*May 2004*

*Tip: look over all
the homework questions.*

1. *Charged particle in a
magnetic field*: know the Hamiltonian _{} and how to solve the
corresponding Schrödinger equation for an electron in a plane with a uniform perpendicular magnetic field. Know the relation between canonical momenta
and velocities, and be able to count states of a given energy per unit area.

2. *Path Integrals*: know the
definition of the propagator as a sum over paths

_{}

Understand the connection between the classical limit of this expression and the classical path. Be able to establish that this propagator satisfies Schrödinger’s equation for a particle in one dimension in a potential.

3. *Spherical Tensors*: read
my web notes, or equivalent. Know by
heart the components of a spherical *vector*,
_{}and be sure you thoroughly understand the Wigner-Eckart
theorem, and in particular what selection rules it implies.

4.* Variational Methods*: know how to
find an upper limit on the ground state energy using a Gaussian or other simple
trial function, for any simple potential, both in one dimension and for
spherically symmetric potentials in three dimensions. Have some idea how to find the next excited
state for a symmetric potential in one dimension.

5. *WKB*: know the
formula

_{}

and when it is valid—be able to give a physical interpretation of the terms. Know how to use it to estimate tunneling amplitudes, and know how the results of the connection formulas lead to the quantization rule (which you should also know—and when it’s valid)

_{}

Know the corresponding formula for spherically symmetric wavefunctions.

6. *Time-Independent Perturbation Theory*:
know by heart (but also know how to derive!) the first- and second-order terms:

_{}

Be familiar with the *Selection Rules*: Parity
and Wigner-Eckart Theorem.

*Degenerate Perturbation Theory*:
understand the essential point that you have to diagonalize the perturbation *H*^{
1} in the subspace of degenerate states of *H*^{ 0}, and be
prepared to do it in a simple case. In
particular, know how to find a diagonal basis for spin-orbit coupling. Read my on-line notes on the Peierls
Transition, a good example of degenerate perturbation theory.

7. *Time-Dependent
Perturbation Theory*: know by heart
(but also know how to derive!) the first-order term:

_{}

You should be able to use this to find transition probabilities in a system when an external potential acts over some time period, and know when this is a valid approximation.

Know the routine for handling a sudden perturbation, and when it is valid to apply it (how sudden is sudden?).

Similarly for an adiabatic approximation: when is *that*
ok?

8. *The Periodic Perturbation*: most
important! Understand the derivation of
Fermi’s Golden Rule, and know the rule by heart. And, know how to handle the delta function in
applying the rule in actual computation.

_{}

You can skip Shankar’s section 18.3 on higher orders, **but** read my web notes instead.

9. *Classical
Electrodynamics*: you should understand how a gauge transformation of the
magnetic field also means changing the phase of an electron wave function, and
this can happen even if the electron stays in a region where there is zero
magnetic field. You should be familiar
with the vector potential formulation of an electromagnetic wave, how the
vector potential is related to the magnetic and the electric fields in a wave,
and the polarization.

10. Read carefully
the whole section on the *Photoelectric Effect* in hydrogen.

Review the relevant sections in Chapter 2 of Sakurai. In particular, understand how he arrives at the expression for the quantized vector potential in a wave, (2.60),

_{}

where the *a*’s are standard simple harmonic oscillator
type annihilation and creation operators. Then skip forward to section 2-4, and
read it as far as equation (2.140). That
is to say, review spontaneous emission in the dipole approximation.