Research Article | Open Access
M. Baradaran, H. Panahi, "Exact Solutions of a Class of Double-Well Potentials: Algebraic Bethe Ansatz", Advances in High Energy Physics, vol. 2017, Article ID 8429863, 13 pages, 2017. https://doi.org/10.1155/2017/8429863
Exact Solutions of a Class of Double-Well Potentials: Algebraic Bethe Ansatz
Applying the Bethe ansatz method, we investigate the Schrödinger equation for the three quasi-exactly solvable double-well potentials, namely, the generalized Manning potential, the Razavy bistable potential, and the hyperbolic Shifman potential. General exact expressions for the energies and the associated wave functions are obtained in terms of the roots of a set of algebraic equations. Also, we solve the same problems using the Lie algebraic approach of quasi-exact solvability through the algebraization and show that the results are the same. The numerical evaluation of the energy spectrum is reported to display explicitly the energy levels splitting.
Double-well potentials (DWPs) are an important class of configurations which have been extensively used in many fields of physics and chemistry for the description of the motion of a particle under two centers of force. Recently, solutions of the Schrödinger equation with DWPs have found applications in the Bose–Einstein condensation , molecular systems , quantum tunneling effect [3, 4], microscopic description of Tunneling Systems , and so forth. Some well-known DWPs in the literature are the quartic potential , the sextic potential , the Manning potential , and the Razavy potential . In addition, it has been found that with some special constraints on the parameters of these potentials, a finite part of the energy spectrum and corresponding eigenfunctions can be obtained as explicit expressions in a closed form. In other words, these systems are quasi-exactly solvable (QES) [9–13]. DWPs in the framework of QES systems have received a great deal of attention. This is due to the pioneering work of Razavy, who proposed his well-known potential for describing the quantum theory of molecules . The fundamental idea behind the quasi-exact solvability is the existence of a hidden dynamical symmetry. QES systems can be studied by two main approaches: the analytical approach based on the Bethe ansatz [14–19] and the Lie algebraic approach [10–13]. These techniques are of great importance because only a few number of problems in quantum mechanics can be solved exactly. Therefore, these approaches can be applied as accurate and efficient techniques to study and solve the new problems that arise in different areas of physics such as quantum field theory [20–22], condensed matter physics [23–25], and quantum cosmology [26–32], whose exact solutions are hard to obtain or are impossible to find. In the literature, DWPs have been studied by using various techniques such as the WKB approximation [33, 34], asymptotic iteration method (AIM) , and the Wronskian method . On the other hand, it is well known that the tunnel splitting which is the differences between the adjacent energy levels is the characteristic of the energy spectrum for the DWPs [37–40]. In this paper, we apply two different methods to solve the Schrödinger equation for three QES DWPs, the Bethe ansatz method (BAM) and the Lie algebraic method, and show that the results of the two methods are consistent. Also, we provide some numerical results of the bistable Razavy potential to display the energy levels splitting explicitly.
This paper is organized as follows: in Section 2, we introduce the QES DWPs and obtain the exact solutions of the corresponding Schrödinger equations using the BAM. Also, general exact expressions for the energies and the wave functions are obtained in terms of the roots of the Bethe ansatz equations. In Section 3, we solve the same problems using the Lie algebraic approach within the framework of quasi-exact solvability and therein we make a comparison between the solutions obtained by the BAM and QES method. We end with conclusions in Section 4. In the Appendix, we review the connection between Lie algebra and the second-order QES differential equations.
2. The BAM for the DWPs
In this section, we introduce the three DWPs that are discussed in this work and solve the corresponding Schrödinger equations via the factorization method in the framework of algebraic Bethe ansatz . The general exact expressions for the energies, the wave functions, and the allowed values of the potential parameters are obtained in terms of the roots of the Bethe ansatz equations.
2.1. The Generalized Manning Potential
First, we consider the three-parameter generalized Manning potential as The parameters , , and are real constants which under certain constraint conditions enable us to obtain the bound-state eigenenergies and associated wave functions exactly. In atomic units (), the Schrödinger equation with potential (1) isXie  has studied this problem and obtained exact solutions of the first two states in terms of the confluent Heun functions. In this paper, we intend to extend the results of  by determining general exact expressions for the energies, wave functions, and the allowed values of the potential parameters, using the factorization method in the framework of the Bethe ansatz. To this end, and for the purpose of extracting the asymptotic behaviour of the wave function, we consider the following transformations:which, after substituting in (2), giveswhereIn order to solve the present problem via BAM, we try to factorize the operator assuch that . Now, we suppose that polynomial solution (Bethe ansatz) exists for (4) as with the distinct roots that are interpreted as the wave function nodes and can be determined by the Bethe ansatz equations. As a result, it is evident that the operator must have the formand then, the operator has the following form: By substituting (8) and (9) into (6), we haveThe last term on the right of (10) is obviously a meromorphic function with simple poles at and . Comparing the treatment of (10) with (4) at these points, we obtain the following relations for the unknown roots (the so-called Bethe ansatz equation), the energy eigenvalues, and the constraints on the potential parameters:As examples of the above general solutions, we study the ground, first, and second excited states of the model in detail. For , by (12) and (3), we have the following relations: for the ground state energy and wave function, with the potential constraint given byFor the first excited state , by (12) and (3), we have for the energy and wave function, respectively. Also, the constraint condition between the parameters of the potential is aswhere the root is obtained from the Bethe ansatz equation (11) asSimilarly, for the second excited state , the energy, wave function, and the constraint condition between the potential parameters are given aswhere the two distinct roots and are obtainable from the Bethe ansatz equationsIn Table 1, we report and compare our numerical results for the first three states. Also, in Figure 1, we draw the potential (1) for the possible values of the parameters , , and .
2.2. The Razavy Bistable Potential
Here, we consider the hyperbolic Razavy potential (also called the double sinh-Gordon (DSHG) potential) defined by where is a real parameter. While the value of is not restricted generally, but according to , the solutions of the first states can be found exactly if is a positive integer. This potential exhibits a double-well structure for with the two minima lying at . Specifically, the Razavy potential can be considered as a realistic model for a proton in a hydrogen bond [42, 43]. The potential (21) has also been used by several authors, for studying the statistical mechanics of DSHG kinks theory . The Schrödinger equation with potential (21) isExact and approximate solutions of the first states for have been obtained via different methods and can be found in [8, 35, 44]. In this and the next section, we extend the solutions of (22) to the general cases of arbitrarily and obtain general exact expressions for the energies and wave functions using the BAM and QES methods. Using the change of variable and the gauge transformationwe obtainNow, we consider the polynomial solutions for (24) aswhere are unknown parameters to be determined by the Bethe ansatz equations. In this case, the operators and are defined asAs a result, we haveNow, evaluating the residues at the two simple poles and , and comparing the results with (24), we obtain the following relations:for the energy eigenvalues and the roots , respectively. For example, for the ground state , from (28) and (23), we have the following relations for energy and wave function:For the first excited state , from (28) and (23), we havewhere . Similarly, the second excited state solution corresponding to is given bywhere the distinct roots and are obtained from the Bethe ansatz equation (29) as follows:The results obtained for the first three levels are reported and compared in Table 2. The Razavy potential and its energy levels splitting are plotted in Figure 2. Also, the numerical results for the eigenvalues and energy levels splitting are presented in Table 3. As can be seen, for a given , the energy differences between the two adjacent levels satisfy the inequality and therefore the energy levels are paired together.
2.3. The Hyperbolic Shifman Potential
Now, we consider a hyperbolic potential introduced by Shifman as where the parameter is a real constant. The Schrödinger equation for potential (34) is given byAccording to the asymptotic behaviours of the wave function at the origin and infinity, we consider the following transformations:Therefore, the differential equation for readsNow, by assumingand defining the operators and aswe obtainComparing the residues at the simple poles and with (37), we obtain the following set of equations for the energy and the zeros :respectively. Here, we obtain exact solutions of the first three levels. For , from (41) and (36), we get and for the first excited state , where the root is obtained from Bethe ansatz equation (42) asSolutions of the second excited state corresponding to are given aswhere the roots and are obtained from (42) as Here, we have taken the parameter . Our numerical results obtained for the first three levels are displayed and compared in Table 4. Also, the Shifman DWP for the parameter values and is plotted in Figure 3. In the next section, we intend to reproduce the results using the Lie algebraic approach in the framework of quasi-exact solvability.
3. The Lie Algebraic Approach for the DWPs
In the previous section, we applied the BAM to obtain the exact solutions of the systems. In this section, we solve the same models by using the Lie algebraic approach and show how the relation with the Lie algebra underlies the solvability of them. To this aim, for each model, we show that the corresponding differential equation is an element of the universal enveloping algebra of and thereby we obtain the exact solutions of the systems using the representation theory of . The method is outlined in the Appendix.
3.1. The Generalized Manning Potential
Applying the results of the Appendix, it is easy to verify that (4) can be written in the Lie algebraic formif the following condition (constraint of quasi-exact solvability) is fulfilled, which is the same result as (12). As a result, the operator preserves the finite-dimensional invariant subspace spanned by the basis and therefore the states can be determined exactly. Accordingly, (48) can be represented as a matrix equation whose nontrivial solution exists if the following constraint is satisfied (Cramer’s rule),which provides important constraints on the potential parameters. Also, from (3), the wave function is aswhere the expansion coefficients obey the following three-term recurrence relation: