A Variation-Perturbation Approach to the Interaction of Radiation with Atoms and Molecules

A variation-perturbation formulation for time-dependent interactions is presented and applied to the interaction between atoms or molecules and a semiclassical radiation field. Details of the treatment through second order are given for the steady-state problem and are compared with the standard perturbation-theory results. It is demonstrated that the usual expression for the time-dependent perturbation solution, which consists of a sum over an infinite set of excited states (including any continuum), can be replaced by a small number of frequency-dependent functions that satisfy certain time-independent differential equations. Since an exact solution to these equations is very difficult for all but the simplest cases, an equivalent variational formulation of more general utility is introduced. The induced time-dependent electric and magnetic moments of atoms and molecules in an oscillating electromagnetic field are evaluated in terms of the frequency-dependent functions. With the usual continuum medium assumptions, the macroscopic polarizations are determined from these moments, and simple formulas for the index of refraction and optical rotatory power are obtained. The method is illustrated by a simple application to the dynamic polarizability of the harmonic oscillator and by a more detailed examination of the dynamic polarizability and index of refraction of the nonrelativistic hydrogen atom. A comparison is made between an exact solution and a variational approximation for this problem. The excellent agreement between the two methods provides an indication of the power of the variational formulation. An indication of its utility for more general applications is given.