## October 17, 2022

### Fine Structure

I’m teaching the undergraduate Quantum II course (“Atoms and Molecules”) this semester. We’ve come to the point where it’s time to discuss the fine structure of hydrogen. I had previously found this somewhat unsatisfactory. If one wanted to do a proper treatment, one would start with a relativistic theory and take the non-relativistic limit. But we’re not going to introduce the Dirac equation (much less QED). And, in any case, introducing the Dirac equation would get you the *leading corrections* but fail miserably to get various non-leading corrections (the Lamb shift, the anomalous magnetic moment, …).

Instead, various hand-waving arguments are invoked (“The electron has an intrinsic magnetic moment and since it’s moving in the electrostatic field of the proton, it sees a magnetic field …”) which give you the wrong answer for the spin-orbit coupling (off by a factor of two), which you then have to further correct (“Thomas precession”) and then there’s the Darwin term, with an even more hand-wavy explanation …

So I set about trying to find a better way. I want use as minimal as possible input from the relativistic theory and get the *leading* relativistic correction(s).