Thermodynamics, Statistical Physics, and Quantum Mechanics

QUANTUM MECHANICS 301

One interaction potential is One can use parity and other

group theory arguments to show that the only nonzero matrix elements are

One can show that and are equal to within a phase factor. We ig-

nore this phase factor and call them equal. The evaluation of this integral

was demonstrated in the previous solution. The result here is

compared to the one in the previous problem.

To first order in the magnetic field, the interaction is given by

In spherical coordinates the three unit vectors for direction are

In these units the vector potential can be written as Similarly,

the momentum operator in this direction is

where the cyclotron frequency is The magnetic field is a

diagonal perturbation in the basis.

Now the state has no matrix elements for these interactions and is

unchanged by these interactions to lowest order. So we must diagonalize

the 3 × 3 interaction matrix for thethreestates

The states are initially fourfold degenerate. The double perturbation
leaves two states with the same eigenvalue while the other two are
shifted by where and Note that
so that, in the absence of the magnetic field, the result is the same as in
Problem 5.48.