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Classical adiabatic angles and quantal adiabatic phase. (English) Zbl 0569.70020
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 \(n=\{n_ l\}\); the classical property is a shift \(\Delta \theta_ l(I)\) in the lth angle variable for motion round a phase-space torus with actions \(I=\{I_ l\}\); the connection is \(\Delta \theta_ l=-\partial \gamma /\partial n_ l.\) Two applications are worked out in detail: the generalised harmonic oscillator, with quadratic Hamiltonian whose parameters are the coefficients of \(q^ 2\), qp and \(p^ 2\); and the rotated rotator, consisting of a particle sliding freely round a non- circular hoop slowly turned round once in its own plane.

MSC:
70H15 Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics
81Q15 Perturbation theories for operators and differential equations in quantum theory
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