Propagator for potential of an oscillator with quadratic and cubic terms

Abstract

In quantum mechanics, the propagator is basically the most important; It contains all physical properties, Feynman's path integration is widely accepted to be the formal method for finding the propagator. However, the method is limited to only some type of problems. One challenging and interesting problem which has not been able to be solved since the time of Feynman is potential with quadratic and cubic terms. The problem is analysed in detail by using Baker - Hausdorff lemma in order to find the possibility of the best approximation; It is found that the best practical one is the first-order semiclassical approximation by using condition the cubic potential term has value very smaller than the quadratic potential term. The approximated propagator has been obtained for quadratic and cubic potential terms when it is analogous to the propagator with quadratic potential term, we have increased prefactor and exponential terms due to the influence of cubic potential term which expresses the anharmonic part of the propagator. The approximated propagator is reduced to the well-known propagator for the simple harmonic oscillator when the anharmomc part is zero.