Problem 1:

Consider a Hamiltonian

H₀ is perturbed by the spin-orbit interaction for a spin ½ particle,

Show that the average perturbation of all states corresponding to a given term (which is characterizes by given L and S) is equal to zero.

Problem 2:

The Hamiltonian of the hydrogen atom may be written as H=H₀+H’, where

We write H’=W_f+W_hf, with

and

or

(a)  Explain the physical origin of the three terms in W_f.

(b)  Which eigenbasis of H₀ is most convenient when evaluating the matrix elements of W_f, and why?

(c)  Explain the physical origin of the terms in W_hf.

(d)  Does the perturbation term H’=W_f+W_hf completely account for all the splittings of the energy levels which are observed in the spectrum of hydrogen?

Problem 3:

Estimate the ground state energy of a system of two interacting particles of mass m₁ and m₂ with interaction energy  using the variational method.