Problem 1:

Let L be the total orbital angular momentum of the electrons of an atom and S be their total spin angular momentum, and let J=L+S be their total angular momentum.  Assume that in the absence of a magnetic field H₀, L², S², J², and J_z form a C.S.C.O. and {|E,l,j,m >} forms an orthonormal basis for the state space.  Rotational invariance implies [H₀,J]=0. Therefore each eigenvalue E is at least 2j+1 fold degenerate.  This is the essential degeneracy.  Assume no accidental degeneracies exist.  (Each 2j+1 fold degenerate eigenvalue E is called a multiplet.)  If the atom is placed in an external magnetic field B pointing in the z direction the Hamiltonian becomes H=H₀+H₁, H₁=w_L(L_z+2S_z). Calculate the effect of the magnetic field on the energy levels of the atom to first order.

Problem 2:

Consider a particle of mass m placed in an infinite two-dimensional potential well of width a.

V(x,y)=0  if 0<x<a  and  0<y<a,    V(x,y)=¥ everywhere else.

The particle is also subject to a perturbation W described by

W(x,y)=w₀  for  0<x<a/2  and  0<y<a/2,   W(x,y)=0 everywhere else.

(a)  Calculate, to first order in w₀, the perturbed energy of the ground state.

(b)  Calculate, to first order in w₀, the perturbed energy of the first excited state.  Give the corresponding wave functions to 0th order in w₀.

Problem 3:

Consider a one dimensional harmonic oscillator of mass m and spring constant k in a uniform gravitational field g:

For g=0 the eigenfunctions of the time independent Schroedinger equation are

The energy eigenvalues are

(a)  For g = 0 show that the matrix element for x, x_nm, equals zero except for the following:

The integral may be easily performed using the recursion relation

(b)  Show, using perturbation theory, that the energy levels are unchanged to first order in g.

(c)  Show, that to order g², all energy levels are shifted equally downward by an amount

(d)  Using a translated coordinate system  show that the result found in (c) is exact.