Origin of the Spin-Orbit Interaction

TL;DR

This paper presents a semi-classical model explaining the origin of the spin-orbit interaction in hydrogen, deriving the correct energy factor without relying on Thomas precession, by analyzing the coupling of induced dipoles and electric fields.

Contribution

It introduces a novel semi-classical approach that naturally derives the 1/2 factor in spin-orbit coupling without Thomas precession.

Findings

Derives the 1/2 factor in spin-orbit energy naturally

Shows the energy change is U/2 during a spin-flip

Provides a semi-classical explanation for spin-orbit interaction

Abstract

We consider a semi-classical model to describe the origin of the spin-orbit interaction in a simple system such as the hydrogen atom. The interaction energy U is calculated in the rest-frame of the nucleus, around which an electron, having linear velocity v and magnetic dipole-moment mu, travels in a circular orbit. The interaction energy U is due to the coupling of the induced electric dipole p=(v/c)x mu with the electric field En of the nucleus. Assuming the radius of the electron's orbit remains constant during a spin-flip transition, our model predicts that the energy of the system changes by Delta_E = U/2, the factor 1/2 emerging naturally as a consequence of equilibrium and the change of the kinetic energy of the electron. The correct 1/2 factor for the spin-orbit coupling energy is thus derived without the need to invoke the well-known Thomas precession in the rest-frame of the electron.