Perturbation Theory for Fine Structure

The calculation of the fine structure of the energies for hydrogen now involves two steps: 1. as one has degenerate states of $\hat{H}_0$, one needs degenerate perturbation theory. 2. This is however simplified by the fact that the corresponding matrix in the subspace of the degenerate eigenstates can be made diagonal in a suitable basis, using the total angular moment

$$\hat{\bf J} = \hat{\bf L} + \hat{\bf S}.$$ (3.15)