Lecture 20.

Handouts:
14. Midterm Review

We argued that angular momentum eigenstates are energy eigenstates;
that noncommutation of the components of \vec{J} meant that we could
not have eigenstates with fixed values of J_x, J_y, and J_z
simultaneously; and that a maximal set of commuting angular momentum
operators (operators that we can simultaneously diagonalize), is given
by J^2 and J_z.

We remembered that, for a rotationally invariant Hamiltonian, the
rotation operators R_{\hat{n}}\ (\theta) commute with the
Hamiltonian (so that the rotated also obeys Schrodinger's
equation). This implies that \vec{J} \dot \hat{n} commutes with the
Hamiltonian, so that \vec{J} \dot \hat{n} is conserved. Invariance
for any rotation axis \hat{n} implies that the angular momentum
vector \vec{J} is conserved, and itself commutes with the
Hamiltonian.

We then proved the fact mentioned earlier: that commuting operators
A and B have simultaneous eigenstates (that is, eigenstates of A
are also eigenstates of B); and that, conversely, noncommuting
operators cannot share all the same eigenstates.

Since J_x, J_y, and J_z don't commute, we cannot find a set of
eigenstates of all three. However, we will do our best, which is to
find the eigenstates of a maximal set of commuting angular momentum
operators. We can choose only one of the incompatible set J_x,
J_y, J_z; so we choose J_z. We then argued that another operator
related to angular momentum, J^2, must commute with J_z, because
the norm of the angular momentum vector cannot change under rotations
about the z-axis --- which are generated by J_z. We manually checked
that this is the case: J^2 commutes with J_z (and any other
component of J, as expected).

So when we look for angular momentum eigenstates, we must look for
eigenstates of the operators J^2 and J_z, the largest set of
incompatible angular momentum measurements we can make. To find the
eigenstates, we will use our simple harmonic oscillator trick of
finding step operators --- or eigenoperators under commutation with the
operator whose eigenstates we seek --- to build a ladder of eigenstates.

---
KB