ON GEOMETRIC PHASES AND DYNAMICAL INVARIANTS

A formalism is developed for calculating Berry phases for non-adiabati c time-periodic quantum systems when a dynamical invariant is known. I t is found that, when the invariant is periodic and has a non-degenera te spectrum, this method allows a convenient way to obtain generalized Berry phases and the proper cyclic initial states. The method is appl ied to the generalized harmonic oscillator and the two-level system, w here the invariant operators are explicitly constructed. Formulae for the conventional Berry's phases are readily obtained by taking the adi abatic limit of the exact results.