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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.

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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