In this section, we will calculate the fine structure corrections to the Hydrogen spectrum. Some of the degeneracy will be broken. Since the Hydrogen problem still has spherical symmetry, states of definite total angular momentum will be the energy eigenstates.
We will break the spherical symmetry by applying a weak magnetic field, further breaking the degeneracy of the energy eigenstates. The effect of a weak magnetic field is known as the anomalous Zeeman effect, because it was hard to understand at the time it was first measured. It will not be anomalous for us.
We will use many of the tools of the last three sections to make our calculations. Nevertheless, a few of the correction terms we use will not be fully derived here.
This material is covered in Gasiorowicz Chapter 17, in Cohen-Tannoudji et al. Chapter XII,and in Griffiths 6.3 and 6.4.