Geometric phases, symmetries of dynamical invariants and exact solution of the Schrodinger equation

Abstract

We introduce the notion of the geometrically equivalent quantum systems (GEQSs) 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 GEQSs. 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 GEQSs includes a system with a T-periodic Hamiltonian. We apply our general results to study the classes of GEQSs that include a system with a cranked Hamiltonian H (t) = e(-iKt)H(0)e(iKt). We show that die 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.