Spherical Bessel functions

A.6 Spherical Bessel functions

Spherical bessel functions are solutions of the following equation:

z^{2} \frac{d^{2} f}{d z^{2}} + 2 z \frac{d f}{d z} + (z^{2} - l (l + 1)) f = 0

for integer $l$ .

The regular solution is denoted $j_{l} (z)$ and the irregular one, $n_{l} (z)$ (or sometimes $y_{l} (z)$ ). The Mathematica functions for obtaining them are SphericalBesselJ[l, z] and SphericalBesselY[l, z]. For $l = 0$ the equation is $\frac{d^{2} g}{d z^{2}} + g = 0$ , where $g = z f$ , and so the solutions are $j_{0} = sin z ∕ z$ and $n_{0} = - cos z ∕ z$ . The general solutions are

j_{l} (z) = z^{l} {(- \frac{1}{z} \frac{d}{d z})}^{l} (\frac{sin z}{z}) and n_{l} (z) = - z^{l} {(- \frac{1}{z} \frac{d}{d z})}^{l} (\frac{cos z}{z}) .

The asymptotic forms are

\begin{array}{rcl} j_{l} (z) \overset{z \to \infty}{\to} \frac{sin (z - l π ∕ 2)}{z} & and & n_{l} (z) \overset{z \to \infty}{\to} - \frac{cos (z - l π ∕ 2)}{z}; \\ j_{l} (z) \overset{z \to 0}{\to} \frac{z^{l}}{(2 l + 1)!!} & and & n_{l} (z) \overset{z \to 0}{\to} - (2 l - 1)!! z^{- l - 1} . \end{array}

(Note “n!!” is like factorial but only including the odd (even) numbers for odd (even) $n$ , eg $7!! = 7 \times 5 \times 3 \times 1$ and $6!! = 6 \times 4 \times 2$ , with $0!! = 0! \equiv 1$ .)

In spherical polar coordinates the Schrödinger equation for a particle in free space ( $V (r) = 0$ ) gives the following equation for the radial wavefuntion:

\frac{d^{2} R_{l}}{d r^{2}} + \frac{2}{r} \frac{d R_{l}}{d r} + (k^{2} - \frac{l (l + 1)}{r^{2}}) R_{l} = 0

where $k^{2} = 2 m E ∕ ℏ^{2}$ . So the solution is

R_{l} (r) = A j_{l} (k r) + B n_{l} (k r)

where $B$ will equal zero if the solution has to hold at the origin, but not if the origin is excluded (for instance outside a hard sphere).

Poisson’s integral representation of the regular spherical Bessel functions

j_{n} (z) = \frac{z^{n}}{2^{n + 1} n!} \int_{- 1}^{1} cos (z x) {(x^{2} - 1)}^{n} d x

together with Rodrigues representation of the Legendre polynomials can be used to show that

j_{n} (z) = \frac{1}{2} {(- i)}^{n} \int_{- 1}^{1} e^{i z x} P_{n} (x) d x

whence follows the expression for the expansion of a plane wave in spherical polars given in section 5.3.

[next] [prev] [up] [top]