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At university we have just derived the expression for the energy split due to spin-orbit coupling in the Hydrogen atom, i.e. what is known as the FINE structure of Hydrogen.

To do this, we considered the situation from the frame of reference of the electron: in this frame, the proton is moving and therefore generates a magnetic field ( because it's a moving charged particle) which couples to the electron's spin.


What if I wanted to consider the situation in the nucleus' (proton's) frame of reference? The moving charged particle is the electron and it is generating a magnetic field (which as it happens couples to the spin of the nucleus and results in the hyperfine structure). But the magnetic field of the electron cannot couple to its own spin? Or can it? How do I explain the energy spit in this reference frame? Physics must be the same in all frames of reference right?


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In such a case, you would couple the magnetic field generated by the electron to the nuclear spin (in this case the proton spin). –  Ignacio Vergara Kausel Jan 23 at 11:38
That's the hyperfine structure of hydrogen, but it's the same argument really: if Physics is the same in all reference frames, then I must be able to get the Energy shifts (both for the fine and hyperfine structures) in the electron as well as in the proton's rest frames –  Harold Jan 24 at 22:01

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But the magnetic field of the electron cannot couple to its own spin? Or can it? How do I explain the energy spit in this reference frame?

In classical EM theory, the common explanation of the LS term from the frame of the electron is not very convincing, because this frame is non-inertial and there are potentially all kinds of non-inertial forces which were not discussed.

Explaining this in the frame of nucleus seems as an easier task. If we imagine charged rotating and orbiting ball in a central electric field of the nucleus, this central electric field affects the orbital motion of the ball; the ball accelerates, which changes its own electric and magnetic field. These fields of the ball then could influence the rotation of the ball. I do not know if it gives something close to the LS term, but it seems possible.

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I think that though the spin orbit coupling is classically motivated through the explanation described above, it is inherently quantum in nature and should not be explained by classical electrodynamics. These corrections were introduced because of discrepancies in the experiments. You should probably look it as just the coupling of two angular momentum operators. The spin orbit coupling in H atom is due to the interaction of the quantum numbers L and S.

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