Abstract
We build the exact solution and the instantaneous diagonalization of a quantum system exhibiting a SU(2) or dynamical symmetry. A generalized displacement operator determines in both cases the temporal evolution and the diagonalization operators of the system. These operators can be used to characterize the exact and the adiabatical evolution of both periodical and cyclic states whose exact and adiabatical phases can be explicitly found. We consider two examples: a charged particle in the presence of a rotating magnetic field and the degenerate optical parametric oscillator. In particular we present and calculate nontrivial geometrical and dynamical phases for the optical parametric oscillator depending upon the coupling parameter that could indeed be measured using an adequate optical experiment.
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