Energy Eigenstates of an System in a B-field

Recall that the Hamiltonian for a magnetic moment in an external B-field is

As usual, we find the eigenstates (eigenvectors) and eigenvalues of a system by solving the time-independent
Schrödinger equation
.
We see that everything in the Hamiltonian above is a (scalar) constant except the operator
, so that

Now if
is an eigenstate of
, then
, thus

Hence the normalized eigenstates must be just those of the operator itself, i.e., for the three values of (eigenvalues of ), we have

and the energy eigenvalues are just the values that takes on for the three values of m i.e.,

Jim Branson 2013-04-22