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 μ, travels in a circular orbit. The interaction energy U is due to the coupling of the induced electric dipole p=(v/c)×μ with the electric field En of the nucleus. According to quantum mechanics, the radius of the electron’s orbit remains constant during a spin-flip transition. Under such circumstances, our model predicts that the energy of the system changes by ΔE=1/2U, 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 introduce the well-known Thomas precession in the rest-frame of the electron.
Gianfranco Spavieri and Masud Mansuripur, "Origin of the spin-orbit interaction (Presentation Recording)," Proc. SPIE 9551, Spintronics VIII, 95510P (Presented at SPIE Nanoscience + Engineering: August 09, 2015; Published: 5 October 2015); https://doi.org/10.1117/12.2187490.4519370526001.
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