Lecture 22.

Handouts:
Midterm Returned

The midterm results were disappointing; they were not representative
of what you seem capable of doing, and what you need to have mastered
before we apply it to quantum fields. I will therefore let you rework
a second copy of the midterm, using your books, notes, and handouts
but NO COLLABORATING. You need rework only those problems that you
want to improve your performance on. I will be available during my
usual office hours Tuesday March 19 for any questions on the material
we've covered, and will collect your reworked exams, along with your
original exam, on Wednesday March 20 at 5:00 p.m. I will average your
in-class score with your improved score to obtain your final first
midterm score.

On to physics: we reviewed the construction of the angular momentum
eigenstates; discussed energy eigenstates and the Zeeman effect; and
discussed representations of angular momentum operators within a basis
of |j,m> eigenstates for fixed j.

Last time we found a truncated ladder of J_z eigenstates, with m
ranging from +j to -j, for each half-integer j. This truncated
ladder for fixed j is called a ``representation''. This is because
no rotation --- even about the x- or y-axis --- ever changes the norm
J^2 of the angular momentum, whose quantum number is j. Rotations
about the x- or y- axis can, however, change the z-component of
angular momentum J_z, with quantum number m. Thus, under
rotations, a state |j, m> can change, but only by mixing with the
2j other states with the same quantum number j. Thus the states of
fixed j --- that is the states in the representation j --- provide
a complete basis of states obtainable from |j, m> by rotation. This
basis of 2j+1 states lets us write all angular momentum operators
and rotations as (2j+1)-dimensional matrices, which describe how the
operators mix states within the truncated j-ladder. These matrices
contain all the information of our original physical description of
the rotation group; the group multiplication table that describes
composition of successive rotations is entirely obeyed by matrix
multiplication of the (2j+1)-dimensional matrices for the rotations.
This is why this (2j+1)-dimensional vector space is called a
representation; it is one of many ways to fulfill the multiplication
table defining the rotation group.

As promised, our knowledge of the angular momentum eigenstates gives
us complete information about the energy eigenstates of a rotationally
symmetric Hamiltonian. Because a rotationally invariant Hamiltonian
commutes with J^2 and J_z, the eigenstates |j,m> that we have
constructed {\em are} the energy eigenstates. Furthermore, we showed
that eigenstates in the same representation --- that is, with the same
j --- are degenerate in energy. Thus we have energy levels that
depend on j and not m, with 2j+1 physically distinct states of
energy E_j.

This also gives us information about almost rotationally symmetric
systems.  If we perturb a rotationally symmetric Hamiltonian (like
that for the valence electron of a hydrogen-like atom) by introducing
an interaction with some preferred direction (like turning on a
magnetic field), we can calculate the shift we cause in the system's
energy levels. We calculated this for the Zeeman effect, seeing that
when we turn on the B field, we lift the degeneracy of atomic energy
levels, making them split apart. We found, for \Delta = e\hbar B/
(2m_e c), that each particular state |j, m> has its original
(degenerate) energy shifted by -m\Delta. Thus each j has 2j+1
energy eigenstates, each with a slightly different energy, and
transitions can be observed between many more pairs of energy levels.

---
KB