2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/185974We 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 $H(t)=e^{-iKt}H_0e^{iKt}$. 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.Accepted for publication in J. Phys. AQuantum PhysicsHigh Energy Physics - TheoryMathematical Physicstext