Mathematical Methods in the Physical Sciences · Problem 7.17

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · schrodinger-equation-separation

Problem

The (time-independent) Schrödinger equation in quantum mechanics is

\nabla^2\psi + (\epsilon - bV)\psi = 0,

where

\epsilon

and

b

are constants and

V

is a given function of

r

\theta

\phi

for each problem. In most simple cases

V

is a function of

r

only (no

\theta

\phi

dependence). (Physically,

V

is the potential energy, and the fact that it depends only on

r

implies that we are dealing with central forces, for example, electrostatic or gravitational forces.) Separate the Schrödinger equation in spherical coordinates for the case

V = V(r)

, and show that the

\theta

\phi

solutions are spherical harmonics (see Problem 16).

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