Monday Math 52

Last week, we looked at solving Laplace’s equation, in three dimensions, specifically Cartesian and cylindrical coordinates. Now, let us consider spherical coordinates (ρ,φ,θ) (we use the physicist’s convention of polar angle θ and azimuthal angle φ).

In spherical coordinates, Laplace’s equation is

Trying a solution of the form f(ρ,φ,θ)=R(ρ)Φ(φ)Θ(θ), we have:

The first bracket depends only on ρ, while the second depends only on the angular coordinates. We choose a constant to which the first bracket equals:


The angular portion is then:

and we have separated φ and θ terms. Expecting solutions that are continuous and periodic in 0≤φ<2π, we set


where m is an integer.
This gives us the remaining differential equation

.

To examine the solutions to this last equation, we first consider the m=0 case. Here, we have:
.
Let us make the change of variables . Then we have


This is amenable to solution by the method of Frobenius. One finds that the resulting power series converges for all -1<x<1 (0<θ<π), but that it diverges at x=±1 (blows up at the polar axis) unless the series terminates. As the recursive formula for the coefficients of the power series is , the polynomial terminates only if k=l(l+1), where l is a non-negative integer. Then the solution is a polynomial with degree l, which is an even polynomial for even l and an odd polynomial for odd l. The initial coefficient, a0 or a1, is arbitrary, and is usually chosen so that . These polynomials are known as Legendre polynomials.
They can be generated via Rodrigues’ formula:
.
The first few polynomials are:




.

Now, for m≠0, we have, using k=l(l+1) (here l is not necessarily an integer),

,
called the “general Legendre equation.” It has solutions called the associated Legendre functions, written , where l is called the degree and m the order of the function. We already have that
, the legendre polynomial of degree l, when l is a non-negative integer.
It can be shown that the associated Legendre functions are well behaved on -1≤x≤1 for l and m integers with |m|≤l, l≥0, with the solutions for non-negative m m in terms of Legendre polynomials being

,
and the negative m solutions given by
.
(I have used here the Condon-Shortley phase.)
So our solutions will generally have a polar angle component
.
If we combine these with the azimuthal angular component written in terms of complex exponentials, we then have:

where l and m are integers with l≥0 and |m|≤l. Choosing our constant to produce the othonormal relation:

(where the integration is over all solid angles, * indicates the complex conjugate, and is the Kronecker Delta),
we get
.
These are called spherical harmonics.
The first few are:







Now, for our radial component, we have ; with k=l(l+1), we get

Trying a solution of the form , we find



or
So our radial component is of the form , and our solutions to Laplace’s equation in spherical coordinates will be of the form
.

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4 Responses to “Monday Math 52”

  1. Monday Math 53 « Twisted One 151’s Weblog Says:

    […] Math 53 By twistedone151 Previously, we introduced the spherical harmonics. We note that as written, the spherical harmonic is real […]

  2. GreenTeaTropical Says:

    thanks for the info !

  3. Physics Friday 59 « Twisted One 151’s Weblog Says:

    […] separation of variables, the angular components will be given by the spherical harmonics (see here and here). We will then have . With this in place, our Schrödinger equation becomes, for the […]

  4. Physics Friday 60 « Twisted One 151’s Weblog Says:

    […] variables in spherical coordinates, with the angular components being the spherical harmonics (see here and here). As in here, when we perform the spherical coordinate separation , we obtain radial […]

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