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Spin-orbital motion and Thomas precession in the classical and quantum theories

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

The motion of a magnetic spin particle in electromagnetic fields is considered on the basis of general principles of the classical relativistic theory. Alternative approaches in derivation of the equations of charge motion and spin precession, the problem of noncollinearity of the momentum and velocity of a particle with spin, the origin and the meaning of Thomas precession in dynamics of the spin particle are also considered. The correspondence principle in the spin theory is discussed. 1

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The motion of a magnetic spin particle in electromagnetic fields is considered on the basis of general principles of the classical relativistic theory. Alternative approaches in derivation of the equations of charge motion and spin precession, the problem of noncollinearity of the momentum and velocity of a particle with spin, the origin and the meaning of Thomas precession in dynamics of the spin particle are also considered. The correspondence principle in the spin theory is discussed. 1
This is the first of a series of papers developing the classical theory of a spinning particle. The equations of motion will be derived from a Lagrangian, and solutions for the classical trajectory and spin precession in external fields will be given. In this paper an abstract spin vector is introduced to characterize the spin of a classical particle. Lagrangians for the classical trajectories and for the motion of the abstract spin vector are derived from corresponding quantum-mechanical Lagrangians by the WKBJ approximation method for nonrelativistic and relativistic particles. The equations of motion for the trajectory and the abstract spin vector following from the extremalization of these Lagrangians are given. The equations of motion for the precession in an external electromagnetic field of the spin vector (or tensor) in space–time is derived from the equations of motion for the abstract spin vector. In the relativistic case, they are equivalent to the Bargmann–Michel–Telegdi equations [Phys. Rev. Lett. 2, 435 (1959)]. The relationship between the ensemble and single-particle points of view is also elucidated.
The classical equations of motion of a charged point-particle with intrinsic spin under the influence of an external electromagnetic field are restated and compared with the Heisenberg equations of motion derived from the Dirac theory. The partition of angular momentum between particle and field in the classical theory is contrasted to the Dirac theory of electron spin. The analogy between the Dirac equation and the theory of parametric amplification is pointed out.A free spinning point particle moving according to the laws of classical relativistic point-particle mechanics may move along a helix. The sum of the intrinsic spin $\ensuremath{\sigma}$ and the angular momentum of the helical motion in this classical analog of zitterbewegung is an effective spin vector S which is a constant of the motion. Because of this internal motion, the effective mass MMM of the particle differs from the mass mmm which is ascribed to it in the equations of motion. Solutions are found in which S is parallel or antiparallel to the momentum, and the sign of MMM is determined by the helicity. When placed in a uniform electromagnetic field, the particle behaves as if it had a rest mass MMM and a magnetic moment $\frac{e\ensuremath{\sigma}}{\mathrm{Mc}}$, in addition to any explicit magnetic moment that may be ascribed to it.
First, the experimental foundations for the existence of a new dynamical variable, the spin of particles, are presented. The Pauli equation is then derived in detail, two applications are given and the energy levels of a particle with spin in a constant magnetic field are studied. This is the analysis of Landau levels, which can be performed by using the known results on the spectrum of harmonic oscillators. In this chapter we also study in detail the addition of orbital angular momentum and spin, as well as the spin–orbit interaction with the resulting Thomas precession. Stern-Gerlach experiment and electron spin The hypothesis that the electron has an intrinsic magnetic moment and an angular momentum, in short a spin , was first suggested in Uhlenbeck and Goudsmit (1926). They noticed, even before the discovery of quantum mechanics, that a complete description of spectra was not possible unless a magnetic moment and a mechanical moment were ascribed to the electron, and hence the concept of an electron as a point charge was insufficient. First, let us therefore try to understand how a magnetic moment can be associated to an atomic system. For this purpose, consider for simplicity the Bohr model of an hydrogen atom, where the electron moves along a circular orbit and rotates with orbital angular momentum L .
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.