Semiclassical approach to perturbation theory

Abstract

The problem of constructing approximate solution methods and finding convenient recurrent algorithms for the study of bound states of quantum mechanical equations is one of the central problems of modern theoretical physics. In this paper an explicit semiclassical treatment of perturbation theory for non-relativistic bound states based on ℏ–­expansions and corresponding quantization conditions of one­dimensional problems is developed. The wave functions are chosen in the same way as in the logarithmic perturbation theory. Due to the introduction of new quantization conditions, this method made it possible to obtain recurrent relations, which are convenient for finding the energy eigenvalues and wave functions of bound states in the case when nodes of wave functions are taken into account. Avoiding the flaws of the standard approach, the obtained recurrent relations have the same form both for the ground and perturbed states.