**Problem 1:**

Consider a hydrogenic atom with nuclear charge Zq_{e} in a **strong** magnetic field **B** = B**k**. Assume
that the Zeeman slitting is much larger than the spin-orbit splitting of the
energy levels, so that to first order the spin-orbit interaction can be ignored.

(a) Find the energies of the 2s and 2p energy levels in the strong magnetic
field. Is the degeneracy completely removed by the Zeeman interaction?

(b) Estimate the magnitude of the magnetic field, B, required to give a Zeeman
splitting in the hydrogenic atom comparable to the binding energy of the ground
state of the hydrogen atom. Can such a magnetic field be created in the
laboratory?

**
Problem 2:**

In the WKB approximation, find the allowed
energies that a ball of mass m, bouncing due to gravity on a perfectly
reflecting surface, can have.

You can use the fact that for this problem
the WKB approximation gives

∮p dq = (n - ¼)h.

where p(q) is the momentum of the ball at the
height q and the integral is over a full periodic path. You may leave your
answer as an integral equation which could be solved to yield the energy levels.

**Problem 3:**

In one dimension, the potential energy of an electron as a function of x is
given by

U(x) = -30 eV exp(-x^{2}/(4 Å^{2})).

Use the variational method to find the energy of the ground state in units of
eV.

**Problem 4:**

A proton and a neutron are confined by a three-dimensional
potential. For this problem assume that the proton and neutron do not interact
with each other, and neglect spin-orbit interactions. Both particles have spin
½. Including the spins, the ground state is four-fold degenerate. To this
system we now add the interaction between the magnetic dipole moments of the
particles described by the interaction Hamiltonian

H' =
k** S**_{p}**∙S**_{n},

where **S**_{p}, **S**_{n} are the spin operators of the proton and neutron,
respectively, and k is a positive constant.

(a) Consider the following operators:

S_{p}^{2}, S_{n}^{2}, S_{pz}, S_{nz},
S^{2}, and S_{z},

where **S** = **S**_{p}
+ **S**_{n}. State which of these operators commute with H'.

(b) Into how many distinct energy levels does the original ground state split
in the presence of H'? Calculate the corresponding energies and state their
degeneracy.

(c) We now place the
system into a uniform external magnetic field, which points in the positive
z-direction, **B** = B **k**. The spin-spin interaction described by H'
continues to be present and the additional interaction Hamiltonian is

H'_{B} = b(S_{pz}
+ S_{nz})B_{z},_{
}where b is a positive constant.

Calculate
the corrections to the energies of the states identified in part (b) due to the
presence of the magnetic field.

(d) Sketch a graph of the energy levels as a function of the external magnetic
field strength, B, including the effects of both H' and H'_{B}.
Identify the curves with the corresponding states identified in part (b).

**Problem 5:**

A harmonic oscillator in two dimensions has the unperturbed Hamiltonian

H = ½m(p_{x}^{2 }+ p_{y}^{2}) + ½mω^{2}(x^{2
}+ y^{2}).

It is subjected to the perturbation H_{1 }= Δxy, where Δ is the strength
of the perturbation.

(a) Write the eigenstates of the unperturbed oscillator in terms of the
eigenstates of the 1-dimensional harmonic oscillator. What are the eigenvalues
of H?

(b) Evaluate the first order corrections to the energies of the three states
lowest in energy
when Δ > 0.