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The angular part of the Laplace operator can be written:

(12.1) 
Eliminating (to solve for the differential equation) one
needs to solve an eigenvalue problem:

(12.2) 
where are the eigenvalues, subject to the condition that the
solution be single valued on
and
.
This equation easily separates in . The equation is
trivial  solutions periodic in are indexed with integer .
The equation one has to work at a bit  there are constraints
on the solutions that can be obtained for any given  but there are
many ways to solve it and at this point you should know that its
solutions are associated Legendre polynomials
where
. Thus the eigensolution becomes:

(12.3) 
where
and
and is
typically orthonormal(ized) on the solid angle .
The angular part of the Laplacian is related to the angular
momentum of a wave in quantum theory. In units where , the
angular momentum operator is:

(12.4) 
and

(12.5) 
Note that in all of these expressions
, etc. are all
operators. This means that they are applied to the
functions on their right (by convention). When you see them appearing
by themselves, remember that they only mean something when they are
applied, so
's out by themselves on the right are ok.
The component of is:

(12.6) 
and we see that in fact satisfies the two eigenvalue
equations:

(12.7) 
and

(12.8) 
The 's cannot be eigensolutions of more than one of the components
of
at once. However, we can write the cartesian components of
L so that they form an first rank tensor algebra of
operators that transform the , for a given , among
themselves (they cannot change , only mix ). This is the
hopefully familiar set of equations:
The Cartesian components of
do not commute. In fact, they form a
nice antisymmetric set:

(12.12) 
which can be written in the shorthand notation

(12.13) 
Consequently, the components expressed as a first rank tensor also do
not commute among themselves:

(12.14) 
and

(12.15) 
but all these ways of arranging the components of
commute
with :

(12.16) 
and therefore with the Laplacian itself:

(12.17) 
which can be written in terms of as:

(12.18) 
As one can easily show either by considering the explict action of the
actual differential forms on the actual eigensolutions or more
subtly by considering the action of on
(and
showing that they behave like raising and lower operators for and
preserving normalization) one obtains:
Finally, note that L is always orthogonal to r where both are
considered as operators and r acts from the left:

(12.22) 
You will see many cases where identities such as this have to be written down
in a particular order.
Before we go on to do a more leisurely tour of vector spherical
harmonics, we pause to motivate the construction.
Next: Magnetic and Electric Multipoles
Up: Vector Multipoles
Previous: Vector Multipoles
Contents
Robert G. Brown
20071228