Magnetism and Electrons — Question 8

Question 8

The electron gyromagnetic sign determines precession direction

A magnetic moment obeys $\boldsymbol\mu=-\gamma\mathbf L$ with positive gamma in a uniform field $B\hat z$. Treat the expectation vector semiclassically with $d\mathbf L/dt=\boldsymbol\mu\times\mathbf B$.

1. Find the angular-velocity vector of precession and determine whether Lz changes.

2. Compare orbital and spin precession angular frequencies for an electron, using orbital gamma=e/(2me) and spin gamma=e/me.

Question 8 - Solution

The electron gyromagnetic sign determines precession direction

(a) Method and calculation

$$\dot{\mathbf L}=-\gamma\mathbf L\times\mathbf B=(\gamma\mathbf B)\times\mathbf L.$$$$\boxed{\boldsymbol\Omega=+\gamma B\hat z,\quad\dot L_z=0}.$$

(b) Interpretation and further calculation

$$\Omega_L=eB/(2m_e),\qquad\Omega_s=eB/m_e.$$$$\boxed{\Omega_s/\Omega_L=2}.$$

Physical check

The torque changes the transverse direction while conserving both the magnitude and the field-axis projection in this ideal model.