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Physical Review

Motion of a Dipole-Quadrupole System

Ralph Schiller

Published May 25, 1968• 4 citations
Publisher

Abstract

Niels Bohr showed that the magnetic moment of an electron cannot be determined by a Stern-Gerlach experiment. His argument is extended to other moments, and general conditions are stated as to when a moment determination is feasible by such an experiment. In addition, the expectation-value equations are given for a stationary, gyrating, nonrelativistic, spin-1 particle in arbitrary, but classical, external electro-magnetic fields. For such a dipole-quadrupole system there is no distinction between the classical and quantal descriptions. Of interest is the fact that $\mathrm{SU}(3)$ is the invariance group of the motion. The theory is extended to the relativistic domain by Lorentz transformation. The group $\mathrm{SU}(3,1)$ appears as the invariance group of the internal motion.

DOI
10.1103/physrev.169.1282

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Niels Bohr showed that the magnetic moment of an electron cannot be determined by a Stern-Gerlach experiment. His argument is extended to other moments, and general conditions are stated as to when a moment determination is feasible by such an experiment. In addition, the expectation-value equations are given for a stationary, gyrating, nonrelativistic, spin-1 particle in arbitrary, but classical, external electro-magnetic fields. For such a dipole-quadrupole system there is no distinction between the classical and quantal descriptions. Of interest is the fact that SU(3)\mathrm{SU}(3)SU(3) is the invariance group of the motion. The theory is extended to the relativistic domain by Lorentz transformation. The group SU(3,1)\mathrm{SU}(3,1)SU(3,1) appears as the invariance group of the internal motion.
In the late 1920s Niels Bohr propagated the idea that the magnetic moment of a free electron could not be observed. This derived from the idea that the spin degree of freedom characterized the electron only when it is bound in an atom. This view initiated a lively discussion, which involved many of the most prominent theoreticians of the time. The independent existence of the electron spin became an issue of principle. In particular it was deemed that quantum effects would destroy the separated classical trajectories in a Stern-- Gerlach type of experiment. We review these discussions and some later developments. Quantum effects do prove to be essential, but they do not overwhelm the magnetic effects of spin. In addition to these arguments, it has been possible experimentally to determine the electron g -factor with high accuracy in electromagnetic traps. In fact no principle seems to prevent the observation of the magnetic moment of the free electron.
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.
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 motion of neutral particles with magnetic moments in an inhomogeneous magnetic field is described in a semiclassical framework. The concept of coherent internal states is used in the formulation of the semiclassical approximation from the full quantum-mechanical expression. The classical trajectories are defined only for certain spin states, that satisfy the conditions for being coherent internal states. The reliability of Stern-Gerlach experiments to measure spin projections is assessed in this framework.
Abstract We propose a phenomenological theory of spin behavior in a magnetic field, which explains from the point of view of classical physics the two-valued result of the Stern-Gerlach experiment. The behavior of the spin and intrinsic magnetic moment of an electron wave of an atom in an external magnetic field is considered. We show that in a weak magnetic field, the intrinsic magnetic moment of an electron wave is always oriented parallel to the magnetic field strength vector, while in a strong magnetic field, depending on the initial orientation of the intrinsic magnetic moment, two orientations are realized: either parallel or antiparallel to the magnetic field strength vector. Within the framework of classical electrodynamics, the calculation of the motion of an atomic beam in an inhomogeneous magnetic field is carried out, which reproduces the results of the Stern-Gerlach experiment.
The behavior of an atom in a nonuniform magnetic field is analyzed, as well as the motion of a classical magnetic dipole (a spinning charged ball) and a rotating charged ring. For the atom it is shown that, while the magnetic field does no work on the electron-orbital contribution to the magnetic moment (the source of translational kinetic energy being the internal energy of the atom), whether or not it does work on the electron-spin contribution to the magnetic moment depends on whether the electron has an intrinsic rotational kinetic energy associated with its spin. A rotational kinetic energy for the electron is shown to be consistent with the Dirac equation. If the electron does have a rotational kinetic energy, the acceleration of a silver atom in a Stern-Gerlach experiment or the emission of a photon from an electron spin flip can be explained without requiring the magnetic field to do work. For a constant magnetic field gradient along the z axis, it is found that the classical objects oscillate in simple harmonic motion along the z axis, the total kinetic energy -- translational plus rotational -- being a constant of the motion. For the charged ball, the change in rotational kinetic energy is associated only with a change in the precession frequency, the rotation rate about the figure axis remaining constant.