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Journal of Physics A Mathematical and General

A stochastic description of a spin-1/2 particle in a magnetic field

Gian Fabrizio De Angelis, Giovanni Jona-Lasinio

Published Jul 1, 1982• 34 citations
Publisher

Abstract

The authors develop the stochastic mechanics of a non-relativistic quantum particle with spin 1/2 in a possibly inhomogeneous magnetic field. They do not make any assumption about the inner structure of the particle, and treat spin components as discrete random variables.
DOI
10.1088/0305-4470/15/7/016

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Following the strategy of stochastic quantization, based on variational principles for processes taking values on a discrete configuration space, we give a detailed and self-contained description of the stochastic processes associated to a two-level quantum system, of interest for the physical description of spin-\textonehalf{} particles and Fermi fields. In particular we investigate the class of processes related to unperturbated quantum evolution and those simulating quantum-measurement procedures leading to mixture formation.
Recently the path integral quantization of the classical spin model was accomplished by Nielsen and Rohrlich. The configuration space of classical spin is the unit sphere with punctures at poles. Motivated by the fact that the Nelson's stochastic mechanics idea was to randomize the configuration space variable of the classical system, we give a review of the probabilistic approaches to the quantization of spin 1/2, with emphasis on the Dankel's problem of finding the probabilistic description of the Pauli equation in terms of stochastic diffusion processes. The original Dankel's analysis referred to the rigid top, and the problem of the point particle limit was left unsolved. Our observation is that if the spin stochastic process refers to the unit sphere with punctures, then Dankel's results provide a solution to the Pauli equation with no need of extra limiting procedures. The effects of the magnetic field can be successfully incorporated into the formalism.
The stochastic theory of quantum mechanics is further developed to include the problem of extended rigid particles, thus allowing the introduction of spin. It is demonstrated that the stochastic equations for the system's center of mass give rise to a generalized Schrödinger equation for integral or half-integral spin; in the particular case of spin ½, upon elimination of the internal variables, Pauli's equation is obtained. A formal simplified relativistic extension of the theory is worked out and shown to lead to Dirac's equation in the case of spin ½ and for a gyromagnetic ratio equal to 2; in the case of arbitrary spin, the theory gives an equation of the Feynman-Gell-Mann type.