- D.14.1 Derivation from the eigenvalue problem
- D.14.2 Parity
- D.14.3 Solutions of the Laplace equation
- D.14.4 Orthogonal integrals
- D.14.5 Another way to find the spherical harmonics
- D.14.6 Still another way to find them

D.14 The spherical harmonics

This note derives and lists properties of the spherical harmonics.

D.14.1 Derivation from the eigenvalue problem

This analysis will derive the spherical harmonics from the eigenvalue problem of square angular momentum of chapter 4.2.3. It will use similar techniques as for the harmonic oscillator solution, {D.12}.

The imposed additional requirement that the spherical harmonics

It is convenient define a scaled square angular momentum by

More importantly, recognize that the solutions will likely be in terms
of cosines and sines of

As you may guess from looking at this ODE, the solutions

If you substitute

Plug in a power series,

Using similar arguments as for the harmonic oscillator, you see that the starting power will be zero or one, leading to basic solutions that are again odd or even. And just like for the harmonic oscillator, you must again have that the power series terminates; even in the least case that

To get the series to terminate at some final power

The rest is just a matter of table books, because with

To normalize the eigenfunctions on the surface area of the unit
sphere, find the corresponding integral in a table book, like
[40, 28.63]. As mentioned at the start of this long and
still very condensed story, to include negative values of ladder operators.

That requires,
{D.64}, that starting from ladder-up operator,

and those for ladder-down operator.

Physicists
will still allow you to select your own sign for the

where

D.14.2 Parity

One special property of the spherical harmonics is often of interest:
their “parity.” The parity of a wave function is 1, or even, if the
wave function stays the same if you replace

To see why, note that replacing

D.14.3 Solutions of the Laplace equation

The Laplace equation

is

Solutions

This is a complete set of solutions for the Laplace equation inside a sphere. Any solution

As you can see in table 4.3, each solution above is a power series in terms of Cartesian coordinates.

For the Laplace equation outside a sphere, replace

To check that these are indeed solutions of the Laplace equation, plug them in, using the Laplacian in spherical coordinates given in (N.5). Note here that the angular derivatives can be simplified using the eigenvalue problem of square angular momentum, chapter 4.2.3.

D.14.4 Orthogonal integrals

The spherical harmonics are orthonormal on the unit sphere:

Further

See the notations for more on spherical coordinates and

To verify the above expression, integrate the first term in the
integral by parts with respect to

and then apply the eigenvalue problem of chapter 4.2.3.

D.14.5 Another way to find the spherical harmonics

There is a more intuitive way to derive the spherical harmonics: they
define the power series solutions to the Laplace equation. In
particular, each

where the coefficients

Even more specifically, the spherical harmonics are of the form

where the coordinates

To get from those power series solutions back to the equation for the spherical harmonics, one has to do an inverse separation of variables argument for the solution of the Laplace equation in a sphere in spherical coordinates (compare also the derivation of the hydrogen atom.) Also, one would have to accept on faith that the solution of the Laplace equation is just a power series, as it is in 2D, with no additional nonpower terms, to settle completeness. In other words, you must assume that the solution is analytic.

D.14.6 Still another way to find them

The simplest way of getting the spherical harmonics is probably the one given later in derivation {D.64}.