The angular part of the Laplace operator can be written: 1r^2 { 1 &thetas#theta;&thetas#theta;(&thetas#theta;&thetas#theta;) + 1^2&thetas#theta;&phis#phi; } = -L^2r^2

Eliminating (to solve for the differential equation) one needs to solve an eigenvalue problem: L^2 &psi#psi;= e &psi#psi; 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:
L^2 = &ell#ell;(&ell#ell;+1)
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:

(14.1) |

and L^2 = L_x^2 + L_y^2 + L_z^2

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:

(14.2) |

and we see that in fact satisfies the

(14.3) |

and

(14.4) |

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:

(14.5) | |||

(14.6) | |||

(14.7) |

The Cartesian components of do not commute. In fact, they form a nice antisymmetric set:

(14.8) |

which can be written in the shorthand notation

(14.9) |

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

(14.10) |

and

(14.11) |

but

(14.12) |

and therefore with the Laplacian itself:

(14.13) |

which can be written in terms of as:

(14.14) |

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:

(14.15) | |||

(14.16) | |||

(14.17) |

Finally, note that **L** is always orthogonal to **r** where both are
considered as *operators* and **r** acts from the left:

(14.18) |

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.