Where The unperturbed solution is . . These staisfy: and . The energy is and

Relativistic corrections:

Start with dirac equation and expand in powers of .

- First correction due to relativistic mass. Correction is of order: .
- Spin orbit interaction. This is also order . Doesn't contribute for S-waves.
- Darwin term-has to do with correction due to only local interactions.
. 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.

source

psfile jl@crush.caltech.edu index

hyperfine_splitting