Spin-orbit coupling and the conservation of angular momentum

TL;DR

This paper explores the conservation of total angular momentum in quantum mechanics with spin-orbit coupling, highlighting differences between classical and quantum treatments and emphasizing the role of canonical momentum.

Contribution

It clarifies how total angular momentum conservation depends on the definition of orbital angular momentum in classical versus quantum frameworks.

Findings

Quantum total angular momentum is conserved with spin-orbit coupling.

Classical orbital angular momentum defined via kinetic momentum is not conserved.

Using canonical momentum, classical total angular momentum is conserved.

Abstract

In nonrelativistic quantum mechanics, the total (i.e. orbital plus spin) angular momentum of a charged particle with spin that moves in a Coulomb plus spin-orbit-coupling potential is conserved. In a classical nonrelativistic treatment of this problem, in which the Lagrange equations determine the orbital motion and the Thomas equation yields the rate of change of the spin, the particle's total angular momentum in which the orbital angular momentum is defined in terms of the kinetic momentum is generally not conserved. However, a generalized total angular momentum, in which the orbital part is defined in terms of the canonical momentum, is conserved. This illustrates the fact that the quantum-mechanical operator of momentum corresponds to the canonical momentum of classical mechanics.