Origin of spin-orbit interaction

Question

The spin-orbit interaction in a hydrogen atom is often explained as arising from an interaction energy $U=-\mathbf{m}\cdot\mathbf{B}$ where $\mathbf{m}$ is a magnetic moment due to the electron’s spin and $\mathbf{B}$ is the magnetic field produced by the proton in the electron’s frame of reference.

Why does one have to switch to the electron’s frame of reference? Can’t one instead use the magnetic field produced by the electron in the usual reference frame where the proton is at rest?

EDIT: I do notice that the magnetic field originating from the electron’s motion vanishes at the instantaneous location of the electron itself ... hence the explanation might be that the electron’s spin only interacts with the magnetic field present at the instantaneous location of the electron.

Nonetheless, the energy of a system is not invariant under Lorentz boosts. How can we add an energy term calculated in the electron’s rest frame (the spin-orbit interaction term) to the total energy in the proton’s rest frame (cfr. the hydrogen atom’s Hamiltonian)?

Answer 1

Well, I should have consulted my own bookshelf before asking this question. In the second edition of Eisberg & Resnick’s Quantum Physics, the following is mentioned:

But this energy has been evaluated in a frame of reference in which the electron is at rest, whereas we are interested in the energy as measured in the normal frame of reference in which the nucleus is at rest. Because of an effect of the relativistic transformation of velocities, called the Thomas precession, the transformation back to the nuclear rest frame results in a reduction of the orientational potential energy by a factor of 2.

The answer hence is that is not correct to add the energy as calculated in the electron’s frame of reference to the hydrogen atom’s Hamiltonian, as I suspected. Sadly the lecture notes I’m studying from do not mention that the factor of 2 popping out of nowhere actually prevents one from adding apples and oranges (energies calculated in different frames of reference).

Answer 2

In the frame of reference of the proton, the proton produces no magnetic field but instead it has an electric field. The electron has a magnetic dipole moment. Due to relativity, a moving magnetic dipole moment behaves similar to an electric dipole (in the frame of the proton). In this case, the origin of the spin-orbit coupling is the electric dipole moment component of the electron due to relativity which can align with the electric field of the proton.

The coupling of the proton with the magnetic field of the electron gives rise to a different term called hyperfine structure (which is usually much weaker than SO).

Transforming magnetic dipoles to electric dipoles is not straightforward and has a few problems related to hidden momentum. Check: https://arxiv.org/abs/1409.4796 and https://arxiv.org/abs/1303.0732

Alternatively, you can use Dirac's equation and take the weakly relativistic limit.

See also this answer: https://physics.stackexchange.com/a/754712/168640