Advances in High Energy Physics

Volume 2018, Article ID 9105825, 7 pages

https://doi.org/10.1155/2018/9105825

## Semiexact Solutions of the Razavy Potential

^{1}Laboratorio de Información Cuántica, CIDETEC, Instituto Politécnico Nacional, UPALM, CDMX 07700, Mexico^{2}Escuela Superior de Ingeniera Mecánica y Eléctrica UPC, Instituto Politécnico Nacional, Av. Santa Ana 1000, México, D. F. 04430, Mexico^{3}Catedrática CONACYT, CIC, Instituto Politécnico Nacional, CDMX 07738, Mexico^{4}Department of Physics and Electronic, School of Science, Beijing University of Chemical Technology, Beijing 100029, China

Correspondence should be addressed to Shi-Hai Dong; moc.oohay@2hsgnod

Received 9 May 2018; Accepted 10 June 2018; Published 28 August 2018

Academic Editor: Saber Zarrinkamar

Copyright © 2018 Qian Dong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

In this work, we study the quantum system with the symmetric Razavy potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun functions. The eigenvalues have to be calculated numerically. The properties of the wave functions depending on are illustrated graphically for a given potential parameter . We find that the even and odd wave functions with definite parity are changed to odd and even wave functions when the potential parameter increases. This arises from the fact that the parity, which is a defined symmetry for very small , is completely violated for large . We also notice that the energy levels decrease with the increasing potential parameter .

#### 1. Introduction

It is well-known that the exact solutions of quantum systems play an important role since the early foundation of the quantum mechanics. Generally speaking, two typical examples are studied for the hydrogen atom and harmonic oscillator in classical quantum mechanics textbooks [1, 2]. Up till now, there are a few main methods to solve the quantum soluble systems. The first is called the functional analysis method. That is to say, one solves the second-order differential equation and obtains their solutions [3], which are expressed by some well-known special functions. The second is called the algebraic method, which is realized by studying the Hamiltonian of quantum system. This method is also related to supersymmetric quantum mechanics (SUSYQM) [4], further closely with the factorization method [5]. The third is called the exact quantization rule method [6], from which we proposed proper quantization rule [7], which shows more beauty and symmetry than exact quantization rule. It should be recognized that almost all soluble potentials mentioned above belong to single well potentials. The double-well potentials have not been studied well due to their complications [8–17], in which many authors have been searching the solutions of the double-well potentials for a long history. This is because the double-well potentials could be used in the quantum theory of molecules to describe the motion of the particle in the presence of two centers of force, the heterostructures, Bose-Einstein condensates, superconducting circuits, etc.

Almost forty years ago, Razavy proposed a bistable potential [18]:which depends on three potential parameters , , and a positive integer . In Figure 1 we plot it as the function of the variables with various , in which we take and . Choose atomic units and also take . Using series expansion around the origin, we havewhich shows that is symmetric to variable . We find that the minimum value of the potential at two minimum values . For a given value , we find that the potential has a flat bottom for , but for it takes the form of a double-well. Razavy presented the so-called exact solutions by using the “polynomial method” [18]. After studying it carefully, we find that the solutions cannot be given exactly due to the complicated three-term recurrence relation. The method presented there [18] is more like the Bethe Ansatz method as summarized in our recent book [19]. That is, the solutions cannot be expressed as one of special functions because of three-term recurrence relations. In order to obtain some so-called exact solutions, the author has to take some constraints on the coefficients in the recurrence relations as shown in [18]. Inspired by recent study of the hyperbolic type potential well [20–28], in which we have found that their solutions can be exactly expressed by the confluent Heun functions [23], in this work we attempt to study the solutions of the Razavy potential. We shall find that the solutions can be written as the confluent Heun functions but their energy levels have to be calculated numerically since the energy term is involved within the parameter of the confluent Heun functions . This constraints us to use the traditional Bethe Ansatz method to get the energy levels. Even though the Heun functions have been studied well, its main topics are focused in the mathematical area. Only recent connections with the physical problems have been discovered; in particular the quantum systems for those hyperbolic type potential have been studied [20–28]. The terminology “semiexact” solutions used in [21] arise from the fact that the wave functions can be obtained analytically, but the eigenvalues cannot be written out explicitly.