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Fortschritte der Physik/Progress of Physics

Randomness in the Quantum Description of Neutral Spin 1/2 Particles

Piotr Garbaczewski

Published Jan 1, 1990• 6 citations
Publisher

Abstract

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.

DOI
10.1002/prop.2190380604

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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.
In stochastic mechanics, the Bopp-Haag-Dankel diffusions on R3×SO⁡(3){\mathbb {R}^3} \times \operatorname {SO} (3)R3×SO(3) are used to represent particles with spin. Bopp and Haag showed that in the limit as the particle’s moment of inertia III goes to zero, the solutions of the Bopp-Haag equations converge to that of the regular Pauli equation. Nelson has conjectured that in the same limit, the projections of the Bopp-Haag-Dankel diffusions onto R3{\mathbb {R}^3}R3 converge to a Markovian limit process. In this paper, we prove this conjecture for spin spin⁡  12\operatorname {spin} \;\tfrac {1} {2}spin21​ and regular potentials, and identify the limit process as the diffusion naturally associated with the solution to the regular Pauli equation.
Classical dynamics of rigid bodies The difficulties in formulating the theory of spin ½ systems outlined at the end of the last chapter may be avoided if we start from a classical theory of rotation in which the translational and orientational coordinates enter on a symmetrical footing, and pass to the quantum theory via the standard canonical quantization procedure. The result is not simply the Pauli theory written in a different matrix representation but a more general physical theory which explicitly incorporates in the wave equation the extended character of the bodies considered, through moment-of-inertia-dependent terms. As expected, the spin ½ character of the wave is manifested through the dependence of the body motion on the total quantum state via quantum potential and torque terms. Spin is not usually approached from this angle in basic quantum mechanics texts so it is worthwhile to build up the theory from scratch (for a more rigorous treatment see Normand (1980) and for further details: Bopp & Haag (1950), Rosen (1951, 1984), van Winter (1954), Dankel (1970), Holland (1988a, b)). We begin by reviewing some basic results in the classical canonical theory of rigid body motion. Our treatment is confined to a rigid homogeneous sphere with moment of inertia I about any axis passing through its centre, where the centre of mass is located. The sphere carries a rigid charge and a magnetic moment. By Chasle's theorem an arbitrary displacement of a rigid body is composed of a translation together with a rotation about a line fixed in the body.
A path integral for spinning particles is developed. It is a one-particle theory, equivalent to the usual quantum mechanics. Our method employs a classical model for spin which is quantized by path integration. The model, the spherical top, is a natural one from a group-theoretic point of view and has been used before in a similar context. The curvature and multiple connectedness of the top coordinate space [SO(3)\mathrm{SO}(3)SO(3)] provide some interesting features in the sum over paths. The Green's function which results from this procedure propagates all spins, and recovery of the usual Pauli spinors from this formalism is achieved by projection to a specific spin subspace.
Abstract The anomalous Zeeman effect made it clear that charged particles like the electron possess a magnetic dipole moment. Classically, this could be understood if the charged particle possesses an eigenrotation, that is, spin. Within Nelson's stochastic mechanics, it was shown that the model of a rotating charged ball is able to reproduce the well‐known spin expectation values. This classically motivated model of intrinsic rotation described in terms of a stochastic process is revisited here within the formalism of stochastic optimal control theory. Quantum Hamilton equations (QHE) for spinning particles are derived, which reduce to their classical counterpart in the zero quantum noise limit. These equations enable the calculation of the common spin expectation values without the use of the wave function. They also offer information on the orientation dynamics of the magnetic moment of charged particles beyond the behavior of the spin averages.