Classical geometric forces of reaction: an exactly solvable model

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 S 2 – I 2 . 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 .