On the Bohr radius relationship to spin-orbit interaction, spin magnitude, and Thomas precession

Abstract

The dynamics of the spin-orbit interaction in atomic hydrogen are studied in a classical electrodynamics-like setting. A Rutherfordian atomic model is used assuming a circular electron orbit, without the quantum principle as imposed arbitrarily in the Bohr model, but with an ad hoc incorporation in the electron of intrinsic spin and associated magnetic dipole moment. Analyzing the motions of the electron spin and orbital angular momenta, it is found that in the presence of Thomas precession, the total angular momentum averaged over the orbit is not generally a constant of the motion. It is noted this differs from the finding of Thomas in a similar assessment of 1927, and the reason for this difference is provided. It is found that although the total orbit-averaged angular momentum is not a constant of the motion, it precesses around a fixed axis similarly to the precession of the total angular momentum vector seen in spin-orbit coupling in quantum theory. The magnitude of the angular velocity of the total orbit-averaged angular momentum is seen to vanish only when the spin and orbital angular momenta are antiparallel and their mutual precession frequencies equate. It is then found, there is a unique radius where the mutual precession frequencies equate. Assuming the electron magnetic moment is the Bohr magneton, and an electron g-factor of two, this radius corresponds to where the orbital angular momentum is the reduced Planck's constant. The orbit radius for stationary total angular momentum for the circular orbit model with nonzero orbital angular momentum is thus the ground-state radius of the Bohr model.

Spin–orbit coupling and the conservation of angular momentum

“…Since Thomas's analysis, little attention seems to have been paid to the problem of angular momentum conservation in semiclassical models of hydrogen-like atoms. Recently, however, Lush has revisited this problem and, using a different approach to ours, concluded that not even the orbit-averaged value of l + s is conserved in this problem [25,26]. Lush attributes the reason for the different finding in [2] to Thomas's omission of the contribution of hidden momentum to the force on a magnetic dipole and suggests the Thomas precession as the source of the nonconservation.…”

Regarding Llewellyn Thomas's paper of 1927 and the "hidden momentum" of a magnetic dipole in an electric field

“…(Alternatively, the torque on the orbit could have been calculated in the electron rest frame based on the Biot-Savart force on the proton transiting the intrinsic magnetic field of the electron, obtaining the same result [23]. The torque must also be obtainable in the laboratory frame.…”