Question: What is the reason for spin-orbit interaction without just assuming that the electron moves on a circular path around a nucleus?

Every explanation for spin-orbit coupling that I have read uses the semi-classical approach of the electron moving on circular trajectories (around the nucleus in the case of hydrogen). I am looking for an explanation from a purely quantum mechanical, or even better from a purely quantum field theoretical, point of view.

My problem is that the circular paths are something that has always just been justified by saying that on a larger scale everything has to look classical. If there is a mathematical proof for the validity of the circular path assumption, that would be fine for me, too. I only don't want to "just assume" the circular trajectories.

  • 2
    $\begingroup$ You can get this properly from the Dirac equation; googling "spin orbit coupling from Dirac equation" gives a bunch of promising results. $\endgroup$
    – knzhou
    Nov 20, 2018 at 15:01
  • $\begingroup$ Thank you very much. I've googled it and will be reading what I find. $\endgroup$
    – Fred
    Nov 20, 2018 at 15:03
  • $\begingroup$ I do not understand what does spin-orbital interaction have to do with circular orbits. There is really no way to "derive" a new interaction, you just have to postulate it in your Hamiltonian and see if the result agrees with the experiment. Are you looking for the technical way spin-orbital coupling appears in the quantum-mechanical Hamiltonian? Are you looking for a way how to derive this term from QFT? $\endgroup$
    – Void
    Nov 20, 2018 at 15:17
  • $\begingroup$ @Void My problem is (or was - I'm reading a source right now which might be just what I was looking for) that the following was always presented to me as the reason for SOI: An electron moves in a circular orbit and thus, due to the charge of the nucleus, feels a magnetic field which interacts with the electron spin. These circular orbits, however, bother me, because an uncertainty in momentum and space (i.e. the electron orbital) clearly isn't the same as an electron moving on a path around the nucleus. A derivation as an expansion term of the Dirac equation seems indeed like what I wanted. $\endgroup$
    – Fred
    Nov 20, 2018 at 15:52
  • $\begingroup$ I know that these orbits need a Hamiltonian in first place to be the solutions to it. However, since SOI is relatively small, I expected it to be a perturbation which in my eyes needs a better reason than circular trajectories. My hope (which might actually be fulfilled) was some QFT expression which in some (classical) limit might give rise to an SOI term as we know it. Currently, I'm looking into a source which seems to do just that by deriving the term from the Dirac equation. But I haven't read that far, yet. $\endgroup$
    – Fred
    Nov 20, 2018 at 15:59