Magnetism and Electrons — Question 7

Question 7

Orbital and spin angular momenta carry different magnetic weights

In an uncoupled product state an electron has $L_z=2\hbar$ and $S_z=-\hbar/2$. Use orbital g=1 and spin g=2, and ignore spin-orbit coupling.

1. Find total angular-momentum projection Jz and total magnetic-moment projection.

2. Find the Zeeman energy in a field B along +z and the ratio $-\mu_z\hbar/(\mu_BJ_z)$. Explain why this ratio is not automatically a Landé factor for a total-J eigenstate.

Question 7 - Solution

Orbital and spin angular momenta carry different magnetic weights

(a) Method and calculation

$$J_z=L_z+S_z=\boxed{3\hbar/2}.$$$$\mu_z=-\frac{\mu_B}{\hbar}(L_z+2S_z)=\boxed{-\mu_B}.$$

(b) Interpretation and further calculation

$$\boxed{U=-\mu_zB=+\mu_BB}.$$$$\boxed{-\mu_z\hbar/(\mu_BJ_z)=2/3};\quad\text{a product state need not have definite }J^2.$$

Physical check

Adding angular-momentum projections before multiplying by a single electron g factor would give an incorrect moment.