Definitions B Lecture Notes Notes on Spherical Bessel Functions We would like to solve the free Schrödinger equation [ h d l(l + ) r R(r) = h k R(r). () m r dr r m R(r) is the radial wave function ψ( x) = R(r)Y m l (θ, φ). By factoring out h /m and defining = kr, we find the equation [ d l(l + ) + R() =. () d The solutions to this equation are spherical Bessel functions. Due to some reason, I don t see the integral representations I use below in books on mathemtical formulae, but I believe they are right. The behavior at the origin can be studied by power expansion. Assuming R n, and collecting terms of the lowest power in, we get There are two solutions, n(n + ) l(l + ) =. (3) n = l or l. (4) The first solution gives a positive power, and hence a regular solution at the origin, while the second a negative power, and hence a singular solution at the origin. It is easy to check that the following integral representations solve the above equation Eq. (): and l () = (/)l i e it ( t ) l dt, (5) + h () l () = (/)l i e it ( t ) l dt. (6)
By acting the derivatives in Eq. (), one finds [ d l(l + ) + d l () = (/)l = (/)l i ± i i [ l(l + ) ( t ) l + ± d dt (l + )it t l(l + ) + dt [ e it ( t ) l+ dt. (7) Therefore only boundary values contribute, which vanish both at t = and t = i for = kr >. The same holds for h () l (). One can also easily see that l () = h () l ( ) by taking the complex conjugate of the expression Eq. (5) and changing the variable from t to t. The integral representation Eq. (5) can be expanded in powers of /. For instance, for l, we change the variable from t to x by t = + ix, and find l () = (/)l ( e i(+ix) x l ( i) l x l idx i) Similarly, we find = i (/)l e i ( i) l = i ei h () l () = i e i ( i) l k (l + k)! k k!(l k)! lc k e x ( x i) k x l dx. (8) k i l k (l + k)! k k!(l k)!. (9) k Therefore both h (,) l are singular at = with power l. The combination j l () = ( l + h () l )/ is regular at =. This can be seen easily as follows. Because h () l is an integral from t = to i, while l from t = + to i, the differencd between the two corresponds to an integral from t = to t = i and coming back to t = +. Because the integrand does not have a pole, this contour can be deformed to a straight integral from t = to +. Therefore, j l () = (/) l e it ( t ) l dt. ()
In this expression, can be taken without any problems in the integral and hence j l l, i.e., regular. The other linear combination n l = ( l )/i is of course singular at =. Note that h () l is analogous to It is useful to see some examples for low l. l () = j l () + i n l () () e i = cos + i sin. () j = sin, j = sin cos, j = 3 n = cos, n = cos sin = i ei, h() = i ( h () = 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) Power Series Expansion Eq. (5) can be used to obtain the power series expansion. We first split the integration region into two parts, l () = (/)l i [ e it ( t ) l dt = (/)l i e it ( t ) l dt. + (4) The first term can be expanded in a power series by a change of variable, t = iτ/, the first term = (/)l = i l l+ = i l l+ = i l l+ e τ ( + τ e τ ) l idτ e τ (τ + ) l dτ lc n n τ l n dτ n!(l n)! n Γ(l n + ) Note that my notation for n l differs from Sakurai s by a sign as seen in Eq. (7.6.5) on page 49. I m sorry for that, but I stick with my convention, which was taken from Messiah. 3
= i l l+ (l n)! n!(l n)! n. (5) On the other hand, the second term can be expanded as the second term = (/)l e it ( t ) l dt = (/)l i n n= n! n t n ( t ) l dt = (/)l i n n t n ( t ) l dt n= n! = (/)l i n n= n! n x (n )/ ( x) l dx = (/) l i n Γ( n + )Γ(l + ) n= n! n Γ( n + l + 3). (6) At this point, it is useful to separate the sum to even n = k and odd n = k +, the second term = ( ) ( l ( = l l+ Because l j l () = l l+ ( ) k (k)! k Γ(k + ) Γ(k + l + 3 ) + ( ) k (k)! k Γ(k + ) Γ(k + l + 3 ) + () = j l () + in l (), we find n l () = l l+ ( ) k (k)! i( ) k Γ(k + ) (k + )! k+ Γ(k + l + ) ) i( ) k k! (k + )! k+ (k + l + )! ) (7) Γ(k + ) Γ(k + l + 3 )k (8) (l n)! n!(l n)! n + l l+ ( ) k k! (k + )!(k + l + )! k+. (9) The expression for j l can be simplified using the identity Γ(n + ) = (n )(n 3) 3 Γ( ) = (n )!! π, n j l () = l ( ) k (k )!! π/ k l+ (k)! (k + l + )!! k π/k+l+ 4
= l = l = l = () l ( ) k (k)! ( ) k (k)! (k + l + )(k + l ) (k + ) k (k + l)(k + l ) (k + )(k)! k (k + l + )! ( ) k l (k + l)! k!(k + l + )! k To write out the first three terms, = ( ) k (k + l)! k!(k + l + )! k. () l [ (l + )!! (l + 3) + 4 8(l + 5)(l + 3). () It suggests that the leading term is a good approximation when l /. Similarly, the expression for n l () can also be simplified, n l () = l l+ = l l+ = l l+ = l l+ = l l+ (l n)! n!(l n)! n + l l+ ( (l n)! n!(l n)! n (l n)! n!(l n)! n + (l n)! n!(l n)! n + ( ) n n! To write out the first three terms, + + ( ) k k! (k + )!(k + l + )! k+ ( ) k k! (k + )!(k + l + )! k+l+ ( ) n+l (n l )! (n l )!n! ( ) n+l (n l)! n (n l)!n! n Γ(l n + ) Γ(l n + ) n. () [ (l )!! = + l+ (l ) + 4 8(l )(l 3) +. (3) It suggests that the leading term is a good approximation when l /. ) 5
3 Asymptotic Behavior Eqs. (8,9) give the asymptotic behaviors of l for : l By taking linear combinations, we also find i ei ( i)l = i ei( lπ/). (4) j l sin( lπ/), (5) n l cos( lπ/). (6) These expressions are good approximations when l. As seen in the next section, there are better approximations when l. 4 Large l Behavior Starting from the integral from Eq. (), we use the steepest descent method to find the large l behavior. Changing the variable t = lτ, j l () = (/) l = (/) l = (/) l /l /l /l /l l l (l l ) e it ( t ) l dt e l(log( l τ )+iτ) ldτ [ exp l + l + l log l(l l ) ( τ i l l l ) + O( τ) 3 ldτ (/) l ( ) πl ll e l e l l e l(l l l ( ) ) π(l l / ) l l l = ( ) e l l l l ( ) l l /. (7) l This expression works very well as long as l and l. 6
Starting from the integral from Eq. (5), we use the steepest descent method to find the large l behavior. Change the variable t = ilτ, l () = (/)l = il (/)l i + i/l l l (l + l ) ie l ( l + l Note that there are actually two saddle points, e it ( t ) l dt [ exp l l + l log l(l + l ) ( τ l + ) l + O( τ) 3 l ) l ( ) l + l /. (8) l τ = l ± l. (9) l In the above calculation, we picked the saddle point with the negative sign with the steepest descent, while the other saddle point is what we picked for j l (). Therefore, n l () ( ) e l l + l l ( ) l + l /. (3) l This expression again works very well as long as l and l (but not too close). On the other hand, for l, the saddle points above become complex. The contribution to the l () is given by the saddle point τ = l i l l, and hence l () ( ) ei l l i l l ( ) l i l / i. (3) l This works very well as long as l and l (but not too close). j l (n l ) is given by the real (imaginary) part of l (). In practice, this form works remarkably well even for l =. 7
.. large l exact j l () l= 4 6 8 -. large Figure : Comparison of the large behavior and the large l behavior of j l () to the exact result. The large l behavior is a very good approximation for 5 > = l, while the large behavior is still a poor approximation unless > O(l ). It is interesting to note that this asymptotic behavior of l () is what you expect from the semi-classical approximation for the free-particle wave function. The classical action for a free particle is r S(r) = h l/k and hence k l r dr = h (kr) l l arctan kr l kr + l (3) e is(r)/ h = e i (kr) l l i (kr) l l, (33) kr which agrees with the large l behavior above except for the last factor which comes from the lowest-order quantum correction. When l, two saddle points collide and I don t know what to do. 5 Recursion Formulae Starting from Eq. (5), we take the derivative d d h() l = l h() l (/)l i e it it( t ) l dt. (34) + The second term can be integrated by parts, and gives = l h() l + (/) l i e it ( t ) l+ dt = l + h() l l+. (35) 8
In fact, other functions j l, n l, and h () l all satisfy the same relation which can be easily checked. Referring to all of them generically as z l (), we find the recursion formula z l = l z l z l+. (36) Because z l () satisfies the differential equation Eq., we can combine it with the above recursion relation and find ( d = d + ) d l(l + ) + z d l = z l l + 3 z l+ + z l+. (37) Relabeling l to l, we obtain z l + z l+ = l + z l. (38) Finally, combining the two recursion relations, we also obtain 6 Plane Wave Expansion The non-trivial looking formula we used in the class z l = z l l + z l. (39) e ikz = (l + )i l j l (kr)p l (cos θ) (4) l= can be obtained quite easily from the integral representation Eq. (). The point is that one can keep integrating it in parts. By integrating e it factor and differentiating ( t ) l factor, the boundary terms at t = ± always vanish up to l-th time because of the ( t ) l factor. Therefore, j l = (/) l ( (i) l eit dt) d l ( t ) l dt. (4) Note that the definition of the Legendre polynomials is P l (t) = d l l dt l (t ) l. (4) 9
Using this definition, the spherical Bessel function can be written as j l = e it P i l l (t)dt. (43) Then we use the fact that the Legendre polynomials form a complete set of orthogonal polynomials in the interval t [,. Noting the normalization P n (t)p m (t)dt = n + δ n,m, (44) the orthonormal basis is P n (t) (n + )/, and hence l= n= n + P n (t)p n (t ) = δ(t t ). (45) By multipyling Eq. (43) by P l (t )(l + )/ and summing over n, i l l + P l (t )j l () = e it l + P l (t )P l (t)dt = e it. (46) n= By setting = kr and t = cos θ, we prove Eq. (4). If the wave vector is pointing at other directions than the positive z- axis, the formula Eq. (4) needs to be generalized. Noting Yl (θ, φ) = (l + )/4π P l (cos θ), we find e i k x = 4π i l j l (kr) l= m= l 7 Delta-Function Normalization Yl m (θ k, φ k )Yl m (θ x, φ x ) (47) An important consequence of the identity Eq. (47) is the innerproduct of two spherical Bessel functions. We start with d xe i k x e i k x = (π) 3 δ( k k ). (48) Using Eq. (47) in the l.h.s of this equation, we find d xe i k x e i k x = (4π) l,m l,m = (4π) l,m dω x drr Y m l (Ω k )Y m (Ω x )Y m l (Ω x)y l m (Ω k )j l (kr)j l (k r) l drr j l (kr)j l (k r)yl m (Ω k )Yl m (Ω k ). (49)
On the other hand, the r.h.s. of Eq. (48) is (π) 3 δ( k k ) = (π) 3 k δ(k k )δ(ω k Ω k ) Comparing Eq. (49) and (5) and noting l,m = (π) 3 k sin θ δ(k k )δ(θ θ )δ(φ φ ). (5) Yl m (Ω k )Yl m (Ω k ) = δ(ω k Ω k ), (5) we find drr j l (kr)j l (k r) = π k δ(k k ). (5) 8 Mathematica In Mathematica, spherical Bessel functions are not defined but the usual Bessel functions are. The j l (z) is obtained by π z BesselJ[l +, z and n l (z) by π z BesselY[l +, z. You may actually want to use etc to get rid of half-odd powers. PowerExpand[ π z BesselJ[l +, z