221B Lecture Notes Notes on Spherical Bessel Functions

Transcription

1 Definitions B Lecture Notes Notes on Spherica Besse Functions We woud ike to sove the free Schrödinger equation [ h d r R(r) = h k R(r). () m r dr r m R(r) is the radia wave function ψ( x) = R(r)Y m (θ, φ). By factoring out h /m and defining = kr, we find the equation [ d + R() = 0. () d The soutions to this equation are spherica Besse functions. Due to some reason, I don t see the integra representations I use beow in books on mathemtica formuae, but I beieve they are right. The behavior at the origin can be studied by power expansion. Assuming R n, and coecting terms of the owest power in, we get There are two soutions, n(n + ) = 0. (3) n = or. (4) The first soution gives a positive power, and hence a reguar soution at the origin, whie the second a negative power, and hence a singuar soution at the origin. It is easy to check that the foowing integra representations sove the above equation Eq. (): and () = (/) i e it ( t ) dt, (5)! + h () () = (/) i e it ( t ) dt. (6)!

2 By acting the derivatives in Eq. (), one finds [ d + d () = (/)! = (/)! i ± i i [ ( t ) + ± d dt ( + )it t + dt [ e it ( t ) + dt. (7) Therefore ony boundary vaues contribute, which vanish both at t = and t = i for = kr > 0. The same hods for h () (). One can aso easiy see that () = h () ( ) by taking the compex conjugate of the expression Eq. (5) and changing the variabe from t to t. The integra representation Eq. (5) can be expanded in powers of /. For instance, for, we change the variabe from t to x by t = + ix, and find () = (/) ( e i(+ix) x ( i) x idx! 0 i) Simiary, we find = i (/) e i ( i)! = i ei h () () = i e i ( i) k ( + k)! k k!( k)! C k 0 e x ( x i) k x dx. (8) k i k ( + k)! k k!( k)!. (9) k Therefore both h (,) are singuar at = 0 with power. The combination j () = ( + h () )/ is reguar at = 0. This can be seen easiy as foows. Because h () is an integra from t = to i, whie from t = + to i, the differencd between the two corresponds to an integra from t = to t = i and coming back to t = +. Because the integrand does not have a poe, this contour can be deformed to a straight integra from t = to +. Therefore, j () = (/) e it ( t ) dt. (0)!

3 In this expression, 0 can be taken without any probems in the integra and hence j, i.e., reguar. The other inear combination n = ( )/i is of course singuar at = 0. Note that h () is anaogous to It is usefu to see some exampes for ow. () = j () + i n () () e i = cos + i sin. () j 0 = sin, j = sin cos, j = 3 n 0 = cos, n = cos sin 0 = i ei, h() = i ( h () 0 = i e i, h() = i ( + i i sin 3 cos, 3, n = 3 cos 3 sin, ) 3 e i = i ( ) 3 3i 3 e i. ) e i h () = i ( ) 3 + 3i 3 e i. (3) Asymptotic Behavior Eqs. (8,9) give the asymptotic behaviors of for : By taking inear combinations, we aso find 3 Pane Wave Expansion i ei ( i) = i ei( π/). (4) j sin( π/), (5) n cos( π/). (6) The non-trivia ooking formua we used in the cass e ikz = ( + )i j (kr)p (cos θ) (7) =0 can be obtained quite easiy from the integra representation Eq. (0). The point is that one can keep integrating it in parts. By integrating e it factor 3

4 and differentiating ( t ) factor, the boundary terms at t = ± aways vanish up to -th time because of the ( t ) factor. Therefore, j = (/) (! (i) eit dt) d ( t ) dt. (8) Note that the definition of the Legendre poynomias is P (t) = d! dt (t ). (9) Using this definition, the spherica Besse function can be written as j = i e it P (t)dt. (0) Then we use the fact that the Legendre poynomias form a compete set of orthogona poynomias in the interva t [,. Noting the normaization P n (t)p m (t)dt = n + δ n,m, () the orthonorma basis is P n (t) (n + )/, and hence n=0 n + P n (t)p n (t ) = δ(t t ). () By mutipying Eq. (0) by P (t )( + )/ and summing over n, n= + P (t )j n () = i n e it P (t )P (t)dt = n=0 i n eit. (3) By setting = kr and t = cos θ, we prove Eq. (7). If the wave vector is pointing at other directions than the positive z- axis, the formua Eq. (7) needs to be generaized. Noting Y 0 (θ, φ) = ( + )/4π P (cos θ), we find e i k x = 4π i j (kr) =0 m= Y m (θ k, φ k )Y m (θ x, φ x ) (4) 4

5 4 Deta-Function Normaization An important consequence of the identity Eq. (4) is the innerproduct of two spherica Besse functions. We start with d xe i k x e i k x = (π) 3 δ( k k ). (5) Using Eq. (4) in the.h.s of this equation, we find d xe i k x e i k x = (4π),m,m = (4π),m dω x drr Y m (Ω k )Y m (Ω x )Y m (Ω x)y m (Ω k )j (kr)j (k r) drr j (kr)j (k r)y m (Ω k )Y m (Ω k ). (6) On the other hand, the r.h.s. of Eq. (5) is (π) 3 δ( k k ) = (π) 3 k δ(k k )δ(ω k Ω k ) Comparing Eq. (6) and (7) and noting,m = (π) 3 k sin θ δ(k k )δ(θ θ )δ(φ φ ). (7) Y m (Ω k )Y m (Ω k ) = δ(ω k Ω k ), (8) we find 0 drr j (kr)j (k r) = π k δ(k k ). (9) 5