Wave Packets & Cross Sections

: |- | ™. -" \ ! WAVE PACKETS, CROSS SECTIONS, BEAMS ,ETC, -O: --|Gontenter - _. _-- -a|“,DiscussionofIncidentStateWavePacketBusiness(typed,3p) “2,ActualMS-typewavepacketcrosssectionderivation. (6p)--—--{-"LAtk*betaiton Momentum-Space Wavefunetione.—aad oO B.-Supportive material (seepage6forcross section idea) HD tame noxnatiog andFerai's GoldenHaleorig, ~~~""1"i,comments onusingfinalstatewavepackets. (typed)- 4-A.TheactualI-calculation. ov See we -;5Experimenter andhisBeams. - aea.Construction ofpackets going inarbitrary direction. a b.how hecomputes the luminosity integral 2. ee c.the very simple case ofthe "big uniform target plate” ' \ 1eee -- - \ i— - os we eee | ae - oe ee a - on ae a ee wee ee EL-by4ee | ~Phit koeWt ;7 Te 10965 — . 1 | | i |Discussion ofthe Wave Packet Cross Section Derivation Oo ~—-}-€Working-Paper Only) === ye298THon a ee weet 1.Inthe beginning, you consider the scattering oftwo particles, where----+ +eachparticle-is-conceived ofusa*titthe ball”travelling throug ——— ~space. Each "ball" isawavepacket following aclassical trajectory eeTT through spaces ee -Each "ball" ischaracterized byamomentum space wavefunction soe - normedized-to-unity.-Tlese wavefunctitons-should “be“thought“ofs—— A+ strongly peaked attheclassical momentum oftheball, butnot .+—infinibely-peaked thereat. Having-an~infinite-peak destroys“ tte~~1 configuration space wavefunction byblurring itout sothat it|~-"#LEs-ald--space withequalprobability density"s Clearly; ~such =aspatial wavefunction has nomeaning, even ifyou “put itinafoboxe -— —-- eT ae -— Now,onceyouhave(theoretically) constructed yourmomentum |~wavefunctions, say with: strong-gaussian peaksy you-may-compute —-. thespatial wavefunction. Thisfunction f(x,t) stiould havethe+=spacetime structure of:oneofthese “balls” wehavebeentalkingabout, travelling along aclassical trajectory. Ofcourse the norm -aldzation-of thespatial: wavefunction depends. on-how-you define~it, and ingeneral will not beunity. seen Once you have -your-spatial: wavefunctions; you-cen derive . densities. Each density should besuch that atanytime thespatial -——- ~-— |——integral isunity. Notice-that: adansity-ie-not a-Lorentz scalars~ ——~ Asyou“flyover"somematterathighspeed,theatomsappearmore O | closely--spaced. in.the-Longitidinal- direction, -so-you-measure™a ~-- higher density than you would ifyour "plane" vere atrest. But, _— | thedensity. divided bythe-energy-of-the-observed-atoms-4s a-scalar -- bevause bothincrease bythesameamount, gemma. | The upshot-of. all-thie-is-that the-spacetime wavefunction - - squared, asdefined byMS, isaLorentz scalar. Soisthe momentum - | space wave-furiction, ome - {2.tlow-that thesevarious-wavefunctions-are established, wecandiscuss~ -- | the matrix elements ofthe T-matrix from an_initial state consisting ; ofthedirect. product-of two-single particle-"distributed"-state ---- vectors, asdiscussed above, into afimal state consisting ofplane‘aves.Wenever-have any.needtoconstruct wave-packets-for-the —- |final states! Now, when this matrix element issquared and multiplied—-s|.by-an-apprépriate_phase space,we-have-s-direct-interpretation: ——— - toe ~ /S/*x.Phasespace -=—probability that-the reaction: tookplace -—- To elaboratealittle, this-is-the puobability—thatywhen we-sent- inthesingle beamparticle andthesingle target particle intheir wh-spatial wavepackets(as described bytheinformation in-themomentum | space wavefunctions and ultimately the momentum state vectors which a appear-intheT-matrix element),when wesent.the-incident particles - I ininthis way, /T/2 xPhasespace ~-the ‘probability that acertain |setoffinalparticles"appear"intherespective momentumconese--—- O‘ So, suppose for each reaction this probability is.0l where we.havesetupvery.small"counters" which spaneachd3ptcone*, Then | ifwedothis experiment 10,000 times, weexpect, teget 100 "events", An"event" dswhen -all -your counters—go~"ding" » ~- ‘ f i aL -3eUsing_our explicit_exprassions forthemomentum_space_incidentstate .— oO t vectors, wemayrewrite /T/?interms ofthestandard plane wave——|-states. everywhere. Thenwe_are..on_faniiliar terrain and-all remaining __\ manipulations except forthecross section areeasy andoldhat.aeee Forstarters, youreplace theTmatrixelement_with the.“alphat: type matrix element bygoing into the "little hilbert space". This _-. —. —ofconree produces, variousd+deltafunctions. Whenell. manipulations __ = are finished, you end upwith this equation: ca t ~ — aa eee >= :Case.=ITTpeaasganseek we ee —~+] =Gn).AGATHOL Soag@Z@— . -.| agLE From comments made earlier, itisclear that this probability is ~ +lorentx -invariants- - mee ri eae heALittle discussion ofthis: equation-is-in order. Wecam-have agenuine picture ofwhat happens inthe scattering process. The two incident~particles -start-in-e-regton-where ‘theyhaveno-spatiat-overlap of >~their density functions. This iswhat "no interaction inthe asymptotic- state”meansy ~ —ee Sateen‘Then, astime progresses, the flying "balls" begin topass through—— =, ~each-gther. Assoon:as-they -overtap; thegeneration ofthe-"finat- ——————e) ‘state begins,Inthetimeintervalttot+dt,thereis"activity" |dithe-overlap-volume-¥e ‘In-that- "spacetime voluie" probability -is i being generated at@level" which isproportional tothé overlap. ~- Té-we-focis down ona-very-small spstial region, then int totedt; -~~|probability isgenerated at2“level” whichisproportional tothe-- -|product -ofthedensities.—To-get- thetotalprobability generated-by— -—| this scattering, youintegrate over allspacetime! Pretty clear!_ So,the-general picture-is-two- balls-of "gunk"-passing-through—each other. Whereever and whenever the gunk ofone ball touches the -— —gunk of-the-other-ball,-new gunk isgenerateds---—— ----- ~~ “In this picture, the particles are definitely not "point" particles. - i.--Quantum-mechanics tells usthat exact-spatial-locelization issimply: - not meaningful. Thus, even ifyou doascattering experiment with —— - ~ am-ineident-electron, that-electron isspread-out-into-a-ball. Sbilly --—~ Iampicturing the “interaction” as"local" inthe sense that product - -4+ is-generated'bnly-where-the-incidents-have-regions-which: occupy the- ~--— | Same spacetime volume. . : More will besaid onthis-subject-later re"field theory%s—— -- * |5+Sinceexperiments—are notdonewitha-single-beam-particle and:a~singde--——~a target, wehave tomake some modifications togettheprobability-==)++ =--You-would-obtain-in-a realexperiment. —- -oe{ Thetiodification iseasy,however. Consider the:scattering of a : -2"beam burst! witha"target". Weneed-not-worry about such-things-as---—— whether the diamter ofthe "beam" islarger orsmaller thah the. target, and.all that.Weneed‘only.conceive of-the-entire targetoO asalarge"particle", andthebeamburst asanother"particle". —Wereplacethe-"single balldensity function" with.a "target -density-fundtion" and a“beam burst density function". These new density + functions are-simply sums-of little balls 4-.s0-they-certainly are 14 t . iS | vee- F7\-- —-}-not-the..smopthdensityfunctionsofmacrophysicse, (e) i ‘$0consider aninstant intimewhenthebeamburst is."in™ the— __—|target. Whereis "product" beinggnerated? Microscopically, product =iisBeinggenerated wherevera"beanball”overlapsa“targetwns, - =- — |.At-one time, such overlapscouldbeoccurring dtseveralplaces,snd T“Gachoverlap mekesacontribution. Ofcoursewehaveinminda—....—-.reasonably small_physical targetsothatwherevertheoverlapoccurs,=__ bl |thefarawaydetectors subtendthegame,counofphasespace.(This +4acalled the"fiducial volume’ Ithink), ___=eatGaent -oe <t*SSteadofexplicitiy showingwhereandwhenthevericis"balis" 7.)ovprlap, weshallow ourmacroscopic density.functions’ keeptrack |ofthat forus. Atanyinstantintime,theoverlapofthesemacro |density functions isreally thesumoflots oflittle ball overlaps. _ .‘All this discussion hasbeendesigned towotivatethissiodifiéd.equations =_ aeeee wee ee ao Qn a. SatjPasGe ITALYale.ORG) . i Sg . ~ weed OEEBAO=Ne we i NT 70.Skcre . -- Now"Prob." isthenumber ofevents obtained perexperiment. Eg,ifprob_ js=27,23bycorrectcalculation (withknowledge ofdetectorsizésetc). CE._.--.tenyouyouldexpectthat,1¢you were toperformtheexperingnt, ‘one fiillion times, the average number ofevents per experiment would he 2183. eei. Thequestion ofhewmanyexperiments youhavetodotomeasure Prob. -to1%isIthinkapurelystatisticalonehaveingnothingtodowith___ |quantim mechanics orphysics. Return tothis later. 7oe 6,Finally wecome tothis matterofthecrosssection. Picktworegions— _.|—..9fphysical space, eachshaped like2,penny.Letthese.regions be... _—very sméll andletthem beflat touching one another, Atsome instanta ;antime,therewillbe.so.manytarget.particlesihthe"targetpenny”,.. | which weimigine tobestrest; "lab". Inthe "bean" penny there are we... |.aknown nymber .ofheamparticles, all_travelling attheclassical. ___|-Meanvelocity. Byfitzing around andfinally ingegrating overall'spage.and time,wefinds 2 « . ; Pro,=ade.Side geegees oo aa |Bycomparison with theabove expressions forprob, wequickly arrive _ - |atthe“standard” cross section forma |7°1aminterested nowtotakethe“planewavelimit”ofalltheseequations — | 49learn_how tocorrectlyinterprettheplanewavederivation, This__ — oO' ‘should be fun! 3 ONE ;aaandapackaleodyoccnHS.” O Wadecaenlatsin, wospacks, Maou, beeen A coe SHttiQT foramen7 ‘ ah |Seley cee), ‘ele oewath»SPRD=cecee) Sole=cesiegy x wy = ee es - ee eeoa|Tk=aiq&s\Telaey -L=S#Etere -| oo, oa _| - . -=) eSGknt) Jere._.flats,Geaaxonruse.awit,—=5dadsesti.PEinCaylee“teoniaLeeuhsealeroperotn,thonSate, _2.DSKupYeamnaatwiaemend : oo OC) eeSeeLSA =iGKeel bD oateens . @ ee ; —G»ooI=Vanda) §.(8.10>. . *Mow,arin&oanaden dusrabeesrin, Sumeleneshaniagaoka)£AKphaOh.Velie:© : ~ 3ACG — aedela=\gaisOH)-2ed8e-9) =a] Aa,7 Ow. aeSela=VEOr |. =e - a gtAcctcrommadiign 2:6)soakLOH>=4) LLLa Manus >Golskb =2G. - woara) Mowidecay fe) nhSe) BLGED QM) CS) ~ ~~ Lo \ at Lv (o)Pins newmnnpesakind: extadaslohes 0.. e7LacedTMeed=OSC a.)CeeTe|xCried> oo. 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WWare etyongettersinApnahake.QeLt © nae myBCon) . oo|° x&|-Segme-- :YFveeOH)SRL1wes _ —ReKuddanphostsl —-— / ~ : - ee ~ - i | | —|1sWhenwe putwavepackets inthefinalstateaswell asthe-initial ©§-_ state, wefind that, assuming proper unit normalization ofall —-— --——|_ -sietrttuted_ataten, theprobability fortheneaction.to takeplaceté:— —— a . wee ee - Pe \MelTelaot- JIL eee ~ Itturns out that Iissimply anintegral ofthe products ofall the + +1Gaveranictfotis' overSpiceandtime.(Notethat’Cz hastodo-with the ~~*s \scaleofthealphastates only). _|2+Here aresome things tonote: ~——="gp ubllike HeCase"whéFepackéts areubedonily Tortheinstate, ~ t the Iweget here contains noexponentials. aa +-- ~p)dWthis “foll-wave case, you'd rotsad-any“phase“space"sThe final state iscompletely preset by your selection of -ft ~“Wavefunctions. ~ ine -- ‘3.Howdoweinterpret’ aninteraction? “Again, two"balls" comeinanit'“"stabttooverlap. During thetimetheyoverlap, finalstateis ——-+-b+ generatetsIe; therets-generated acertain probubilityfor-m—~ —- =~ "balls" tobegradually created inthe interaction vilume. Theseneo +~ballsthenexit~alongtheirweltdefined classical trajectories ~ | and end upinvarious detectors. Asbe6fre, "final ball probability” ~ "|>“Hergenerated inail-regions ofovertaput"any tineinstantsThe”~~~~ @) totalprobsbility isobtainedbyintegrating overallspacetime. |In-practice;thisintegration is-restricted-to thefidieial-volume-——~ — and the experiment duration. |~—*Werdealwithstrongly-peaked-wavefunctions ‘sothat,“the~T=mattix- element can becomputed using the classical plane wave momenta.-—— “Thus,atl-theinformation" about“suchthings~ws-whether-the ‘beam -|hitsthetarget, orwhether somefinalparticle hits,adetector, |aH this information is-contained-in-the-master overlap integral T-——— ‘Asusual, asingle experiment cannot tell you anything. You -_—-~ + have-to repeat; theexperiment lots~of-times so:that “the-concept of~~~ "final bell probability" gets replaced with “average number of-——|+-dingsinthe-detéctor set": ~~ wonttee me -|hecInpractice; ifyouwanted-to: compute-this "number -of-dings"; you-—| could dooneoftwo‘equivalent things. (1)youcould work with - 1~-#-"standard"-final-state wavepacket say much smaller-than the-— -|detector. Thenincomputation youwouldsumoverallpossiblesoot ot|-—finai-wavepackets-which hittheareaof-yourdetectors. (2)You - | could. "custom design" afinal wavepacket that exactly covers - |~your-detector, andjust useit.Ofcourse-we areassuming the ad. |detector issmalllsotheaboveaverage TapproxisOK. '5.Actually, the use ofthe "final state wave packets picture" isnot _ —1 =very useful-experimentally-“It ts‘easiest-to-use’ the-plane wave . "idealized" packets and integrate them over your detector areas. }Ie,it-isbest‘toshow-the"phase: space"-and' thenintegrate-as needed” (@)| over whatever observation areas ordetectors you are using. Ineffect, - -~!—“this-is the-"custom wavefunction" method described’ abdve, butwe-=~~ ‘never have toconstruct the complete wavefunction. Ineffect weonly an constructitatoneinstant-in-time {te,atthedetector)"and“we say~=~ where it came from, on * i \ ~2- 1 -O--- ‘64..0ntheotherhand,foranunderstnading_of a“oneshotexperiment, _ “T“the "final state wave packets picture” isthe only way togo. It - - | says "there isacertain specific probability forthefinal ==| particles toexit inthose specified wave packets asopposed tos—-}...some, other_set_of wavepackets_( ie,someother_star). Inevery =_ -experiment, some thing always happens, sosome probability was _+-=:- =+—Realized_into"an_"event". Thisleadsustowonder alittle about. __.__ . Many Worlds veCopehhagen, but now isnot the time orplace. . 1],‘Although wecanexperimentally dispense with these final state wave “-.+—Packets, theinitial statepackets areindespensible, Thetheorist. a isable togenerate across section for comparison with experiment, --- _-4--..but_the_experinenter doesnot"measure"crosssectiondirectly.He_ |Mueasures" the number ofdings inhis various detectors for a3 -—- ~~|—hour runofsomespecified beam ________ ___.When Isay that the initial wave packets are indispensible, -- that needs some qualification. No experimenter hasactyally_ | -_ had full knowledge of“the spatial wavefunction” inall its complexity; - _ie,itsspreading properties, etc.Heonly_needs toknowthe __spatial shape (density function) ofthepacket near thetime of _|AMteraction. Duringtheinteraction, hewouldgscunethatno | ~“ehange ofshape occurs, the bursts just propagate like rigid bodies. _ - -~~2 ee pe + So, inacrossed beasis experiment, one needs very precies imowledge a —. --—-1._. oftheburstdensity functions, including knowledgeof"wherethe” in‘bursts areandwhen".Itisnoteasytoknowexactlywhereyour e)_bursts are all_thetime,michlesstolmowtheirshape.Thetiewhy. there issomuch huffeh about knowing your luminosity”. -a«.Infact,atSLAConereallygivesuponcalculating the= 1”Iuminosity directiy inthisway.Youendupcomputing itindirectly-|by_simulteneoubly measuring Bhabha rates. Sinceyouknowfromthe _|theory theBhabha cross section, youcandeduce theluminosity from _ | the Bhabha rate. oe Le ee i | cor oo we eee ee ' 1 .‘ 1 a{+ . wee ee _- .j-- -. waeeeee oe e)* t - ~ = ee ee eee ee ee t yw EE —_T nn —e—— . a 4 OC-—- Grilled = Sk © Dakine : re oe (Rae SeldaeaaaptfSeee abEyeaejek | . ©SptBT SGLmpgad =BdedenyeM LeTees. 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