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

We introduce the notion of the geometrically equivalent quantum systems (GE QSs) 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 GE QSs. These systems have a common dynamical invariant, and their Hamiltonian s and evolution operators are related by symmetry transformations of the in variant. If the invariant is T-periodic, the corresponding class of GEQSs i ncludes a system with a T-periodic Hamiltonian. We apply our general result s to study the classes of GEQSs that include a system with a cranked Hamilt onian 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 independ ent, it leads to nontrivial cyclic evolutions and geometric phases. Our ana lysis allows for an explicit construction of a complete set of nonstationar y cyclic states of any time-independent simple harmonic oscillator. The per iod of these cyclic states is half the characteristic period of the oscilla tor.