Geometric Phases, Symmetries of Dynamical Invariants, and Exact Solution of the Schr'odinger Equation

摘要

We introduce the notion of the geometrically equivalent quantum systems(GEQS) as quantum systems that lead to the same geometric phases for a givencomplete set of initial state vectors. We give a characterization of the GEQS.These systems have a common dynamical invariant, and their Hamiltonians andevolution operators are related by symmetry transformations of the invariant.If the invariant is $T$-periodic, the corresponding class of GEQS includes asystem with a $T$-periodic Hamiltonian. We apply our general results to studythe classes of GEQS that include a system with a cranked Hamiltonian$H(t)=e^{-iKt}H_0e^{iKt}$. We show that the cranking operator $K$ also belongsto this class. Hence, in spite of the fact that it is time-independent, itleads to nontrivial cyclic evolutions and geometric phases. Our analysis allowsfor an explicit construction of a complete set of nonstationary cyclic statesof any time-independent simple harmonic oscillator. The period of these cyclicstates is half the characteristic period of the oscillator.