Assignment 10

Problem 1:

Consider a hydrogenic atom with nuclear charge Zqe in a strong magnetic field B = Bk.  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(-x2/(4 Å2)).
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 Sp∙Sn,
where SpSn are the spin operators of the proton and neutron, respectively, and k is a positive constant.
(a)  Consider the following operators:
Sp2, Sn2, Spz, Snz, S2, and Sz,
where S = Sp + Sn.  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(Spz + Snz)Bz,
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(px2 + py2) + ½mω2(x2 + y2).
It is subjected to the perturbation H1 = 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.