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American Journal of Physics

Application of the variational method to one-dimensional infinite wells with internal c x n potentials

David D. Keeports

Published Mar 1, 1990• 5 citations
Publisher

Abstract

The variational method is applied to the problem of a cxn potential within an infinite one-dimensional well. Approximate ground- and excited-state wavefunctions are constructed as linear combinations of two-term even and odd polynomials. Comparisons are made between calculated approximate wavefunctions and energies, and results predicted from perturbation theory and from the behavior of related systems.

DOI
10.1119/1.16488

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The variational method is applied to the problem of a cxn potential within an infinite one-dimensional well. Approximate ground- and excited-state wavefunctions are constructed as linear combinations of two-term even and odd polynomials. Comparisons are made between calculated approximate wavefunctions and energies, and results predicted from perturbation theory and from the behavior of related systems.
The authors propose a new method of solving the Schrodinger equation for a quantum system enclosed in a box with infinite potential walls. The method combines the variational technique with boundary perturbation theory and can be applied to a general non-separable case. When the Schrodinger equation for an enclosed system separates into an ordinary differential equation, the method gives the exact energy and wavefunction. As an application of the method the authors obtain the ground-state energy of the hydrogen atom placed off-centre in a spherical cavity. A generalisation of the variational boundary perturbation technique for finite potential walls is suggested.
A new variational approach is proposed for the study of the many-body problem with the purpose of combining the advantages of perturbation-theoretic and variational approaches, while avoiding their disadvantages. A trial wave function is constructed so that a general expansion for the expectation value of an arbitrary physical quantity can be derived. Each term in the expansion can be calculated with the use of the techniques of Feynman diagrams, but in contrast with usual perturbation-theoretic approaches, every term in the expansion always converges. The procedure for applying the method to real systems is illustrated by a calculation of the ground-state energy of the electron gas at high densities.
Solving the tunneling problem in a double-well potential requires knowing the wave function not just of the ground state, but also of the first excited state. A method is described that allows one to construct potentials together with these two wave functions. No approximations are made. To illustrate the method several examples are given. These include potentials that either diverge for large \ensuremath{\Vert}x\ensuremath{\Vert} or become asymptotically flat, with one or two wells, which may be symmetric or asymmetric.
Both single-well and multiwell one-dimensional anharmonic oscillators have provided an extremely fruitful testing ground for various modern techniques of microscopic quantum many-body theory and quantum field theory. They have served both as (0+1)-dimensional field-theory models of their important counterparts in a higher number of dimensions, and as algebraically simple yet highly nontrivial problems on which various approximation or truncation schemes could be tested to very high orders. Such techniques have included perturbation theory and its many possible resummation schemes, and the coupled-cluster method. In the present paper the spotlight is focused on variational methods, and we concentrate on their application to the particularly demanding double-well quartic anharmonic oscillator. We examine in detail both single-state and multistate variational calculations based on either orthogonal (uncorrelated) or nonorthogonal (correlated) basis functions. Particular attention is paid to the question of the sensitivity and accuracy of the results with respect to any nonlinear parameters that characterize the basis states. Practical schemes are suggested for their choice and are shown to be capable of very high accuracy. Use of this simple model also allows detailed comparisons to be made with other techniques of a perturbative or perturbative-variational hybrid character.
We apply the variational method to the ground-state and first-excited-state energies of a Gaussian well and show that the expansion of the approximate energy when the potential strength approaches zero does not agree with the exact result. We argue that a recent semiclassical approach to tunneling through a Gaussian barrier does not agree with an accurate calculation of the transmission probability. We point out the advantages of converting the Schrödinger equation into a dimensionless eigenvalue equation.