Lecture 21.

Below, \pm = ``plus or minus''.

We used step operators J_\pm to find the eigenstates | j,m >.

We first showed that the step operators J_\pm = J_x \pm i J_y are
eigenoperators under commutation with J_z, calculating the
commutation relations [J_z, J_\pm] = \pm \hbar J_z.  In analysis
that should be becoming familiar, we considered an initial J_z
eigenstate |m> (with eigenvalue J_z |m> = m\hbar |m>). We 
showed that the state J_\pm |m> is also an eigenstate, with
eigenvalue (m \pm 1)\hbar. Thus J_+ increases a state's J_z
value by \hbar, while J_- decreases it by \hbar.

We next showed that J^2 commutes with J_\pm (since it is just a
linear combination of J_x and J_y).  Thus when we consider a J^2
eigenstate (whose eigenvalue J^2 |j> = f(j) |j> is unknown), acting
on that eigenstate with a step operator J_\pm gives back an
eigenstate of J^2, with the same eigenvalue f(j).

We thus know how the step operators act on the fully specified
eigenstates |j, m>: they leave j invariant, while raising or
lowering m. This gives us a ladder of J_z eigenstates for fixed
j. As in the simple harmonic oscillator case, the ladder does not
continue forever. Again the constraint that raised and lowered states,
J_\pm |v>, have nonnegative norm restricts the allowed states on the
ladder. We found that the constraint resulted in a bound J_z^2 <
J^2, placing both upper and lower bounds on m. This means that the
ladder has both an upper and a lower rung; for consistency, J_+ must
annihilate the highest state, while J_- annihilates the lowest. We
showed by explicitly calculating the norm of J_+ |j, m_{max}> that
this occurs only if m_{max} = j, with f(j) = j(j+1) \hbar^2 giving
the eigenvalue of J^2. J_- |j, m_{min}> = 0 imposes the same
eigenvalue on J^2, with m_{min} = -j.

Finally, the ladder goes from |j, j> to |j, -j> in integral
steps. This means that there must be 2j+1 steps that go from the
highest to lowest state, which can occur only if 2j+1 is an integer
(otherwise, we do not hit the lowest state as we descend the ladder).

We have thus found all the angular momentum eigenstates |j,m>. For
each half-integer j, we have the truncated ladder of states from m=
j to m=-j, with J^2 eigenvalue j(j+1) \hbar and J_z
eigenvalues m\hbar.

---
KB