  Quantum mechanical phases arising from a periodically varying Hamiltonian are
considered. These phases are derived from the eigenvalues of a stationary,
``dressed'' Hamiltonian that is able to treat internal atomic or molecular
structure in addition to the time variation. In the limit of an adiabatic time
variation, the usual Berry phase is recovered. For more rapid variation,
non-adiabatic corrections to the Berry phase are recovered in perturbation
theory, and their explicit dependence on internal structure emerges. Simple
demonstrations of this formalism are given, to particles containing interacting
spins, and to molecules in electric fields.
