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

Since the two states have no matrix elementswith the other two

states, we can omit them from the remaining steps in the calculation. Thus

we must find the eigenvalues of a 2 × 2 matrix for the states and

Thismatrix has eigenvalues The perturbation splits the fourfold

degenerate state into states with eigenvalues

Since is proportional to the electric field, the energies split linearly with
.
The matrix element can be evaluated by using the explicit represen-
tation for the eigenstates of the hydrogen atom:

yielding

The angular integral gives 2/3, and

5.49 Hydrogen with Electric and Magnetic

Fields (MIT)

We use the same notation as in Problem 5.48 to describe the four orbital

states: thes-state is and the three are

Here again,spin is not affected by this perturbation. As in Problem5.48,

we must evaluate the 10 different matrix elements which

occur in the symmetric 4 × 4 matrix.

300 SOLUTIONS