Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES

Helmholtz’s equation in spherical polars is thus

F(r, θ, φ)=r−^1 /^2 [AJ+1/ 2 (kr)+BY+1/ 2 (kr)](Ccosmφ+Dsinmφ)

×[EPm(cosθ)+FQm(cosθ)]. (21.57)

For solutions that are finite at the origin we requireB= 0, and for solutions

that are finite on the polar axis we requireF= 0. It is worth mentioning that

the solutions proportional tor−^1 /^2 J+1/ 2 (kr)andr−^1 /^2 Y+1/ 2 (kr), when suitably

normalised, are calledspherical Bessel functionsof the first and second kind,

respectively, and are denoted byj(kr)andn(μ) (see section 18.6).

As mentioned at the beginning of this subsection, the separated solution of

the wave equation in spherical polars is the product of a time-dependent part

(21.54) and a spatial part (21.57). It will be noticed that, although this solution

corresponds to a solution of definite frequencyω=kc, the zeros of the radial

functionj(kr) are not equally spaced inr, except for the case= 0 involving

j 0 (kr), and so there is no precise wavelength associated with the solution.

To conclude this subsection, let us mention briefly the Schrodinger equation ̈

for the electron in a hydrogen atom, the nucleus of which is taken at the origin

and is assumed massive compared with the electron. Under these circumstances

the Schr ̈odinger equation is

−

^2

2 m

∇^2 u−

e^2

4 π 0

u

r

=i

∂u

∂t

.

For a ‘stationary-state’ solution, for which the energy is a constantEand the time-

dependent factorTinuis given byT(t)=Aexp(−iEt/
), the above equation is

similar to, but not quite the same as, the Helmholtz equation.§However, as with

the wave equation, the angular parts of the solution are identical to those for

Laplace’s equation and are expressed in terms of spherical harmonics.

The important point to note is that foranyequation involving∇^2 , providedθ

andφdo not appear in the equation other than as part of∇^2 , a separated-variable

solution in spherical polars will always lead to spherical harmonic solutions. This

is the case for the Schrodinger equation describing an atomic electron in a central ̈

potentialV(r).

21.3.3 Solution by expansion

It is sometimes possible to use the uniqueness theorem discussed in the previous

chapter, together with the results of the last few subsections, in which Laplace’s

equation (and other equations) were considered in polar coordinates, to obtain

solutions of such equations appropriate to particular physical situations.

§For the solution by series of ther-equation in this case the reader may consult, for example, L.

Schiff,Quantum Mechanics(New York: McGraw-Hill, 1955), p. 82.