An alternative to the magnus expansion in time-dependent perturbation theory

Abstract

An alternative to the Magnus expansion in time-dependent quantum mechanical perturbation theory is presented. An exponential form of the time-development operator, given as the exponential of a sum of products of integrals well known in time-dependent quantum mechanics, is derived from the standard perturbation expansion. The derivation is simple and the form of the terms in the exponential is so straightforward that the Nth order term can be written by inspection. The standard Magnus terms can be obtained by rearrangement of the terms of the new expansion probably easier than they can be obtained by iteration of the original equations. A proof by induction is presented to establish the form of the Nth term and it is shown how the new expansion terms relate to the Magnus terms. Some apparent differences between Magnus terms in the literature are resolved.