A method of deriving exact and approximate solutions for time-evolution problems of quantum mechanics is discussed. The cyclic evolution of quantum systems for periodic in time Hamiltonians is studied. A class of periodic time-dependent Hamiltonians with cyclic solutions is constructed in a closed analytic form. The periodic time-dependent Hamiltonians are generated whose expectation values for cyclic solutions do not depend on time. It is shown, the spin-expectation values and probability density in a given point of space-time are not dependent on time, too. As a consequence, this approach can be used for modelling quantum dynamic wells and wires with the effect of the particle localization. Nonadiabatic geometric phases are calculated in terms of obtained cyclic solutions. A time-dependent periodic Hamiltonian admitting exact solutions is applied to construct a set of universal gates for quantum computer. A way of obtaining entanglement operators is discussed, too.

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