A quantum-statistical Monte Carlo method; path integrals with boundary conditions

TLDR

The approach builds on Feynman/Wiener path integrals, modified to handle hard‑core boundary conditions. The paper introduces a Monte Carlo method for quantum‑statistical mechanics and outlines its extension to higher‑dimensional, many‑body, and realistic systems. It employs iterated short‑time Green’s functions with image approximations and is tested on one‑dimensional hard‑walled box problems. The method yields results that agree excellently with exact quantum mechanics across temperature ranges, from classical to ground‑state dominated regimes.

Abstract

A new Monte Carlo method for problems in quantum-statistical mechanics is described. The method is based on the use of iterated short-time Green’s functions, for which ’’image’’ approximations are used. It is similar to the use of Feynman or Wiener path integrals but with a modification to take account of hard-core boundary conditions. It is applied to two one-dimensional test problems: that of a single particle in a hard-walled box and that of two hard particles in a hard-walled box. For these test problems, the results are in excellent agreement with exact quantum-mechanical results both at high temperatures (near the classical limit) and at very low temperatures such that essentially only the ground state is occupied. Generalizations to three-dimensional systems, to many-body systems, and to more realistic potentials are discussed briefly.