Putting everything together

Apart from the corrections $E'_{\rm SO}$, one also has to take into account the relativistic corrections dur to $\hat{H}_{\rm KE}$ and $\hat{H}_{\rm Darwin}$ from section II.3.1. It turns out that the final result for the energy eigenvalue in first order perturbation theory with respect to $\hat{H}_1 = \hat{H}_{\rm KE} + \hat{H}_{\rm Darwin} + \hat{H}_{\rm SO}$, Eq. (II.3.1), is given by the very simple expression

$$E_{nlsjm} = E_n^{(0)} + \frac{(E_n^{(0)})^2}{2mc^2}\left[3-\frac{4n}{j+\frac{1}{2}}\right],\quad j=l\pm \frac{1}{2}.$$ (3.26)

For a detailed derivation of this final result (though I haven't checked all details), cf. James Branson's page,

http://hep.ucsd.edu/ branson/http://hep.ucsd.edu/ branson/

or Weissbluth [4], cf. 16.4. Gasiorowicz [3] 12-16 seems to be incorrect.

Figure: Fine-Splitting of the hydrogen level $E_{n=2}$, from Gasiorowicz[3]

Final remark: we do not discuss the effects of a magnetic field (anamalous Zeeman effect) or the spin of the nucleus (hyperfine interaction) here. These lead to further splittings in the level scheme.