Spin dynamics and Larmor precession

[Lunar_Lander]

Homework Statement

Consider an electron with spin, which should be in a homogenous magnetic field B=B₀e_z. This situation is described by the Hamiltonian of the shape $\hat{H}=g\frac{\mu_B}{\hbar}\textbf{BS}$.

Consider now the time dependent state $|\psi(t)>$ of the electron in spin space. The spatial structure of $|\psi(t)>$ is not considered here at all!

(a) Show that the expectation values of the spin operators satisfy the equations of motion:

$\frac{d}{dt}<S_x>=-\omega_0<S_y>$
$\frac{d}{dt}<S_y>=\omega_0<S_x>$
$\frac{d}{dt}<S_z>=0$.

What is the value of $\omega_0$?

(b) Show that because of the equations of motion, the expectation value of the spin rotates in the xy-Plane (~Larmor precession)

Homework Equations

When looking up the Larmor precession I found this: $\omega=\gamma\cdot|B|=\frac{gq}{2m}\cdot|B|$.

The Attempt at a Solution

Is the formula I gave in (2) already the solution for $\omega_0$? Being naive I would think that the rotation cannot happen in the z-Direction, as the temporal derivative of the z-Component of the spin operator is 0. Can that be?

 

[BigJDubs]

It would mean that the B-Field is not really homogenous as the z-axis is the only axis with a non-time-dependent expectation value? I'm totally lost here, any help would be appreciated!