Completeness

Massachusetts Institute of Technology Department of Physics Physics 8.07 Fall 2003 Completeness Relations c/circlecopyrt2003 Edmund Bertschinger. All rights reserved. 1 Introduction Two mathematical principles underly the solution of Laplace ’s equation with basis func- tion expansions such as those used in the separation of variables: o rthonormality and completeness. The treatment here will be brief and non-rigoro us as the aim is to give an understanding of completeness relations without proving the m. The mathematical basis for completeness relations is given by Sturm-Liouville theo ry of second-order differential equations. The interested student may find a complete presentati on in a course such as 18.075 or 18.152 at MIT. The mathematical problem we are trying to solve is Laplace’s e quation ∇2V= 0 in some volume bounded by a surface Son which boundary conditions are imposed on V or its normal gradient, ∂V/∂n ≡/vector n·/vector∇V. In applying separation of variables, we obtain eigenvalue problems such as L2u=−k2u (1) subject to boundary conditions. Here k2is a positive constant and L2is a second- order differential operator (ordinary or partial) that come s from separating variables in Laplace’s equation. For example, in Cartesian coordinates, ∇2=L2(x) +L2(y) +L2(z) =∂2 ∂x2+∂2 ∂y2+∂2 ∂z2, (2) while in spherical coordinates ∇2=1 r2∂ ∂r/parenleftBigg r2∂ ∂r/parenrightBigg +1 r2L2(θ,φ), L2(θ,φ)≡1 sinθ∂ ∂θ/parenleftBigg sinθ∂ ∂θ/parenrightBigg +1 sin2θ∂2 ∂φ2.(3) The corresponding eigenvalue problems are d2u dx2=−k2u (4) 1 and 1 sinθ∂ ∂θ/parenleftBigg sinθ∂F ∂θ/parenrightBigg +1 sin2θ∂2F ∂φ2=−l(l+ 1)F , (5) subject to appropriate boundary conditions in each case. The c onstants −k2and−l(l+1) are called eigenvalues, and the corresponding solutions of the differential equations are called eigenfunctions. Imposition of appropriate boundary conditions makes equati on (4) and (5) Sturm- Liouville problems, for which orthogonality and completene ss may be rigorously estab- lished. There are many more Sturm-Liouville problems than th ese two cases, but they arise so frequently in electromagnetism that these notes focus o n them. Equations (4) and (5) are eigenvalue problems. If one applies appropriate boundary conditions, then it turns out that these equations can only be sa tisfied for certain values of the constants kandl. Sturm-Liouville theory shows that the eigenfunctions form a complete, orthogonal set of functions satisfying the boundar y conditions. There is a strong analogy here with ordinary vector algebra. This analo gy will be exemplified in the sections that follow. 2 Orthonormality and Completeness in 1-D We start with eigenvalue problems like equation (4) supplemen ted by boundary condi- tions. The functions u(x) satisfying equation (4) form a set of basis functions for all functions satisfying the same boundary conditions. We assume that boundary condi- tions are imposed at x=aandx=b(which may be ±∞). We assume that there is a scalar product operation that takes two functions and maps th em to a single number. In Sturm-Liouville theory, the inner product of two functio nsf(x) and g(x) is (f,g) =/integraldisplayb af(x)g(x)w(x)dx , (6) where w(x) is a given function that depends on the differential operato r in the eigenvalue problem. Now suppose, for simplicity, that the eigenvalues kare discrete: k∈ {kn}with n= 1,2,.... Eigenvalue nhas corresponding eigenfunction un(x). Then, any function V(x) satisfying the same boundary conditions as unmay be written V(x) =∞/summationdisplay n=1Vnun(x) (7) where the coefficients Vnare unique constants. The functions un(x) are akin to basis vectors; equation (7) is the expansion of a vector in basis vecto rs. This analogy is possible because functions define a linear vector space, albeit o ne with infinitely many dimensions. (Recall that the dimensionality of a vector space eq uals the number of linearly independent basis vectors.) 2 Equation (7) is useful only if two conditions are satisfied. First , we must be able to determine the coefficients Vnthat solve a given problem. Second, the series expansion must converge to the solution V(x). The first condition is made possible by orthonormality of the basi s functions. Now, Sturm-Liouville theory shows that eigenfunctions with diffe rent eigenvalues are orthog- onal with respect to the inner product: ( um,un) = 0 if m/negationslash=n. Ifm=n, on the other hand, the Sturm-Liouville scalar product is positive-definit e, so that we may normalize the eigenfunctions: (um,un) =δmn(Orthonormality) . (8) Taking the scalar product of equation (7) with unand using orthonormality, we obtain Vn= (un,V). (9) The second condition expresses the completeness of the eigenfun ctions. In Sturm- Liouville theory, completeness is established by showing that ( f,f)≥0 for any function f(x), and (δV,δV ) = 0 ,where δV(x)≡V(x)−∞/summationdisplay n=1(un,v)un(x). (10) This condition implies that the series given by equation (7) c onverges (with respect to the inner product norm) to V(x) if the expansion coefficients are given by equation (9). Combining these two equations, we require V(x) =∞/summationdisplay n=1/bracketleftBigg/integraldisplayb aV(x/prime)un(x/prime)w(x/prime)dx/prime/bracketrightBigg un(x) =/integraldisplayb a/bracketleftBigg∞/summationdisplay n=1un(x)un(x/prime)w(x/prime)/bracketrightBigg V(x/prime). (11) It turns out that for Sturm-Liouville problems this holds fo r anyVin the function space, which requires ∞/summationdisplay n=1un(x)un(x/prime)w(x/prime) =δ(x−x/prime) (Completeness) . (12) The following subsections give the eigenfunctions, inner prod uct, orthonormality and completeness relations for several common 1-D cases. 2.1 Fourier Integrals The first case is d2u/dx2=−k2uon the infinite domain −∞< x < ∞with boundary conditions |u(x)|is finite as x→ ±∞ . The eigenvalues know are the set of all real numbers, so that the eigenfunctions are labelled not by a discre te index nbut instead by a continuous variable k: u(x;k) =eikx. (13) 3 The inner product is defined with respect to weight w(x) = (2 π)−1. Because the eigen- value is continuous rather than discrete, orthonormality is expressed with the Dirac delta function: /integraldisplay∞ −∞u∗(x;k/prime)u(x;k)w(x)dx=1 2π/integraldisplay∞ −∞ei(k−k/prime)xdx=δ(k−k/prime). (14) Note that the eigenfunctions are complex and the inner produc t takes one complex conjugate. Equation (14) is the Fourier integral representa tion of the delta function. It is proven in math courses that discuss Fourier analysis. The Fourier completeness relation is given by an integral rat her than sum over the continuous eigenvalue: /integraldisplay∞ −∞u∗(x/prime;k)u(x;k)w(x/prime)dk=1 2π/integraldisplay∞ −∞eik(x−x/prime)dk=δ(x−x/prime). (15) This follows at once from equation (14). 2.2 Fourier Sine Series Often one has to solve boundary value problems in a finite rathe r than infinite domain. In this case the eigenvalues are discrete. If the boundary cond itions on d2u/dx2=−k2u areu= 0 at x= 0 and x=a, then the eigenfunctions are un(x) = sin( nπx/a ),(n= 1,2,...), (16) the weight function is w(x) = 2/a, and orthonormality is (um,un) =/integraldisplaya 0um(x)un(x)w(x)dx=2 a/integraldisplaya 0sin(mπx/a ) sin(nπx/a )dx=δmn.(17) This integral is elementary and may easily be checked. However , the completeness rela- tion is not so obvious: ∞/summationdisplay n=1un(x/prime)un(x)w(x) =2 a∞/summationdisplay n=1sin(nπx/prime/a) sin(nπx/a ) =δ(x−x/prime). (18) Although the proof of equation (18) is beyond these notes, we can easily enough verify it numerically. If we integrate both sides over x/primefromx/prime= 0 to x/prime=x0, and then exchange xandx0, we should get the Heaviside step function: θ(x−x0) =2 π∞/summationdisplay n=11 n[1−cos(nπx/a )] sin(nπx0/a). (19) Figure 1 shows this sum evaluated with 20 and 200 terms. The series converges slowly to the correct result everywhere except at the discontinuity x=x0, where there is an 4 1 0.20.8 0.40.6 0 x1 0.8 0.4 0 0.6 0.2 Figure 1: Numerical approximation to the Heaviside step functio nθ(x−0.6) computed using 20 (green) and 200 (red) terms of the Fourier ser ies in eq. (19). The overshoot and oscillation arise from the Gibbs phenom enon of Fourier series. undershoot and overshoot. This phenomenon is well known with F ourier series, which converge to the correct value for continuous functions but n ot necessarily at discontinu- ities. This means that equation (18), and more generally equa tion (12), satisfy the delta function rule f(x0) =/integraldisplay∞ −∞f(x)δ(x−x0)dx (20) only for smooth functions f(x) without discontinuities. Fortunately, solutions of Laplace ’s equation are smooth, so the Gibbs phenomenon is of no concern in this case. 2.3 Fourier Cosine Series If the boundary conditions are u= 1 at x= 0 and x=a, then the eigenfunctions of d2u/dx2=−k2uare un(x) =/braceleftBigg 1/√ 2, n = 0 , cos(nπx/a ), n= 1,1,2,...,(21) 5 the weight function is w(x) = 2/a, and orthonormality is (um,un) =/integraldisplaya 0um(x)un(x)w(x)dx=2 a/integraldisplaya 0cos(mπx/a ) cos(nπx/a )dx=δmn.(22) The only difference with the Fourier sine series is the inclusion of the n= 0 term. The completeness relation is ∞/summationdisplay n=0un(x/prime)un(x)w(x) =2 a∞/summationdisplay n=0cos(nπx/prime/a) cos(nπx/a ) =δ(x−x/prime). (23) If this relation is integrated to give the Heaviside function, and the series plotted, the results are very similar to Fig. 1. I leave it as an exercise for the reader to write down the relati ons when the boundary conditions differ from the two simple cases considered here (sine and cosine series). Whatever the boundary conditions, one simply chooses the eigen functions of d2u/dx2 that obey the boundary conditions. Note that the hyperbolic fu nctions which result when−k2=κ2>0 donotform an orthonormal basis. The corresponding boundary value problem is not of Sturm-Liouville type. 2.4 Legendre Series The Legendre differential equation arises from the equation ( 5) when ∂F/∂φ = 0. Defin- ingx= cos θ, equation (5) reduces to an ordinary differential equation, d dx/bracketleftBigg (1−x2)du dx/bracketrightBigg =−l(l+ 1)u , (24) which is to be solved on the domain 0 ≤x≤1. For any lthere are two solutions to the differential equation, denoted Pl(x) and Ql(x). For most applications, the boundary conditions are that u(x) be finite everywhere in 0 ≤x≤1. This condition is satisfied if and only if lis an integer. It is sufficient to consider non-negative integer s only, because the eigenvalue −l(l+ 1) is invariant under l→ −(l+ 1). Even with lrestricted to {0,1,2,...}, there are still two solutions to equation (24). One solution ( Pl) is finite at x=±1 while the other ( Ql) has logarithmic singularities. Only the Pl(x) with integer lcan satisfy the physical boundary conditions that u(x) be finite everywhere. The functions Pl(x) are the Legendre polynomials. The functions Ql(x) are called the Legendre functions of the second kind. The orthonormality relation for Legendre polynomials is (Pl,Pl/prime) =2l+ 1 2/integraldisplay1 −1Pl(x)Pl/prime(x)dx=δll/prime (25) and the completeness relation is ∞/summationdisplay l=02l+ 1 2Pl(x/prime)Pl(x) =δ(x−x/prime). (26) 6 1 0.20.8 0.40.6 0 x1 0.5 -0.5 -1 0 Figure 2: Numerical approximation to the Heaviside step functio nθ(x− 0.2) computed using 20 (green) and 90 (red) terms of the Legendre series. The Gibbs phenomenon is a general feature of Sturm-Liouvill e eigenfunction expansions. Integrating the completeness relation gives a series represen tation of the Heaviside func- tion. The results are shown in Figure 2. 3 Orthonormality and Completeness on the Sphere Equation (5), with boundary conditions stating that F(θ,φ) is finite everywhere on the sphere, defines an eigenvalue problem. The dependence on the t wo variables separates, and the eigenfunctions are F(θ,φ) =Ylm(θ,φ)≡/radicaltp/radicalvertex/radicalvertex/radicalbt2l+ 1 4π(l−m)! (l+m)!Pm l(cosθ)eimφ. (27) The functions Ylm(θ,φ) are called spherical harmonics. Apart from a normalization constant, they are products of the associated Legendre functio nsPm l(cosθ) and a complex exponential. The boundary condition F(θ,φ+ 2π) =F(θ,φ) requires that mbe an integer. The boundary condition that Fbe finite everywhere on the sphere requires that 7 lbe an integer; as before, we can take it to be a non-negative in teger without loss of generality. Finally, the boundary conditions also require −l≤m≤l. Thus, for each degree l, there are 2 l+ 1 values of the order m. The spherical harmonic functions obey the following orthono rmality condition: /integraldisplay Ylm(θ,φ)Y∗ l/primem/prime(θ,φ)dΩ =δll/primeδmm/prime, (28) where dΩ≡sinθdθdφ and the integration is taken over the sphere, 0 ≤θ≤π, 0≤φ≤ 2π. The complex conjugate is necessary because of the exp( imφ) factor just as with the complex Fourier integral. The completeness relation for sphe rical harmonics is ∞/summationdisplay l=0l/summationdisplay m=−lYlm(θ,φ)Y∗ lm(θ/prime,φ/prime) =1 sinθδ(θ−θ/prime)δ(φ−φ/prime). (29) Many of the properties of spherical harmonics are presented an d discussed in Classical Electrodynamics by J.D. Jackson. We will not use them in 8.07 but they are used extensively in advanced physics. 8