Testing Monte Carlo methods for path integrals in some quantum mechanical problems (original) (raw)
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Path integral Monte Carlo on a lattice: Extended states
Physical Review E, 2014
The equilibrium properties of a single quantum particle (qp) interacting with a classical gas for a wide range of temperatures that explore the system's behavior in the classical as well as in the quantum regime is investigated. Both the quantum particle and atoms are restricted to the sites of a one-dimensional lattice. A path-integral formalism is developed within the context of the canonical ensemble in which the quantum particle is represented by a closed, variable-step random walk on the lattice. Monte Carlo methods are employed to determine the system's properties. For the case of a free particle, analytical expressions for the energy, its fluctuations, and the qp-qp correlation function are derived and compared with the Monte Carlo simulations. To test the usefulness of the path integral formalism, the Metropolis algorithm is employed to determine the equilibrium properties of the qp for a periodic interaction potential, forcing the qp to occupy extended states. We consider a striped potential in one dimension, where every other lattice site is occupied by an atom with potential ǫ, and every other lattice site is empty. This potential serves as a stress test for the path integral formalism because of its rapid site-to-site variation. An analytical solution was determined in this case by utilizing Bloch's theorem due to the periodicity of the potential. Comparisons of the potential energy, the total energy, the energy fluctuations and the correlation function are made between the results of the Monte Carlo simulations and the analytical calculations.
User's guide to Monte Carlo methods for evaluating path integrals
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.
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
Asymptotic properties of path integral Monte Carlo calculations
Il Nuovo Cimento B, 1994
We analyse the problem of the convergence and errors in calculations of the path integral in quantum Monte Carlo method. The behaviours of different estimators of potential, kinetic and total energies are compared through the analysis of the con-esponding correlation functions.
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...
Numerical path integral approach to quantum dynamics and stationary quantum states
Applicability of Feynman path integral approach to numerical simulations of quantum dynamics in real time domain is examined. Coherent quantum dynamics is demonstrated with one dimensional test cases (quantum dot models) and performance of the Trotter kernel as compared with the exact kernels is tested. A novel approach for finding the ground state and other stationary sates is presented. This is based on the incoherent propagation in real time. For both approaches the Monte Carlo grid and sampling are tested and compared with regular grids and sampling. We asses the numerical prerequisites for all of the above.
Path Integrals in Quantum Physics
Lecture Notes in Physics Monographs
These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, manybody physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions etc.) as well as specialized topics (scattering, dissipative systems, spin & color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions. Content 0. Contents First, an overview over the planned topics. The subsections marked by ⋆ are optional and may be left out if there is no time available whereas the chapters printed in blue deal with basic concepts. Problems from the optional chapters or referring to "Details" are marked by a ⋆ as well.
Journal of Experimental and Theoretical Physics, 1998
We show how the worldline quantum Monte Carlo procedure, which usually relies on an artificial time discretization, can be formulated directly in continuous time, rendering the scheme exact. For an arbitrary system with discrete Hilbert space, none of the configuration update procedures contain small parameters. We find that the most effective update strategy involves the motion of worldline discontinuities (both in space and time), i.e., the evaluation of the Green's function. Being based on local updates only, our method nevertheless allows one to work with the grand canonical ensemble and non-zero winding numbers, and to calculate any dynamic correlation function as easily as expectation values of, e.g., total energy. The principles found for the update in continuous time generalize to any continuous variables in the space of discrete virtual transitions, and in principle also make it possible to simulate continuous systems exactly.
Physical Review E, 2006
A detailed description is provided of a new Worm Algorithm, enabling the accurate computation of thermodynamic properties of quantum many-body systems in continuous space, at finite temperature. The algorithm is formulated within the general Path Integral Monte Carlo (PIMC) scheme, but also allows one to perform quantum simulations in the grand canonical ensemble, as well as to compute off-diagonal imaginary-time correlation functions, such as the Matsubara Green function, simultaneously with diagonal observables. Another important innovation consists of the expansion of the attractive part of the pairwise potential energy into elementary (diagrammatic) contributions, which are then statistically sampled. This affords a complete microscopic account of the long-range part of the potential energy, while keeping the computational complexity of all updates independent of the size of the simulated system. The computational scheme allows for efficient calculations of the superfluid fraction and off-diagonal correlations in space-time, for system sizes which are orders of magnitude larger than those accessible to conventional PIMC. We present illustrative results for the superfluid transition in bulk liquid 4 He in two and three dimensions, as well as the calculation of the chemical potential of hcp 4 He.