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arXiv:physics/9910034

On the form of Lorentz-Stern-Gerlach force

Sameen Ahmed Khan, M. Pusterla

Published Oct 22, 1999• 0 citations
arXivHTMLPDF

Abstract

In recent times there has been a renewed interest in the force experienced by a charged-particle with anomalous magnetic moment in the presence of external fields. In this paper we address the basic question of the force experienced by a spin-1/2 point-like charged-particle with magnetic and electric moments in the presence of space-and time-dependent external electromagnetic fields, when derived from the Dirac equation {\em via} the Foldy-Wouthuysen transformation technique. It is interesting to note that the force thus derived differs from the ones obtained by various other prescriptions.

DOI
10.48550/arxiv.physics/9910034

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In recent times there has been a renewed interest in the force experienced by a charged-particle with anomalous magnetic moment in the presence of external fields. In this paper we address the basic question of the force experienced by a spin-1/2 point-like charged-particle with magnetic and electric moments in the presence of space-and time-dependent external electromagnetic fields, when derived from the Dirac equation {\em via} the Foldy-Wouthuysen transformation technique. It is interesting to note that the force thus derived differs from the ones obtained by various other prescriptions.
A relativistic spin operator cannot be uniquely defined within relativistic quantum mechanics. Previously, different proper relativistic spin operators have been proposed, such as spin operators of the Foldy-Wouthuysen and Pryce type, that both commute with the free-particle Dirac Hamiltonian and represent constants of motion. Here we consider the dynamics of a relativistic electron spin in an external electromagnetic field. We use two different Hamiltonians to derive the corresponding spin dynamics. These two are: (a) the Dirac Hamiltonian in the presence of an external field, and (b) the semirelativistic expansion of the same. Considering the Foldy-Wouthuysen and Pryce spin operators we show that these lead to different spin dynamics in an external electromagnetic field, which offers possibilities to distinguish their action. We find that the dynamics of both spin operators involve spin-dependent and spin-independent terms, however, the Foldy-Wouthuysen spin dynamics additionally accounts for the relativistic particle-antiparticle coupling. We conclude that the Pryce spin operator provides a suitable description of the relativistic spin dynamics in a weak-to-intermediate external field, whereas the Foldy-Wouthuysen spin operator is more suitable in the strong field regime.
The Foldy-Wouthuysen transformation for relativistic spin-1 particles interacting with nonuniform electric and uniform magnetic fields is performed. The Hamilton operator in the Foldy-Wouthuysen representation is determined. It agrees with the Lagrangian obtained by Pomeransky, Khriplovich, and Sen'kov. The classical and quantum formulae for the Hamiltonian agree. The validity of the Corben-Schwinger equations is confirmed. However, it is difficult to generalize these equations in order to take into account the quadrupole moment defined by a particle charge distribution. The known second-order wave equations are not quite satisfactory because they contain non-Hermitian terms. The Hermitian second-order wave equation is derived.