Time Development of an $\ell=1$ 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 $+\hbar$, $\psi^{(x)}_+$, what is the state at a later time $t$? What is the expected value of $L_x$ 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 $L_z$ 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 $t=0$.

$$\bgroup\color{black}\psi(t=0)=\pmatrix{\psi_+ \cr \psi_0\cr \psi_-}.\egroup$$

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

$$\bgroup\color{black}\psi(t)=\pmatrix{\psi_+e^{-iE_+t/\hbar} \... ...i\mu_BBt/\hbar} \cr \psi_0\cr \psi_-e^{i\mu_BBt/\hbar}}.\egroup$$

As a concrete example, let's assume we start out in the eigenstate of $L_x$ with eigenvalue $+\hbar$.

$$\begin{eqnarray*} \psi(t=0) & = & \psi_{x+}=\pmatrix{{1\over 2}\cr {1\over\sqr... ...r 4}(4\cos({\mu_BBt\over\hbar}))=\hbar\cos({\mu_BBt\over\hbar}) \end{eqnarray*}$$

Note that this agrees with what we expect at $t=0$ and is consistent with the angular momentum precessing about the z axis. If we checked $\langle\psi\vert L_y\vert\psi\rangle$, we would see a sine instead of a cosine, confirming the precession.