Past exam of the atomic and laser physics course of the University of Oxford / 2021 / 1 / b / Solution

Within a fixed $L,S,J$ angular momentum multiplet, the component of $\mathbf{L}$ along $\mathbf{J}$ follows from $2\mathbf{L}\cdot \mathbf{J}={\mathbf{J}}^2+{\mathbf{L}}^2-{\mathbf{S}}^2$. The corresponding result for $\mathbf{S}$ follows by interchanging $L$ and $S$. For the quantum state $|LSJ{M}_J⟩$,These are dimensionless expectation values in the convention of part (a). Substituting into the first-order time-independent perturbation theory formula $\Delta E=⟨H_Z⟩$ givesFor $g_s=2$, the Landé g-factor becomesThe magnetic sublevels must first be diagonalized within a degenerate multiplet; choosing $M_J$ along the magnetic field already does this. For $J=0$ the displayed fraction is undefined, but all vector expectation values vanish and there is no first-order Zeeman splitting. A higher-order shift may remain.