Quantum mechanics of systems periodic in time

Some expressions for the time evolution of quantum-mechanical systems with Hamiltonians periodic in time, derivable from the work of Shirley and applied by Young, Deal, and Kestner and Haeberlen and Waugh---all for finite-basis-set systems---are derived for a general system (possibly infinite Hilbert space). These results suggest a new type of approximation to the time-evolution operator, one which is exact at multiples of the period of the Hamiltonian. Comparison is made to an exactly soluble problem, namely, a nonrelativistic hydrogen atom in a circularly polarized monochromatic field.