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Next: Angular Momentum Operators Up: Multipole Expansion Previous: Introduction

Multipole Expansion of Scalar Wave Equation

Before considering the vector wave equation, let us consider the somewhat simpler scalar wave equation. A scalar field $ \psi({\bf r},t)$ satisfying the homogeneous wave equation,

$\displaystyle \nabla^{\,2}\psi - \frac{1}{c^{\,2}}\frac{\partial^{\,2}\psi}{\partial t^{\,2}}=0,$ (1409)

can be Fourier analyzed in time,

$\displaystyle \psi({\bf r}, t) = \int_{-\infty}^{\infty} \psi({\bf r}, \omega)\, {\rm e}^{-{\rm i}\,\omega\, t} \,d\omega,$ (1410)

with each Fourier harmonic satisfying the homogeneous Helmholtz wave equation,

$\displaystyle (\nabla^{\,2} + k^{\,2})\,\psi({\bf r}, \omega) = 0,$ (1411)

where $ k^{\,2}=\omega^{\,2}/c^{\,2}$ . We can write the Helmholtz equation in terms of spherical coordinates $ r$ , $ \theta$ , $ \varphi$ :

$\displaystyle \left(\frac{1}{r^{\,2}}\frac{\partial}{\partial r}\,r^{\,2} \frac...
...^2\theta}\frac{\partial^{\,2} }{\partial \varphi^{\,2}}+k^{\,2}\right)\psi = 0.$ (1412)

As is well known, it is possible to solve this equation via separation of variables to give

$\displaystyle \psi({\bf r}, \omega) = \sum_{l=0,\infty}\sum_{m=-l,+l} f_{lm}(r)\, Y_{lm}(\theta,\varphi).$ (1413)

Here, we restrict our attention to physical solutions that are well-behaved in the angular variables $ \theta$ and $ \varphi$ . The spherical harmonics $ Y_{lm}(\theta,\varphi)$ satisfy the following equations:

$\displaystyle -\frac{\partial^{\,2} Y_{lm}}{\partial\varphi^{\,2}}$ $\displaystyle = m^{\,2}\, Y_{lm},$ (1414)
$\displaystyle -\left(\frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \,\sin...
...rac{1}{\sin^2\theta}\frac{\partial^{\,2} }{\partial \varphi^{\,2}}\right)Y_{lm}$ $\displaystyle = l\,(l+1)\,Y_{lm},$ (1415)

where $ l$ is a non-negative integer, and $ m$ is an integer that satisfies the inequality $ \vert m\vert\leq l$ . The radial functions $ f_{lm}(r)$ satisfy

$\displaystyle \left[\frac{d^{\,2}}{dr^{\,2}} + \frac{2}{r}\frac{d}{dr} + k^{\,2} - \frac{l\,(l+1)}{r^{\,2}} \right]f_{lm}(r) =0.$ (1416)

With the substitution

$\displaystyle f_{lm}(r) = \frac{u_{lm}(r)}{r^{\,1/2}},$ (1417)

Equation (1418) is transformed into

$\displaystyle \left[\frac{d^{\,2}}{dr^{\,2}} + \frac{1}{r}\frac{d}{dr} +k^{\,2} - \frac{(l+1/2)^{\,2}}{r^{\,2}}\right] u_{lm}(r) =0,$ (1418)

which is a type of Bessel equation of half-integer order, $ l+1/2$ . Thus, we can write the solution for $ f_{lm}(r)$ as

$\displaystyle f_{lm}(r) = \frac{A_{lm}}{r^{\,1/2}}\,J_{l+1/2}(k\,r) + \frac{B_{lm}}{r^{\,1/2}} \,Y_{l+1/2}(k\,r),$ (1419)

where $ A_{lm}$ and $ B_{lm}$ are arbitrary constants. The half-integer order Bessel functions $ J_{l+1/2}(z)$ and $ Y_{l+1/2}(z)$ have analogous properties to the integer order Bessel functions $ J_m(z)$ and $ Y_m(z)$ . In particular, the $ J_{l+1/2}(z)$ are well behaved in the limit $ \vert z\vert\rightarrow 0$ , whereas the $ Y_{l+1/2}(z)$ are badly behaved.

It is convenient to define the spherical Bessel functions, $ j_l(r)$ and $ y_l(r)$ , where

$\displaystyle j_l(z)$ $\displaystyle = \left(\frac{\pi}{2 \,z}\right)^{1/2} J_{l+1/2}(z),$ (1420)
$\displaystyle y_l(z)$ $\displaystyle = \left(\frac{\pi}{2 \,z}\right)^{1/2} Y_{l+1/2}(z).$ (1421)

It is also convenient to define the spherical Hankel functions, $ h_l^{(1)}(r)$ and $ h_l^{(2)}(r)$ , where

$\displaystyle h_l^{(1)} (z)$ $\displaystyle = j_l(z)+ {\rm i}\,y_l(z),$ (1422)
$\displaystyle h_l^{(2)} (z)$ $\displaystyle = j_l(z)- {\rm i}\,y_l(z).$ (1423)

Assuming that $ z$ is real, $ h^{(2)}_l(z)$ is the complex conjugate of $ h^{(1)}_l(z)$ . It turns out that the spherical Bessel functions can be expressed in the closed form

$\displaystyle j_l(z)$ $\displaystyle = (-z)^{\,l} \left(\frac{1}{z}\frac{d}{dz}\right)^l \left(\frac{\sin z}{z}\right),$ (1424)
$\displaystyle y_l(z)$ $\displaystyle = -(-z)^{\,l} \left(\frac{1}{z}\frac{d}{dz}\right)^l \left(\frac{\cos z} {z}\right).$ (1425)

In the limit of small argument,

$\displaystyle j_l(z)$ $\displaystyle \rightarrow \frac{z^{\,l}}{(2\,l+1)!!}\left[1+ {\cal O}(z^2)\right],$ (1426)
$\displaystyle y_l(z)$ $\displaystyle \rightarrow -\frac{(2\,l-1)!!}{z^{\,l+1}}\left[1+ {\cal O}(z^2)\right],$ (1427)

where $ (2l+1)!! = (2l+1)\,(2l-1)\, (2l-3)\cdots 5\cdot 3\cdot 1$ . In the limit of large argument,

$\displaystyle j_l(z)$ $\displaystyle \rightarrow \frac{\sin(z-l\,\pi/2)}{z},$ (1428)
$\displaystyle y_l(z)$ $\displaystyle \rightarrow -\frac{\cos(z-l\,\pi/2)}{z},$ (1429)

which implies that

$\displaystyle h_l^{(1)} (z)$ $\displaystyle \rightarrow (-{\rm i})^{\,l+1}\, \frac{ {\rm e}^{+{\rm i}\,z} }{z},$ (1430)
$\displaystyle h_l^{(2)} (z)$ $\displaystyle \rightarrow (+{\rm i})^{\,l+1}\, \frac{ {\rm e}^{-{\rm i}\,z} }{z}.$ (1431)

It follows, from the above discussion, that the radial functions $ f_{lm}(r)$ , specified in Equation (1421), can also be written

$\displaystyle f_{lm}(r) = A_{lm}\,h_l^{(1)}(k\,r) + B_{lm}\,h_l^{(2)}(k\,r).$ (1432)

Hence, the general solution of the homogeneous Helmholtz equation, (1413), takes the form

$\displaystyle \psi({\bf r},\omega) = \sum_{l=0,\infty}\sum_{m=-l,+l}\left[A_{lm}\,h_l^{(1)}(k\,r) + B_{lm}\,h_l^{(2)}(k\,r)\right]Y_{lm}(\theta,\varphi).$ (1433)

Moreover, it is clear from Equations (1412) and (1432)-(1433) that, at large $ r$ , the terms involving the $ h_l^{(1)}(k\,r)$ Hankel functions correspond to outgoing radial waves, whereas those involving the $ h_l^{(2)}(k\,r)$ functions correspond to incoming radial waves.

next up previous
Next: Angular Momentum Operators Up: Multipole Expansion Previous: Introduction
Richard Fitzpatrick 2014-06-27