A Guide to Physics Problems. Part 2. Thermodynamics, Statistical Physics, and Quantum Mechanics

5.45 Spin Interaction (Princeton)

In first-order perturbation theory the change in energy is

since and the matrix element of is zero for the ground state
The first excited state is three-fold degenerate: denotethese states as

In this notation thematrix elements are

In second-order perturbation theory

where where the unit matrix is Each spin state has the same

energy, to second order.

5.46 Spin–Orbit Interaction (Princeton)

a) In three dimensions the lowest eigenvalue of the harmonic oscillator is

which can be viewed as from each of the three dimensions.

The ground state hass-wave symmetry. The lowest excited states have

eigenvalue There are three of them. They have symmetry

and are the states and

where

QUANTUM MECHANICS 297