January 21, 1998

PHY 741 - Problem Set # 1

1.

In class, we showed that a constant magnetic field ${\bf{B}}=B_0{\bf{\hat{z}}}$ in the symmetric gauge contributes the following terms to the Hamiltonian:

$${\cal{H}}^{\prime} = \frac{e B_0}{2 m c}(L_z + g S_z) + \frac{e^2 B_0^2}{8 m c^2} (x^2+y^2).$$ (1)

Using perturbation theory, determine the corrections to the n = 1 and n = 2 energy levels of a hydrogen-like ion (single electron in the Coulomb field of a nucleus of charge Z) as a function of B₀ and Z, including terms up through 2nd order in the magnetic field strength.