Angular Momenta and Magnetic Dipole Moments — Question 6

Question 6

Spin-orbit energies follow from the square of total angular momentum

Use $H_{\mathrm{so}}=A\boldsymbol L\cdot\boldsymbol S/\hbar^2$ with $A>0$. A p electron has $l=1,s=1/2$ and total $\boldsymbol J=\boldsymbol L+\boldsymbol S$.

1. Derive the energy for each allowed j.

2. Find the splitting and the degeneracy-weighted mean energy.

Question 6 - Solution

Spin-orbit energies follow from the square of total angular momentum

(a) Method and calculation

$$2\boldsymbol L\cdot\boldsymbol S=J^2-L^2-S^2\Rightarrow E_j=\tfrac A2[j(j+1)-2-3/4].$$$$\boxed{E_{3/2}=A/2,\quad E_{1/2}=-A}.$$

(b) Interpretation and further calculation

$$\boxed{E_{3/2}-E_{1/2}=3A/2}.$$$$\bar E=\frac{4(A/2)+2(-A)}{6}=\boxed{0}.$$

Physical check

The weighted centroid remains at the unsplit energy in this model. Reversing the sign of A reverses the level ordering.