Variational path-integral approach to optimizing the effective classical potentials

Abstract

The main purpose of this work is the presentation of the methods of approximating the effective classical potential for calculating an optimal quantum partition function of a quantum statistical mechanical system by path integrals. This shows the systematic development of the Feynman variational path integral as an approach to the exact result. We see that by means of the Feynman-Kleinert approximation, we can obtain a good result. Furthermore, this work also presents an improvement to Feynman-Kleinert method for increasing the accuracy of the approximation. These present methods make use of the Jensen-Peierls inequality and the cumulant expansion. The most interesting application is illustrated by application to the anharmonic oscillator with a potential V(x) = x²/2+gx[superscript 4]/4 where it yields a quite accurate approximation for any coupling constant even in the low temperature limit.