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European Journal Of Physics Education

Systematic Convergence in Applying Variational Method to Double-Well Potential.

Wai‐Ning Mei

Published Jan 1, 2016• 0 citations
Publisher

Abstract

In this work, we demonstrate the application of the variational method by computing the ground- and first-excited state energies of a double-well potential. We start with the proper choice of the trial wave functions using optimized parameters, and notice that accurate expectation values in excellent agreement with the numerical results can be aquired by cautious systematic improvement of trial wave functions.

DOI
10.20308/ejpe.04084

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In this work, we demonstrate the application of the variational method by computing the ground- and first-excited state energies of a double-well potential. We start with the proper choice of the trial wave functions using optimized parameters, and notice that accurate expectation values in excellent agreement with the numerical results can be aquired by cautious systematic improvement of trial wave functions.
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In applying the variational method, six different sets of trial wave functions are used to calculate the ground state and first excited state energies of the strongly bound potentials, i.e. V(x) = x 2 m , where m = 4, 5 and 6. It is shown that accurate results can be obtained from thorough analysis of the asymptotic behaviour of the solutions. Consequently, it was found that there is a progressive qualitative change in the wave functions from the harmonic to anharmonic and then strongly bounded region, which explains why using a harmonic oscillator wave function together with the conventional perturbative approach to calculate the energy correction, was unsuccessful.
Three sets of trial wave functions were applied to calculate the ground state and first excited state energies of the x 4 and x 6 anharmonic oscillator potentials. We demonstrate that, even with a rather crude initial guess, results can be improved by proper adjustment of the trial wave functions after careful study of their general features. Meanwhile, we found some interesting and unexpected results when applying the second set of trial wave functions. Once again, we suggest using together with computer algebra software packages, these examples when teaching the variational method in a quantum mechanics course.
The Schrödinger equation serves as a pivotal differential equation, capturing the fundamental duality of particles as both matter and waves. Upon solving it, we derive a particle's wave function within a given potential, delineating the confines within which the particle operates. This wave function enables us to ascertain the "most probable" values for parameters such as position, momentum, and energy within the specified potential. In this study, we focus on the double-well potential in two dimensions, resembling a bowl-like structure. Specifically tailored equations, termed Equation 1 and Equation 2, stem from the general Schrödinger equation when applied to the prescribed conditions. This research aims to utilize these equations to describe a particle’s behavior within the ground and the first excited states of the double well potential, employing two distinct methodologies. The first methodology involves leveraging the variational method on Equations 1 and 2 to obtain approximate solutions, providing crucial inputs for our subsequent approach. The second methodology serves as entails an analytical solution of Equations 1 and 2. A comparative analysis of the outcomes from both methods is anticipated to ensure the coherence and validity of the variational method. This project integrates key principles and proficiencies from both physics and mathematics disciplines, offering a platform for honing practical skills in applying these methodologies within a real-world context.
An improved variational path integral approach is developed and applied to the quantum double-well potential, in which part of the quartic term of the potential is included in the trial action. The expression of the effective classical potential (ECP) under a non-Gaussian expectation is obtained. Here the frequency and fourth-order derivative of the potential are treated as two variational parameters, determined by the minimization of the ECP at each point. We calculate the ECP, the free energy and the level splitting of a symmetrical double-well potential. It is shown that the present results are better than those of the Feynman–Kleinert Gaussian variational method.
Following previous work in which a variational method for the many-body problem was proposed wherein an effective potential $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ was introduced as a variational parameter to construct a trial wave function, we apply the method to the two-dimensional electron gas. In the method, every physical quantity is expanded in powers of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ instead of the bare potential as in the usual perturbation-theoretic approach. Because of this choice of expansion parameter, the expansion series converges rapidly even in the strong-coupling region. The result for the correlation energy in the two-dimensional electron gas agrees very well with that given by the variational Monte Carlo method even in the lowest-order calculation (i.e., to second order in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$). The difference is within several percent for 1<rs<1001<{r}_{s}<1001<rs​<100.