Calculation of equilibrium averages with Feynman-Hibbs effective classical potentials and similar variational approximations

A number of years ago, Feynman and Hibbs derived an insightful expression for a quasiclassical effective potential in order to estimate the free energy for a quantum-mechanical system that obeys Boltzmann statistics. In the present paper, it is demonstrated how these ideas can be extended to calculate an approximate equilibrium average of any general coordinate-dependent operator. The central feature of this analysis is the replacement of the operator by an effective Gaussian-averaged quasiclassical function.