Steady States and Quasienergies of a Quantum-Mechanical System in an Oscillating Field

A general formalism is presented for a system whose Hamiltonian is periodic in time. The formalism is intended to deal with the interactions between bond electrons and an external electromagnetic field, which can be treated semiclassically, such as electric and magnetic polarizations, optical rotation, and transitions among discrete levels. A particular bound-state solution of the Schr\"odinger equation which belongs to an irreducible representation of the time-translation symmetry group is defined as a steady state, and the characteristic number of the irreducible representation as a quasienergy. It is shown that the defined steady states and quasienergies behave in a newly constructed Hilbert space like stationary states and energies of a conservative system in many respects. It is also shown that for a resonant case the unperturbed quasienergy becomes degenerate and the transitions among discrete levels can be accounted for by the familiar degenerate perturbation procedure. Using a suitable Hilbert space, the steady states are established as firmly as the stationary states stand in the theory of a conservative system.