Thermodynamics, Statistical Physics, and Quantum Mechanics

the interaction potential acts only between the electrons, it is natural to

write the orbital part in center-of-mass coordinates, where

and

The problem stated that the totalmomentum was zero, so set We

must now determine the form for the relative eigenfunction It obeys

the Schrödinger equation with the reduced mass where is the

electron mass:

We have reduced the problem to solving the bound state of a “particle” in

a box. Here the “particle” is the relative motion of two electrons. However,

since the orbital part of the wave function must have odd parity, we need

to find the lowest energy state which is antisymmetric,

Bound states have where the bindingenergy Define

two wave vectors: for outside the box, and

when the particle is in the box, The lowest

antisymmetric wave function is

We match the wave function and its derivative at one edge, say

which gives two equations:

We divide these two equations, which eliminates the constants A and B.

The remaining equation is the eigenvalue equation for

Since and are both positive, the cotangent of must be negative,
which requires that This imposes a constraint for the existence

270 SOLUTIONS