Time Development of an System in a B-field: Version I

We wish to determine how an angular momentum 1 state develops with time, develops with time, in an applied B field. In particular, if an atom is in the state with x component of angular momentum equal to , , what is the state at a later time ? What is the expected value of as a function of time?

We will choose the z axis so that the B field is in the z direction.
Then we know the energy eigenstates are the eigenstates of
and are the
basis states for our vector representation of the wave function. Assume that
we start with a general state which is known at
.

But we know how each of the energy eigenfunctions develops with time so its easy to write

As a concrete example, let's assume we start out in the eigenstate of with eigenvalue .

Note that this agrees with what we expect at and is consistent with the angular momentum precessing about the z axis. If we checked , we would see a sine instead of a cosine, confirming the precession.

Jim Branson 2013-04-22