Spin–orbit coupling and the conservation of angular momentum

Abstract

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.

Global dynamical analysis of an electronic spin–orbit coupling system

“…Suppose that a charged particle moves in a non-relativistic semi-classical coulomb field with mass m, charge q, spin s = a × b (In non-relativistic limit, the spin is parameterized in terms of a and b), and coulomb potential f. In the inertial frame, the Lagrangian formulation is written in the form [14]…”

“…They are the non-relativistic dynamical evolution model of the hydrogen atom with electron spin-orbit coupling, given in [14].…”

“…At once, a Lagrange formulation of the system is obtained in the nonrelativity limit, and the spin-evolution equations can be derived from the spin kinetic energy in the Lagrange function. It is shown in [14] that the total angular momentum in the Lagrange system is not conserved if the orbital angular momentum is defined as the kinetic angular momentum, but it is conserved when the canonical momentum including the general momentum and the hidden momentum.…”

“…Due to the above-mentioned points, a main motivation of this paper is to improve the work of [14], and to establish a complete canonical Hamiltonian formalism of a non-relativistic coulomb field with semi-classical electronic spin-orbit coupling in terms of the canonical spinning variables presented in [25]. In this way, the concepts of the hidden momentum and the hidden angular momentum are not necessary to be used, and the spin evolution equations are not necessary to be derived from the spin kinetic energy.…”

“…First, let us introduce a Lagrangian formulation of classical charged particles with spin-orbit coupling, which can be found in [14]. Then a Hamiltonian formulation is used instead of the Lagrangian one, and all conserved quantities of the Hamiltonian system are given.…”