Stern-Gerlach Experiment: Spin Orbit Coupling

[eep]

Hi,
We're looking at the consequences of the Stern-Gerlach experiment which has lead us to the idea of spin orbit coupling in the hydrogen atom. In class, the coupling term of the Hamiltonian was derived by assuming the electron has a magnetic moment, then viewing the hydrogen atom in the electron's frame. We then said that the proton orbits the electron, which is effectively a current, thereby producing a magnetic field which interacts with the moment. Obviously, this idea of the proton having a circular trajectory is totally crap in quantum mechanics, but everyone I've asked hasn't been able to give me a different derivation. All I've gotten is that this is one of those cases where classical mechanics gives a result which happens to work in the quantum case. Anyone know how to get the result without this idea of an orbiting proton?

 

[jtbell]

It looks like you have to start with the (relativistic) Dirac equation instead of the (non-relativistic) Schrödinger equation, and solve it for the hydrogen atom. In doing so, you have to introduce the electric and magnetic fields together by way of the four-vector potential which includes both the electric scalar potential and the magnetic vector potential.

http://zopyros.ccqc.uga.edu/lec_top/rltvt/node5.html

This automatically gives you the effects of spin-orbit coupling in the energy levels. If you then take the non-relativistic limit of the Dirac equation, you get the Schrödinger equation with a correction term which is the same as the one that you derived using the "orbiting proton" picture.

http://zopyros.ccqc.uga.edu/lec_top/rltvt/node6.html