Classical adiabatic angles and quantal adiabatic phase

Abstract

A semiclassical connection is established between quantal and classical properties of a system whose Hamiltonian is slowly cycled by varying its parameters round a circuit. The quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n=(nl); the classical property is a shift Delta theta l(I) in the lth angle variable for motion round a phase-space torus with actions I=(Il); the connection is Delta theta l=- delta gamma / delta nl. Two applications are worked out in detail: the generalised harmonic oscillator, with quadratic Hamiltonian whose parameters are the coefficients of q2, qp and p2; and the rotated rotator, consisting of a particle sliding freely round a non-circular hoop slowly turned round once in its own plane.