Evaluation of Feynman path integrals by Monte Carlo methods

Abstract

Details of a project suitable for an advanced undergraduate course in computational physics, involving the evaluation of functional integrals, are presented. The square of the ground state wavefunction for a simple system may be formulated as a Feynman path integral, and such path integrals evaluated using Monte Carlo methods on a microcomputer. Application to a system interacting through a Morse potential or through a simple harmonic potential is described, while the possible employment of over-relaxation in the latter case is explained.

Review of computer simulations of isotope effects on biochemical reactions: From the Bigeleisen equation to Feynman's path integral

“…[15][16][17][18][33][34][35][36] Our ab initio path-integral method can be exactly reduced to the Bigeleisen equation in EIE and KIE calculations, and can also methodically go beyond this equation, e.g., by systematically including (non-parabolic) quantum tunneling effects, as well as anharmonic corrections to harmonic zero-point and vibrational energies. Further, in contrast to the commonly-used path-integral Monte Carlo (PIMC) [37][38][39][40][41][42] and molecular dynamics (PIMD) [43][44][45] simulations, calculated values using our ab initio path-integral method can be as precise as (though not as accurate as) the numerical precision of the computing machine. Illustrations of our ab initio path-integral method by some applications on reactions in solution and in protein enzymes, together with some other biologically relevant reactions, will follow.…”

Review of Feynman’s Path Integral in Quantum Statistics: from the Molecular Schrödinger Equation to Kleinert’s Variational Perturbation Theory

“…Similar to the complementary interplay between the rapidly growing quantum Monte Carlo simulations [146][147][148][149] and the well-established ab initio or density-functional theories (DFT) for electronic structure calculations [4,5,[25][26][27]29], non-sampling/non-stochastic pathintegral methods can complement the conventional Fourier or discretized path-integral Monte-Carlo (PIMC) [131,136,[139][140][141] and molecular dynamics (PIMD) [87,88] [e.g., Eqs. (2.46) and (2.47)] simulations which have been widely used in condensed phases.…”

“…The success of path-integral calculations in various disciplines of science is partially owing to the emergence of Monte Carlo (MC) simulations, which started being widely used at about the same time around the birth of path integrals [87,88,[131][132][133][134][135][136][137][138][139][140][141][142][143][144][145]. In practice, each path in space-time is conventionally either discretized into a set of virtual 'beads' on a sliced time axis or represented in Fourier space.…”

The Feynman Path Goes Monte Carlo

“…By the mid-eighties, path integral simulations of simple quantum mechanical problems had become both conceptually and technically "easy." Indeed, the exposition by Creutz and Freedman was already written in an introductory, didactic manner, and in 1985 the simulation of the one-particle harmonic oscillator was explicitly proposed as an undergraduate project, to be handled on a Commodore CBM3032 microcomputer, in a paper published in the American Journal of Physics [19].…”