Where 
The unperturbed solution is 
. 
. These staisfy: 
and 
.
The energy is 
and 
Where The unperturbed solution is . . These staisfy: and .
The energy is and
Relativistic corrections:
Start with dirac equation and expand in powers of 
.
Correction is of order:
.
This is also order . Doesn't contribute
for S-waves.
. This is oreder . Only contributes to S-waves.
These are the fine structure.
In uncoupled basis, only spin-orbit term is nondiagonal. In coupled basis, everything is diagonal.
Then 
which gives 
. Then for the relativistic mass, 
For the spin-orbital interaction: 
For the Darwinian interaction: 
The sum of the terms is then: 
. There is only j angular dependence.
For n=2, you get, 
splitting from 
by about 
.
hyperfine structure and lamb shift add in.