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Joint and angle-covariant spin measurements with a quadrupole magnetic field

Hans Martens, Willem M. de Muynck

Published May 1, 1994• 0 citations
OpenAlex

Abstract

We study a Stern-Gerlach type setup, with a quadrupole magnetic field, for neutral particles of arbitrary spin. The Hamiltonian is of a form proposed for joint measurements of the incompatible observables. The measurement results are discussed, showing the limitation of such Hamiltonians. Some remarks are made on the relevance of covariance as a criterion for measurement schemes.

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We study a Stern-Gerlach type setup, with a quadrupole magnetic field, for neutral particles of arbitrary spin. The Hamiltonian is of a form proposed for joint measurements of the incompatible observables. The measurement results are discussed, showing the limitation of such Hamiltonians. Some remarks are made on the relevance of covariance as a criterion for measurement schemes.
The measurement of spin-1/2 observables using a Stern-Gerlach type device is studied. A magnetic field with a dominant dipole component leads to a measurement of spin in one direction. The measurement is shown to be feasible (in principle) for electrons as well as for neutral particles. A quadrupole field leads to a joint measurement of two incompatible spin observables. Again, both the electron and the neutral case are presented. The accuracy of the joint measurement is compared to the limit imposed by the uncertainty principle.
We present a state estimation scheme for spins using a modified setup of the Stern-Gerlach experiment in which a beam of neutral spin-1/2 point particles interacts with a quadrupolar magnetic field. The proposed estimation procedures, based either on a quadrant or a continuous intensity distribution detection, require a suitable initial spatial state of the beam. The statistical characterization of the estimators of the initial spin state allows us not only to associate an error to the estimated parameters, but also to define a measure for comparing estimation procedures corresponding to different Stern-Gerlach setups.
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 quantum mechanical framework. The validity of the semiclassical approximations which are generally used to describe these phenomena is discussed. Approximate expressions for the evolution operator are derived and compared to the exact calculations. Focusing and spin-flip phenomena are predicted. The reliability of Stern-Gerlach experiments to measure spin projections is assessed in this framework.
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.