Geometric Phases, Symmetries of Dynamical Invariants, and Exact Solution of the Schrödinger Equation

Published Jan 1, 2001• 0 citations

Abstract

We introduce the notion of the geometrically equivalent quantum systems (GEQS) as quantum systems that lead to the same geometric phases for a given complete set of initial state vectors. We give a characterization of the GEQS. These systems have a common dynamical invariant, and their Hamiltonians and evolution operators are related by symmetry transformations of the invariant. If the invariant is T-periodic, the corresponding class of GEQS includes a system with a T-periodic Hamiltonian. We apply our general results to study the classes of GEQS that include a system with a cranked Hamiltonian H(t) = e −iKt H0e iKt. We show that the cranking operator K also belongs to this class. Hence, in spite of the fact that it is time-independent, it leads to nontrivial cyclic evolutions and geometric phases. Our analysis allows for an explicit construction of a complete set of nonstationary cyclic states of any time-independent simple harmonic oscillator. The period of these cyclic states is half the characteristic period of the oscillator. PACS numbers: 03.65.Bz 1

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