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Consider the emission and scattering of electromagnetic
radiation. This type of problem involves solving the
vector wave equation. The solutions of this equation in free space
are conveniently written as an expansion in
orthogonal spherical waves.
This expansion is known as the multipole expansion. Let us
examine this expansion in more detail.
Before considering the vector wave equation, let us
consider the somewhat simpler
scalar wave equation. A scalar field
satisfying the homogeneous wave equation

(1159) 
can be Fourier analyzed in time

(1160) 
with each Fourier harmonic satisfying the Helmholtz wave equation

(1161) 
where
. We can write the Helmholtz equation in
terms of spherical polar coordinates (, , ):

(1162) 
As is well known, it is possible to solve this equation via the
separation of variables:

(1163) 
Here, we restrict our attention to physical
solutions which are well behaved in the angular variables
and .
The spherical harmonics
satisfy
the following equations:
where is a nonnegative integer, and is an integer which
satisfies the inequality . The radial functions
satisfy

(1166) 
where there is no dependence on .
With the substitution

(1167) 
Eq. (7.7) is transformed into

(1168) 
It can be seen, by comparison with Eq. (5.39), that this is a type of
Bessel's equation of halfinteger order . Thus, we can write
the solution for as

(1169) 
where and are arbitrary constants.
The halfinteger order Bessel functions and
have analogous properties to the integer order
Bessel functions and . In particular,
the are well behaved in the limit
,
whereas the are badly behaved. The asymptotic
expansions (5.43) remain valid when
.
It is convenient to define the spherical Bessel functions
and , where
It is also convenient to define the spherical Hankel functions

(1172) 
For real , is the complex conjugate of
.
It turns out that
the spherical Bessel functions can be expressed
in the closed form
In the limit of small argument
where
.
In the limit of large argument
and

(1179) 
The inhomogeneous Helmholtz equation is conveniently solved using
the Green's function
, which satisfies
(see Eq. (2.109))

(1180) 
The solution of this equation, subject to the Sommerfeld radiation
condition, which ensures that sources radiate waves instead of absorbing
them, is written (see Section 2.13)

(1181) 
The spherical harmonics satisfy the completeness relation

(1182) 
Now the three dimensional delta function can be written

(1183) 
It follows that

(1184) 
Let us expand the Green's function in the form

(1185) 
Substitution of this expression into Eq. (7.17) yields

(1186) 
The appropriate boundary conditions are that is finite at the
origin and corresponds to an outgoing wave at infinity
(i.e.,
in the limit
). The solution of the above equation which
satisfies these boundary conditions is

(1187) 
where and are the greater and the lesser of
and , respectively. The correct discontinuity
in slope at is assured if
,
since

(1188) 
Thus, the expansion of the Green's function is

(1189) 
This is a particularly useful result,
as we shall discover, since it easily allows
us to express the general
solution of the inhomogeneous wave equation as a multipole expansion.
It is well known in quantum mechanics that Eq. (7.6b) can be written
in the form

(1190) 
The differential operator is given by

(1191) 
where

(1192) 
is times the orbital angular momentum operator of
wave mechanics.
The components of can be conveniently written in the combinations
We note that operates only on angular variables and is independent
of . From the definition (7.29) it is evident
that

(1196) 
holds as an operator equation. It is easily
demonstrated from Eqs. (7.30) that

(1197) 
The following results are well known in quantum mechanics:
In addition,
where

(1204) 
Next: Multipole expansion of the
Up: The multipole expansion
Previous: The multipole expansion
Richard Fitzpatrick
20020518