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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Wave mechanics and inertial guidance

I. M. Benn, R.W. Tucker

Published Mar 15, 1989• 28 citations
Publisher

Abstract

The adiabatic classical dynamics of certain constrained nonautonomous systems is discussed in terms of the transport of vectors on a surface in parameter space. The classical analogue of Berry's phase is related to the integrated geodesic curvature of a curve on such a surface and the motion of the system analyzed in terms of the relative rotation rates of different vector fields. The classical dependence of the motion on the topology of the parameter space motivates a particular quantum description of a particle confined to a closed space tube.

DOI
10.1103/physrevd.39.1594

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arXiv:0905.1596•May 11, 2009•1 citation
The adiabatic classical dynamics of certain constrained nonautonomous systems is discussed in terms of the transport of vectors on a surface in parameter space. The classical analogue of Berry's phase is related to the integrated geodesic curvature of a curve on such a surface and the motion of the system analyzed in terms of the relative rotation rates of different vector fields. The classical dependence of the motion on the topology of the parameter space motivates a particular quantum description of a particle confined to a closed space tube.
We consider particles constrained to move in the close vicinity of a space curve through a steep quadratic potential in the plane normal to the curve. As is known, the effective 1D Hamiltonian that governs the motion along the curve involves both its curvature and torsion. Thus, the adiabatic cyclic deformations of the curve might give rise to geometric phases being accumulated by the particle. We report that this is indeed, in general, the case, and analyze in detail the origin of the phases, as described in both an inertial Cartesian frame and a noninertial one, adapted to the curve. We extend previous work to include non-locally arclength preserving deformations, and derive general formulas for Berry’s curvature in this case.
When quantum mechanical motion of a particle constrained to a curved surface in the Euclidean space is considered, three different approaches may be adopted to describe the system. One is the confining approach which enforces a particle to stay in the surface by a physical confining potential. Both of the others follow Dirac's general prescription to treat a constrained system, but employ different constraints, namely the usual and the conservative constraints, respectively. The three approaches lead to results which do not in general agree with each other, though they give the same classical equation of motion. It is emphasized that the arbitrariness in the ordering of operators cannot be avoided when the latter two approaches are adopted. In the approach with the conservative constraint, however, there is a physical ordering, which leads to the result in agreement with that of the confining approach. We interpret these discrepancy and agreement in simple physical terms.
We are accustomed to think the phase of single particle states does not matter. After all, the phase cancels out when calculating physical observables. However, the geometric phase can cause interference even in single particle states and can be measured. Berry’s phase is a geometric phase the system accumulates as its time-dependent Hamiltonian is subjected to closed adiabatic excursion in parameter space. In this report, we explore how Berry’s phase manifests itself in various fields of physics, both classical and quantum mechanical. The Hannay angle is a classical analogue to Berry’s phase and they are related by a derivative. The Aharonov-Bohm effect is a manifestation of Berry’s phase. Net rotation of deformable bodies in the language of gauge theory can be translated as a Berry phase. The well-known BornOppenheimer approximation is a molecular Aharonov-Bohm effect and is another manifestation of Berry’s Phase.
We study a slow classical system [particle] coupled to a fast quantum system with discrete energy spectrum. We adiabatically exclude the quantum system and construct an autonomous dynamics for the classical particle in successive orders of the small ratio εεε of the characteristic times. It is known that in the order ε0ε^0ε0 the particle gets an additional [Born-Oppenheimer] potential, while in the order $\eps^1$ it feels an effective magnetic field related to the Berry phase. In the order ε2ε^2ε2 the motion of the classical particle can be reduced to a free [geodesic] motion on a curved Riemannian manifold, with the metric generated by the excluded quantum system. This motion has a number of unusual features, e.g., it combines subspaces of different (Riemannian and pseudo-Riemannian) signature for the metric tensor. In the order ε3ε^3ε3 the motion of the classical particle is still described by a Lagrangian, but the latter linearly depends on the particle's acceleration. This implies the existence of a spin tensor [non-orbital angular momentum] for the particle. This spin tensor is related to the momentum via an analogue of the zitterbewegung effect. The Hamiltonian structure of the system is non-trivial and is defined via non-linear Poisson brackets. The linear dependence of the effective classical Lagrangian on higher-order derivatives is seen as well in the higher orders εnε^nεn.