Vector Spherical Harmonics

Transcription

1 Vector Spherica Haronics Lecture Introduction Previousy we have seen that the Lapacian operator is different when operating on a vector or a scaar function. We avoided this probe by etting the Lapacian operate on a a Cartesian coordinate. We wi now deveop a set of generaized functions that represent soutions to the vector wave equation in spherica coodinates. Consider first the geoetry of the radiation probe. The source is coposed of oving charges of sa reative diensions, whie the observation point is paced essentiay at infinity. Therefore we use a utipoe expansion in spherica coordinates. We begin by finding the soution to the scaar wave equation in spherica coordinates, in order to obtain a set of eigenfunctions that span (r, θ, φ) space. In spherica coordinates the scaar Lapacian is; 2 = (/r 2 ) r [r2 r ] + [ r 2 sin(θ) We define the operator; L = i r θ (sin(θ) θ ) + r 2 sin 2 (θ) 2 φ 2 ] Now ; L x = i(y z z y ) L y = i(z x x z ) L z = i(x y y x ) Then use x = r sin(θ) cos(φ) y = r sin(θ) sin(φ) x = r cos(φ)

2 By transforing to the spherica set of coordinates; L x = i[sin(φ) θ L y = i[ cos(φ) θ L z = i φ Then note that ; + cot(θ) cos(φ) φ ] + cot(θ) sin(φ) φ ] L L = L 2 = [ sin(θ) θ (sin(θ) θ ) + sin 2 (θ) 2 φ 2 ] Finay the scaar wave equation in spherica coordinates is ; [ 2 + k 2 ]V = [(/r 2 ) r [r2 r ] + (/r2 )L 2 ]V = 0 2 Eigenfunctions We seek a soution of the above equation in the for V = f (kr) Y M (θ, φ). The Y M spherica haronic eigenfunctions; are the Y L 2 Y = ( ) [ = ( + ) Y (2 + )( )! ] π( + )π /2 P (cos(θ)) e iφ L z Y = Y The adder operators can aso be defined as ; L ± = L x ± il y = e ±iφ ( θ + ±i cot(θ) φ ) and shown to produce an eigenfinction of L z with ± L z [L ± Y ] = ( ± ) Y The boundry condition that the soution be continuous between φ and φ + 2π requires be an integer. The boundry condition that the Legendre function P reain finite at θ = 0, π requires that be an integer. A soution to the associated Legnedre equation requires that. Substitution into the scaar wave equation gives the radia d.e. 2

3 [ d2 dr 2 + (2/r) d dr + k2 ( + ) r2 ]F = 0 The soution is coposed of spherica Besse functions, which for our purposes are cobined to for the spherica Hanke functions, h and h 2. These are. F (kr) = h (, 2) (kr) = (π/2x) /2 [J +/2 (x) ± in +/2 (x)] The asyptotic fors of the Hanke functions are ; h,2 i x ( i) +e±ix x 3 Expansion of a vector fied We wi use the definition of the vector operator defined above. L = i r = ˆr r (i/r2 )[ r L] Then any vector fied can be expanded in a for; V = ψ + Lψ 2 + ( L)(ψ 3 /i) where ψ i i =, 2, 3 are arbitrary scaar functions. This is proved by using the fact than any vector can be decoposed in an irrotationa vector (cur = 0) and a soenoida vector (divergence = 0). We do not show the proof of the above expansion but this can be deonstrated by obtaining the vaues of the ψ i functions for an arbitrary vector. Given this expansion we write; B = ψ + Lψ 2 + ( L)(ψ 3 /i) Now we aso ust have that B = 0 and fro the definition L = 0 thus we can satisfy the expansion of the vector fied and the divergence condition independenty of ψ,2 Therefore we write; B = Lψ 2 A siiar deveopent for E can be obtained when we set E = 0, a condition we can obtain even in the presence of charge as wi be observed ater. 3

4 For the oent we consider a genera exape of the above expansion. Consider the agnetic fied of a current oop. Fro Apere s aw in steady state, B = µ J = A Choose a gauge where A = 0. the equation to be soved is then ; 2 A = µ J The above equation invoves the vector Lapacian which we have previousy seen can be quite copicated. In genera we define the 3 expansion vectors for above the proposed expansion by the foowing; V = ˆr[ [ ]/2 Y ] + ˆθ[ Y M [( + )(2 + )] /2 θ ] + ˆφ[ i [( + )(2 + )] /2 sin(θ) Y W = ˆr[[ 2 + ]/2 Y ] + ˆθ[ Y M [(2 + )] /2 θ ] + ˆφ[ i [(2 + )] /2 sin(θ) Y ] chi = ˆθ[ [(2 + )] /2 sin(θ) Y M ] + ˆφ[ i Y [(2 + )] /2 sin(θ) θ ] These 3 functions span the 3-D space and are an orthorgona set such that; ] A B dω = δ ABδ δ In the above A, B are any of the above functions. We aso note that if θ = π θ. and φ = π + φ then V (θ, φ ) = ( ) + V (θ, φ) W(θ, φ ) = ( ) + W(θ, φ) χ(θ, φ ) = ( ) χ(θ, φ) we aso find that; [F(r) V ] = ( )/2 [ df dr + + r 2 F]Y [F(r) W ] = ( 2 + )/2 [ df dr r [F(r) χ ] = 0 F]Y Then the condition that A = 0 reoves the projection of A onto V or W. We aso 4

5 note that L and 2 coute. L 2 L = LL 2 L i 2 = 2 L i The equation to be soved is then; 2 [R(r) χ ] = [ d2 R dr 2 + (2/r) dr dr ( + ) r 2 ] χ Because in this exape there is no azutha dependence, = 0 and we ust take the boundry conditions finite as r we expect a soution of the for; A r (+) [ i Y 0 [( + )] /2 θ ]ˆφ when 2 A = 0. these soutions can be used to construct the Green function fro which a soution for J 0 can be obtained. There are other possibe sets of vector spherica haronics. Morse and Feshbach use B, C, and P in which the anguar coponents are entirey in P. 5