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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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