ABSTRACT
A small parameter is used to identify the orders of perturbation. The unperturbed Hamiltonian is assumed time-independent and can be diagonalized. The interaction of an electron with an applied electromagnetic field is governed by the gauge-invariant Hamiltonian. If both the initial and final energy levels are discretized, the delta function indicates that the transition happens only when the energy difference between the two levels matches the frequency of the perturbation. The delta function suggests that the absorption or stimulated emission happens only when the energy difference between the final and initial states matches precisely the frequency of the applied electromagnetic wave. The quantum transition induced by a monochromatic perturbation is determined by the matrix element between the initial and final states.
