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Physical Review

Classical Motions of Spin-½ Particles

Kenneth Rafanelli, Ralph Schiller

Published Jul 13, 1964• 54 citations
Publisher

Abstract

From the WKB solutions to the squared Dirac equation we derive the classical trajectories and spin precessions originally postulated by Bargmann, Michel, and Telegdi for charged spin-\textonehalf{} particles. Identical equations of motion are obtained from the WKB solutions to the Dirac equation.
DOI
10.1103/physrev.135.b279

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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 functional form of the wave function with respect to variables appropriate for the WKB approximation is chosen by taking into account the transformation properties of the spinor wave-function.This representation makes it possible, using only the wave equation, to pursue directly the time changes in the variables, i.e. the spin direction, velocity and density, etc., characterizing an electron.These changes correspond straightforwardly to a classical model of the non-relativistic Pauli electron with the proper magnetic moment, and the covariant form of this model, which is just "the Frenkel-Kramers model of the Dirac electron" introduced recently by the author.In this derivation, the existence of freedom of • rotation around the .spinaxis and certain definite angular velocity of this rotation are required.These results and the correspondence obtained may clarify the classical-mechanical significance of the spinor wave-function and also of the spin itself.An explanation is appended of the hyperbolic functions of the velocity metric, which help in the complicated calculations for the Dirac electron. § I. Introduction*) The main part of this paper appeared previously in Rep. Seisanken.No. 22 (1957), 55, in Japanese.
Abstract. We study the Dirac equation with slowly varying external potentials. Using matrix-valued Wigner functions we prove that the electron follows with high precision the classical orbit and that the spin precesses The high precision experiments of the anomalous magnetic moment of the electron or the muon use in their interpretation the BMT equation for spin precession [1]. The derivation of Bargmann, Michel, and Telegdi [2] does not even mention quantum mechanics, however. Rather their arguments are