#### Date of Award

6-1-2001

#### Document Type

Thesis

#### Degree Name

Master of Arts (MA)

#### Department

Mathematics

#### First Advisor

Dr. Wai-Ning Mei

#### Abstract

One dimensional double well potentials usually refer to those potentials which have two equivalent or non-equivalent minima. They have been used to model diverse systems ranging from the electronic structure of diatomic molecules to electron-lattice coupling in high temperature superconductors [1][2]. However, one difficulty to the utility of these models is that exact analytical solutions for almost all double well potentials have not been found, in spite of the fact that this type of potential is constantly utilized as an illustrative example in the literature. Despite intensive research into this problem, only a few types of double wells have been exactly solved to date [3]. The most common way to estimate the eigenvalues of the double well potential is through numerical methods. The usefulness of the system would be greatly enhanced if the true eigenfunctions could be determined for every case. The goal of this thesis is to find a good analytical scheme to obtain the eigenfunctions for the Schrodinger equation with polynomial double well potential (V(x) = x4 - {3x2 where {3 is a positive integer). To achieve this goal, a combination of two methods are used: the linear combination of atomic orbitals to form molecular orbitals (LCAO-MO) method, and the Rayleigh-Ritz ·method using variational parameters. Specifically, the trial wavefunctions are expressed in terms of two linear combinations, namely: \Jl(x, a, /3, 81, ... , 8n )Left and \Jl(x, a, /3, 81, ... , 8n )right· It is found that coupling occurs in Gerard functions and anticoupling in un-Gerard. In each method, w contains a complete set of wave functions known to be the exact solutions of the harmonic oscillator. Then the variational parameters are optimized to get the approximate eigenfunction and eigenvalues. The results show that the calculated energy agrees with the numerical results over the entire range of the coupling parameters. The greater the number of terms used to represent 'l/J the more accurate the eigenvalues, position and expectation values. . We also use the resultant wave functions to verify the Heisenburg uncertainty relation by calculating the expectation values of < x2 > and < p2 >, i.e. the position and momentum operators. Thus we determine an approximation method that is useful for revealing the properties of the physical systems that can be modeled with these potentials.

#### Recommended Citation

Fulk, Chad, "Approximating Eigenvalues for Double Well Potentials." (2001). *Student Work*. 3546.

https://digitalcommons.unomaha.edu/studentwork/3546

## Comments

A Thesis Presented to the Department of Mathematics and the Faculty of the Graduate College University of Nebraska In Partial Fulfillment of the Requirements for the Degree Master of Arts University of Nebraska at Omaha. Copyright 2001 Chad Fulk