 Spin-orbit energy shift in a central potential

For a central-potential Hamiltonian operator with the spin-orbit interaction $H_{\mathrm{so}}=[r^{-1}{V}^{\prime }(r)/(2m^2{c}^2)]\mathbf{L}\cdot \mathbf{S}$, use $\mathbf{J}=\mathbf{L}+\mathbf{S}$ andA coupled angular momentum eigenstate makes this angular factor a number. The radial operator commutes with all angular factors, giving the energy shift in the title. It is independent of the magnetic projection $m_j$, so a fixed-$j$ multiplet retains its $2j+1$ degeneracy.

For spin one-half, the angular bracket is $l$ when $j=l+1/2$ and $-(l+1)$ when $j=l-1/2$. Only the former exists at $l=0$, where it is zero. For the Coulomb potential energy, $r^{-1}{V}^{\prime }(r)=e^2/(4\pi {ϵ}_0{r}^3)$; the radial normalization convention must be respected when computing its expectation value.