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European Journal of Physics

The simple harmonic oscillator ground state using a variational Monte Carlo method

Shawn Pottorf, Aaron Pudzer, M. Y. Chou, Javier E. Hasbun

Published Jan 1, 1999• 20 citations
Publisher

Abstract

A Monte Carlo method is used to evaluate the ground-state energy of a quantum particle in a harmonic oscillator potential. The use of a trial wavefunction illustrates the process by means of which the Monte Carlo method approximates with great accuracy the well known ground-state energy of a quantum mechanical harmonic oscillator. A FORTRAN program incorporating the numerical approach that is suitable for a personal computer is available in the PDF version of this article.

DOI
10.1088/0143-0807/20/3/311

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A Monte Carlo method is used to evaluate the ground-state energy of a quantum particle in a harmonic oscillator potential. The use of a trial wavefunction illustrates the process by means of which the Monte Carlo method approximates with great accuracy the well known ground-state energy of a quantum mechanical harmonic oscillator. A FORTRAN program incorporating the numerical approach that is suitable for a personal computer is available in the PDF version of this article.
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Variational calculations play an important role in quantum mechanics, particularly in determining the ground-state energy of physical systems. Didactic examples in which one can see how the exact energy and wave function are approached by using a family of trial functions are not very numerous. An example was given in a recent book [M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon and Breach, New York, 1969)] which discusses the ground state of the hydrogen atom using as trial wave function a linear combination of harmonic-oscillator states. In the present paper we carry out a similar analysis for the ground state of a particle of mass m in a three-dimensional square-well potential. We discuss not only the approach to the exact energy when we vary the frequency and the number of oscillator states but also analyze the overlap of the exact and variational wave functions and compare the exact and approximate form factors.
This thesis aims to give an understanding of quantum dots trapped in different potentials by computationally finding the ground state energies of the system.The potentials studied are the single and double harmonic oscillator wells and the finite square well.To achieve this, we have implemented a robust Variational Monte Carlo framework that allows for different single-particle wavefunctions.This allowed us to study the use of a linear expansion of the single harmonic oscillator basisfunctions as an approximation to the single-particle wavefunctions of the other potential wells.We investigated how many of these basisfunctions needed to achieve sufficiently good single-particle wavefunctions.The framework has several optimizations that drastically reduce the computational time of the Monte Carlo method.The framework has a rigorous testing regime that ensures the authenticity of the results.To verify the results, we have compared them to the results of other Variatonal Monte Carlo computations, as well as Diffusion Monte Carlo, the Coupled Cluster method, the Similarity Renormalization Group method and the Full Configuration Interaction method.In addition, we have verified the results of the finite square well by comparing with a general series solution.
Introduction In Chapters 1 to 4 we studied methods for solving the Schrödinger equation for many-electron systems. Many of the techniques described there carry over to other quantum many-particle systems, such as liquid helium, and the protons and neutrons in a nucleus. The techniques which we discussed there were, however, all of a mean-field type and therefore correlation effects could not be taken into account without introducing approximations. In this chapter, we consider more accurate techniques, which are similar to those studied in Chapter 10 and are based on using (pseudo-)random numbers – hence the name ‘Monte Carlo’ for these methods. In Chapter 10 we applied Monte Carlo techniques to classical many-particle systems; here we use these techniques for studying quantum problems involving many particles. In the next section we shall see how we can apply Monte Carlo techniques to the problem of calculating the quantum mechanical expectation value of the ground state energy. This is used in order to optimise this expectation value by adjusting a trial wave function in a variational type of approach, hence the name variational Monte Carlo (VMC). In the following section we use the similarity between the Schrödinger equation and the diffusion equation in order to calculate the properties of a collection of interacting quantum mechanical particles by simulating a classical particle diffusion process. The resulting method is called diffusion Monte Carlo (DMC).