier forces andmeton many, Surening ofnuclear hare, 289,292tee Selectionales,251,303Rareats, 306 Tororal anal momencum, 353Rayligieans blackbodyradiation aw. forpurty change, 358Reali ofexpctaton value 32 forspin change, 34RecoemissionQMoutuvereffec),369forecompons ofgularmomentum,Reducedas,145,187 333 tectonpct197 solnZemanfet,213, Refactonbypote step,76 2eozeo waning 356 Relatconectiontorogen Shadowseateng388spectrom, 271 Shell mod ofmils, 188Reuters, 461 Simultaneoussentnctons, 6,71,117 Relaisansarmaion beoweenabandcondifonsonopeors19 “enterofmasfumes462 Sigestate,behaiorerparticx= Resonanceenpy,rlation topos of changes38), enegykel,390 Statedettinant,149,302 Resonanteter,289 SomefaeWibon quantire,19 ‘BreeWigoeorl392 Specientsofmole,322 Resonanstatsinelu,295;s0¢cso “Yabatonleffects334 ‘Astoateaton Spectrumofhydrops,17,197Resonate inpate physic 432 poiationn 974‘Ricmann-Leshoge homme, 362 Spectum ofhelm, 285ivationprincipe92,29,318Spectrumsfmomentumoperator,8 Kotationlmosonofmolecu,36 Spectruminparcepyar,39 Rotationalatesand thePaulleinspe, Spee Bene!fnchoae, 182 Ea ronI? oem squaion 170 Sheil Hane!function, 182 ‘withSintonparis Seteharmon17 Ruther, pantry model 14 Shel Syme ofdomedshel303‘tostection forCoulomb catering, 400. Spheal wove ib Hncoming andouting, 184,380 Swesting, 392 spin operators 252‘non teen amplitude andbound Spincompocn expectation a, 235atepostion 393 Spirdependence ofseston eng 396Scalarproduc,13 Spindependent penal246,395,101Seg St ay Sindeentent ohhci,230 . Seeing em indeviecon photodinseaion,‘Scatteringlength,394,396, 385 moen SuIndex l‘SpinandIntensityrales,358 “Twosexperiment, 20,34Spinmatrices, 232Spin-obitcompling, 272,304 Uncertainty relations 3,62‘Spinsagletwavefunctions,244 ‘ispersion,119‘Spinprestoninmagneticfield,236 ‘eforestimates,39 Spinstaitic connection, 148 (generalproof,497Spinrips wavefunction,244 {infinitebox,62 Spinors,233, ‘ucearrooiaMossbauereffect,372 ‘Spreadingofwavepacket,31 fhadowscattering,386 ‘Square wel,78 Unitary symmetry, $42‘boundstains,185 ‘cleianddecuplets463 ‘eeppotentialmit,186 ‘iscoveryof7,444 resonant scattering, 390Swave,186,392 Valencebonds,329 ___inthvesdimensions, 188,189 Variationalpincpl,foratoms,299‘squareintegrablefunctions,46,111 forhelm,292 Starkeffect,259 formolecules,318,328 Convergenceofperturbationseries,266Vectorpotenti208, forn=2ates,263, forconstantmagneticed,213 andpasty,260 ‘Vestorpotentialforemisionandabsorptiontecond order, 261 ofphotons, 343,StefanBoltzmannlaw,6 Vectorspaces,13 Strange particle production anddecays, 437. Virkltheorem (problem), 208 Strangeness, 436,440‘Sumrules,363,368 Watermolecule,336,337 Superconducting exp,#9 Waveequation, seSchrodinger equation Subctpocition ofwares,sceWavepacketsWavemechani,peeralstructure,111 Symimetzy ofHamiltonian, 6 Wavepackets 27,69 q gaussian, 28 : ‘Thomas precession effect, 272 Finitations onwidth, 28‘ThomasReiche-Kuhn imrale(problem), andnon-normalizale sates, 69268 propasation,30 ‘Tomedependenceofexpectationales,120Inscattering,379 Timedependenceofoperons,135 spreading,31 ‘Timedependenceofwavefonctions,3Wavo-particle duality,20 ‘Timedevelopmentofsystems,136WentzelKramersBrillouin(VKB)approx ‘Timedevelopment ofdecayingsate,473, ‘mation,85,469 ‘Timeencigy uncertainty ratio, 36,366 Wienlaworblackbodyradiation, 2“Tranaton rate,347 ‘Work function, 8,87 relation tolifetime, 365-Thanumission coefficient foruae wel, Yukawa theory ofnuclear forces, 39,481"6 ‘Yakawa formofnuclear potent, 434 “Trammisson coefficient inWKB approxi‘mation,86 ‘eamaneffect,normal,213,‘Tripletstat,bebaviorunderparticleex- ‘anomalous,275, change, 250 forstrongfields,277‘Tunneling,85,86 Zeyo-pointenergy,105‘Twopartie system,146 Zerotansionsectionrule,356