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AIP conference proceedings

The Relativistic Stern-Gerlach Interaction and Quantum Mechanics Implications

P. Cameron

Published Jan 1, 2003• 2 citations
Publisher

Abstract

The time varying relativistic Stern‐Gerlach force, which acts over a charged particle endowed with a magnetic moment, is deduced from the Dirac Hamiltonian finding its coincidence with the classical expression. Possible drawbacks related to the Heisenberg uncertainty principle are discussed.
DOI
10.1063/1.1607242

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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.
We have studied a relativistically covariant Stern-Gerlach (SG) experiment for a relativistic spin and a Dirac spin. We have obtained the relativistic spin in an arbitrary frame by using the classical spin dipole tensor, which gives the covariant spin dipole interactions, and the relation between a spin and a spin magnetic dipole moment. The relativistic spin is shown to have problems to become a proper spin operator for a massive relativistic particle because of two reasons. First, the relativistic spin three-vector operators cannot satisfy the spin algebra. Second, the SG experiment for the relativistic particle provides a paradox between two observers in the particle rest frame and the laboratory frame, in which the particle is moving. We have shown that the paradox in the SG experiment is resolved by the Dirac spin, which is covariantly defined by a Lorentz transformation in the Dirac spinor representation. The Dirac spin three-vector operators satisfy the spin algebra. It is shown that the SG experiment for the Dirac spin in the inertial frame, where there is only magnetic field, can determine the spin without the information of the momentum of the particle. This shows that the reduced spin density matrix for the Dirac particle can be well-defined by integrating out the momentum degrees of freedom.
The conflict between Bohr's assertion that the magnetic moment of the electron cannot be measured with experiments based on the concept of classical trajectories, and the measurement of the magnetic moment of electrons in a modified Penning trap by Dehmelt et al. has led us to reevaluate other implications of Bohr's assertion. We show that, contrary to the analysis of Bohr and Pauli, the assumption of classical trajectories in a Stern-Gerlach--like device can result in a high degree of spin separation for an electron beam. This effect may persist within a fully quantum-mechanical analysis. The magnetic fields considered are such that a tabletop Stern-Gerlach electron spin filter is feasible.