Asymptotic properties of path integral Monte Carlo calculations (original) (raw)
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American Journal of Physics, 2018
We give an introduction to the calculation of path integrals on a lattice, with the quantum harmonic oscillator as an example. In addition to providing an explicit computational setup and corresponding pseudocode, we pay particular attention to the existence of autocorrelations and the calculation of reliable errors. The over-relaxation technique is presented as a way to counter strong autocorrelations. The simulation methods can be extended to compute observables for path integrals in other settings.
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Journal of Physics: Conference Series, 2008
A recently developed method systematically improved the convergence of generic path integrals for transition amplitudes, partition functions, expectation values and energy spectra. This was achieved by analytically constructing a hierarchy of discretized effective actions indexed by a level number p and converging to the continuum limit as 1/N p. Here we apply the above general method to numerical calculations using Metropolis Monte Carlo simulations of energy expectation values and energy spectra. We analyze and compare the ensuing increase in efficiency of several orders of magnitude.
Path integral Monte Carlo method for the quantum anharmonic oscillator
European Journal of Physics, 2020
The Markov chain Monte Carlo (MCMC) method is used to evaluate the imaginary-time path integral of a quantum oscillator with a potential that includes a quadratic term and a quartic term whose coupling is varied by several orders of magnitude. This path integral is discretized on a time lattice on which calculations for the energy and probability density of the ground state and energies of the first few excited states are carried out on lattices with decreasing spacing to estimate these quantities in the continuum limit. The variation of the quartic coupling constant produces corresponding variations in the optimum simulation parameters for the MCMC method and in the statistical uncertainty for a fixed number of paths used for measurement. The energies and probability densities are in excellent agreement with those obtained from numerical solutions of Schrödinger’s equation. The theoretical and computational framework presented here introduces undergraduates to the path integral for...
The principle of least action has been one of the sacred pillars of Theoretical Physics. The path integral formalism developed by Feynman to explain the quantum mechanical motion was based on the sum over paths method. Over the years, this approach has been successfully extended to many other areas of Physics such as Quantum Field Theory and also to problems of black hole and quantum gravity. Even though this formalism is clearly more intuitive than others, it has not been explored or used to solve problems in many standard textbooks. The main reason is the mathematical rigour involved due to horrendous integrals present in calculating the propagator. This can be successfully solved only for few simple cases like free particle, harmonic oscillator or electron in some special potentials. In this project, we will discuss about the theoretical background of this formalism and numerically solve the ground state wave function and the propagator for harmonic oscillator using lattice discretization and Markov Chain Monte Carlo (MCMC) using the Metropolis-Hastings algorithm. It is certainly possible to extend this procedure to variety of other problems as well which needs further research. 4 5 CONTENTS
The Feynman Path Goes Monte Carlo
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Path integral Monte Carlo (PIMC) simulations have become an important tool for the investigation of the statistical mechanics of quantum systems. I discuss some of the history of applying the Monte Carlo method to non-relativistic quantum systems in path-integral representation. The principle feasibility of the method was well established by the early eighties, a number of algorithmic improvements have been
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The conventional second-order Path Integral Monte Carlo method is plagued with the sign problem in solving many-fermion systems. This is due to the large number of anti-symmetric free fermion propagators that are needed to extract the ground state wave function at large imaginary time. In this work, we show that optimized fourth-order Path Integral Monte Carlo methods, which use no more than 5 free-fermion propagators, can yield accurate quantum dot energies for up to 20 polarized electrons with the use of the Hamiltonian energy estimator.
Journal of Computational Physics, 2015
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