The Path-Integral Simulation of Quantum Systems

Abstract

We explain how Feynman’s path-integral formulation of quantum statistical mechanics gives a natural way of simulating quantum systems in thermal equilibrium. The theory is first outlined for a simple system, consisting of a single particle in one dimension acted on by an external potential. Starting from the standard basic expressions for the partition function and the thermal averages of observables, it is shown how a simple sequence of mathematical operations allows these expressions to be brought into Feynman’s path-integral form. In this form, the quantities of interest are expressed as an integral over cyclic paths of the particle, which can be evaluated by classical simulation methods. This approach, generalized to many particles in three dimensions, gives a simulation technique for quantum many-body systems. We discuss how to calculate some important observables such as the energy and the radial distribution function. In the application of path-integral techniques to systems like liquid helium, the inclusion of quantum exchange is crucial, and we indicate how this can be achieved. We then illustrate the use of the technique by describing simulations that have been performed on (i) an electron dissolved in a molten salt; (ii) hydrogen in metals; and (iii) liquid helium-four.