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 1. Introduction Consider a quantal or classical system with N freedoms, whose Hamiltonian H (q , p ; X (t)) depends on a set of slowly changing parameters X = { X w } as well as dynamical variables or operators q = { q,}, p = {p,} (i S j Q N). The evolution of the system is governed by an adiabatic theorem. In the quantal case (Messiah 1962), this states that a system originally in an eigenstate, labelled by one or more parameters n = {n,}, will1 remain in the same eigenstate In; X (t)) , with energy E , (X (t)) , as the X vary. In the classical case (Dirac 1925), the theorem states that an orbit initially on an N-dimensional phase-space torus with actions I = {I,} (Arnold 1978) will continue to explore the tori with the same values of I (adiabatic invariants), in spite of the changing Hamiltonian corresponding to X (t) , provided such tori continue to exist (for example if the system remains integrable for all parameters X). These well known adiabatic theorems fail to describe an important feature of the evolution, which manifests itself if the Hamiltonian returns to its original form after a (long) time T, i.e. X (T) = X (0). We shall describe such changes as taking the system round a circuit C in the space of parameters X. Quantally, the feature is a geometricaIphasefactor exp(i y,(C)) accumulated round C by a system in the nth state: if the state is initially lUr(O)), then the state at T is (The second factor contains the familiar dynamical phase, and is present even if the parameters …