General Perturbation and Variational Methods for Responses to Time-Dependent Interactions in Quantum Mechanics

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Abstract
The perturbed energy of a system in the presence of a generally time-dependent external perturbation, in the limit where all transient effects have disappeared, but still in a range where eventual real transitions can be neglected, contains a term which is proportional to the so-called "response function" giving the response to the external field. This term can be identified and separated quite generally from the total energy perturbation, and the perturbed wave function can be written in the form of a product of a time-dependent amplitude and a time-dependent phase factor involving only this part of the perturbed energy. Using this expression for the wave function, we are able to develop a time-dependent perturbation formalism in which the response to a given perturbation is obtained from just that term of the energy identified above, rather than from the expectation value of the operator corresponding to the response to be determined. A well-known formal difficulty inherent in the Dirac method of variation of constants is avoided in this way. The present method enables us to construct a variational principle by introducing a certain time-averaged energy functional, whose stationary value is precisely the average of the energy term which depends on the response function only. It can therefore be used for variational derivations of approximations for responses to dynamical perturbations. In the case of a static perturbation, the present perturbation-variation formalism is equivalent to the ordinary time-independent one. These new techniques are illustrated by a detailed discussion of the polarization of free atoms or molecules in an oscillating electric field.