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Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences

Classical geometric forces of reaction: an exactly solvable model

Michael Berry, Jonathan M. Robbins

Published Sep 8, 1993• 68 citations

Abstract

Abstract We illustrate the effects of the classical ‘magnetic’ and ‘electric’ geometric forces that enter into the adiabatic description of the slow motion of a heavy system coupled to a light one, beyond the Born–Oppenheimer approximation of simple averaging. When the fast system is a spin S and the slow system is a massive particle whose spatial position R is coupled to S with energy (fast hamiltonian) S · R, the magnetic force is that of a monopole of strength I (= adiabatic invariant S · R/R) centred at R = 0, and the electric force is inverse-cube repulsion with strength S2 – I2. Confining the slow particle to the surface of a sphere eliminates the Born–Oppenheimer and electric forces, and generates motion with precession and nutation exactly equivalent to that of a heavy symmetrical top. In the adiabatic limit the nutation is small and the averaged precession is precisely reproduced by the magnetic force. Alternatively, choosing the exactly conserved total angular momentum to vanish eliminates the Born–Oppenheimer and magnetic forces, and generates as exact orbits a one-parameter family of curly ‘antelope horns’ coiling in from infinity, reversing hand, and receding to infinity. In the adiabatic limit the repulsion of the ‘guiding centre’ of these coils is exactly reproduced by the electric force. A by-product of the ‘antelope horn’ analysis is a determination of the shape of a curve with a given curvature k and torsion T in terms of the evolution of a quantum 2-spinor driven by a planar ‘magnetic field’ with components k and T.

DOI: 10.1098/rspa.1993.0126

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Abstract We illustrate the effects of the classical ‘magnetic’ and ‘electric’ geometric forces that enter into the adiabatic description of the slow motion of a heavy system coupled to a light one, beyond the Born–Oppenheimer approximation of simple averaging. When the fast system is a spin S and the slow system is a massive particle whose spatial position R is coupled to S with energy (fast hamiltonian) S · R, the magnetic force is that of a monopole of strength I (= adiabatic invariant S · R/R) centred at R = 0, and the electric force is inverse-cube repulsion with strength S2 – I2. Confining the slow particle to the surface of a sphere eliminates the Born–Oppenheimer and electric forces, and generates motion with precession and nutation exactly equivalent to that of a heavy symmetrical top. In the adiabatic limit the nutation is small and the averaged precession is precisely reproduced by the magnetic force. Alternatively, choosing the exactly conserved total angular momentum to vanish eliminates the Born–Oppenheimer and magnetic forces, and generates as exact orbits a one-parameter family of curly ‘antelope horns’ coiling in from infinity, reversing hand, and receding to infinity. In the adiabatic limit the repulsion of the ‘guiding centre’ of these coils is exactly reproduced by the electric force. A by-product of the ‘antelope horn’ analysis is a determination of the shape of a curve with a given curvature k and torsion T in terms of the evolution of a quantum 2-spinor driven by a planar ‘magnetic field’ with components k and T.

Abstract We study the dynamics of a heavy (slow) classical system coupled, through its position, to a classical or quantal light (fast) system, and derive the first-order velocity-dependent corrections to the lowest adiabatic approximation for the reaction force on the slow system. If the fast dynamics is classical and chaotic, there are two such first-order forces, corresponding to the antisymmetric and symmetric parts of a tensor given by the time integral of the force–force correlation function of the fast motion for frozen slow coordinates. The antisymmetric part is geometric magnetism, in which the ‘magnetic field’ is the classical limit of the 2-form generating the quantum geometric phase. The symmetric part is deterministic friction, dissipating slow energy into the fast chaos; previously found by Wilkinson, this involves the same correlation function as governs the fluctuations and drift of the adiabatic invariant. In the ‘half-classical’ case where the fast system is quantal with a discrete spectrum of adiabatic states, the only first-order slow force is geometric magnetism; there is no friction. This discordance between classical and quantal fast motion is explained in terms of the clash between the semiclassical and adiabatic limits. A generalization of the classical case is given, where the slow velocity, as well as position, is coupled to the fast motion; to first order, the symplectic form in the lowest-order hamiltonian dynamics is modified.

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

ε

of the characteristic times. It is known that in the order

ε^0

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

ε^2

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

ε^3

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

ε^n

.

The adiabatic motion of a charged, spinning, quantum particle in a two-dimensional (i.e. of constant direction) magnetic field is studied. A suitable set of operators generalizing the kinematical momenta and the guiding centre operators of a particle moving in a homogeneous magnetic field is constructed. This allows us to separate the two degrees of freedom of the system into a fast and a slow one which are, in the classical limit, the rapid rotation of the particle around the guiding centre and the slow guiding centre drift. In terms of these operators the Hamiltonian of the system can be rewritten as a power series in the magnetic length , and the fast and slow dynamics separates. The effective guiding centre Hamiltonian is obtained to second order in the adiabatic parameter and reproduces correctly the classical limit.

Abstract It is known that, within the Born–Oppenheimer approximation, the slow modes of the nuclear motion are altered by three effects that emerge from integrating out the fast modes of the electronic motion. The first is an effective scalar potential V d y n coming from the eigenvalue of the electronic state, the second is an effective magnetic field coming from the Berry phase vector potential A . The third term is an additional potential V g e o m originating in the geometry of ray space and the Fubini–Study metric. In this article, we illustrate these effects and their geometric origin in the context of a simple toy model of a slow neutron interacting with a strong, spatially varying magnetic field. Regarding the neutron spin as a fast degree of freedom, we work out the slow dynamics of the motion of the neutron. Our treatment is geometrical and brings out the effects originating in the Kähler geometry of ray space and the Fubini–Study metric. We then give examples of magnetic field configurations which isolate these three separate effects. Finally we apply these ideas to the trapping of cold atoms. Our main result is that the geometric electric potential V g e o m dominates for smaller traps and can be used to confine cold atoms in static traps. This observation could result in better and smaller atomic clocks. This paper is dedicated to Michael Berry on his 80th birthday.