Classical adiabatic angles and quantal adiabatic phase

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 y,, associated with an eigenstate with quantum numbers n = {n,}: the classical property is a shift A@, ( I) in the Ith angle variable for motion round a phase-space torus with actions I ={I,}; the connection is At?, =-icy/dn,. Two applications are worked out in detail: the generalised harmonic oscillator, with quadratic Hamiltonian whose parameters are the coefficients of q2, qp and p'; and the rotated rotator, consisting of a particle sliding freely round a non-circular hoop slowly turned round once in its own plane.