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