Consider a charged particle on a ring of unit radius with flux f/f =a passing through the ring, where f0=hc/e is the flux quantum. The Hamiltonian operator can be written as H=H0+V, where
q is the
angular coordinate. We have chosen units with 
(a) Find the complete set
of eigenvalues and eigenfunctions of H0.
(b) Use perturbation theory to find the first and second order corrections to the
ground state energy E0 of H0 due to the perturbation V
for 0<a<˝ .
(c) For a=˝ the ground state energy of H0
is degenerate. Find the first order correction to E0 for this case.
 | Solution: Concepts, principles, relations that apply to the problem: Why do they apply? How do they apply? (a)
(b) Let 0<a<˝. Then all states are non degenerate and the ground state has n=0.
Details of the calculations: (c) If a=˝ then all states are degenerate. The ground
state is any linear combination of n=0 and n=1. We have to diagonalize
the matrix of V in the subspace spanned by
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Try
Consider a rigid rotator with moment of inertia I and
electric moment p constrained to rotate in a plane about the z - axis
through its center of mass. A uniform electric field E is directed along x,
so that the wave equation has the form 
where f is the angle of rotation. Treat 
as a
perturbation term.
(a) Calculate the energy levels W and wave functions y
of the unperturbed system.
(b) Calculate all perturbed energy levels through first order and the ground state
through second order.
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An electron of mass m is constrained to move on a 2D
surface in the presence of a localized potential due to a defect. We will assume that the
electron is bound in an energy eigenstate and can be described by the single-particle
Schroedinger equation,
.
(a) Since the potential is a function of the radial coordinate r only, the wavefunction can be separated into a product form. For angular momentum Lz=M, what is the separated wavefunction ( give the angular dependence explicitly ) and what is the radial Schroedinger equation for yM(r)?
Now suppose we model the defect by an attractive 2D square well of radius a and
depth -v, 
and assume that the electron moves freely outside the well.
(b) Consider the ground state y0(r) of the electron, which is independent of the polar angle q. Without specifying constants, write the most general form of the radial wavefunction inside the well, in terms of special functions.
(c) Based on dimensional arguments, the most general form the ground state energy
can have is 
where 
is a dimensionless combination of parameters. Use dimensional analysis to
find a suitable choice for . ( It is not unique. )
(d) Now suppose we estimate the ground state binding energy of the electron
variationally. If we take as our Ansatz the Gaussian form 
, first show that the
normalization is 
Now
determine the expected kinetic and potential energies <T> and <V>
and the total energy <E>, sketch them as functions of l,
and give an intuitive explanation of their qualitative behavior in the small and large l limits.
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