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 sl(2) 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.
1. Introduction
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 [1], molecular systems [2], quantum tunneling effect [3, 4], microscopic description of Tunneling Systems [5], and so forth. Some well-known DWPs in the literature are the quartic potential [6], the sextic potential [7], the Manning potential [2], and the Razavy potential [8]. 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 [8]. 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) [35], and the Wronskian method [36]. 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 sl(2) 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 [15]. 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 [36](1)Vx=-v1sech6x-v2sech4x-v3sech2x.The parameters v1, v2, and v3 are real constants which under certain constraint conditions enable us to obtain the bound-state eigenenergies and associated wave functions exactly. In atomic units (m=ħ=c=1), the Schrödinger equation with potential (1) is(2)-d2dx2-v1sech6x-v2sech4x-v3sech2xψx=Eψx.Xie [36] 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 [36] 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:(3)z=tanh2x,ψx=1-z-E/2ev1/2zϕz,which, after substituting in (2), gives(4)Hϕz=0,H=-zz-1d2dz2-v1z2+32+-E-v1z-12ddz-14λz+ε,where(5)λ=3+2-Ev1+v1+v2,ε=-v1-v2-v3-E+-E-v1.In order to solve the present problem via BAM, we try to factorize the operator H as(6)H=An+An,such that Anϕn(z)=0. Now, we suppose that polynomial solution (Bethe ansatz) exists for (4) as (7)ϕnz=∏k=1nz-zkn≠01n=0,with the distinct roots zk 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 An must have the form(8)An=ddz-∑k=1n1z-zk,and then, the operator An+ has the following form: (9)An+=-ddz-12z+1+-Ez-1+v1-∑k=1n1z-zk.By substituting (8) and (9) into (6), we have(10)An+An=-d2dz2-2v1z2+3+2-E-2v1z-12zz-1ddz+12z+1+-Ez-1+v1+∑j≠kn2zj-zk∑k=1n1z-zk.The last term on the right of (10) is obviously a meromorphic function with simple poles at z=0,1 and zk. Comparing the treatment of (10) with (4) at these points, we obtain the following relations for the unknown roots zk (the so-called Bethe ansatz equation), the energy eigenvalues, and the constraints on the potential parameters:(11)∑j≠kn2zk-zj+12zk+1+-Enzk-1+v1=0,(12)3+2-Env1+v1+v2+4nv1=0,(13)-v1-v2-v3-En+-En-v1+4v1∑k=1nzk+4nn-1+432+-En-v1n=0.As examples of the above general solutions, we study the ground, first, and second excited states of the model in detail. For n=0, by (12) and (3), we have the following relations: (14)3+2-E0v1+v1+v2=0,ψ0z=1-z-E0/2ev1/2z,for the ground state energy and wave function, with the potential constraint given by(15)-v1-v2-v3-E0+-E0-v1=0.For the first excited state n=1, by (12) and (3), we have (16)3+2-E1v1+v1+v2+4v1=0,ψ1z=1-z-E1/2ev1/2zz-z1,for the energy and wave function, respectively. Also, the constraint condition between the parameters of the potential is as(17)-v1-v2-v3-E1+4v1z1+5-E1-5v1+6=0,where the root z1 is obtained from the Bethe ansatz equation (11) as(18)z1=-3/2--E1+v1±3/2+-E1-v12+2v12v1.Similarly, for the second excited state n=2, the energy, wave function, and the constraint condition between the potential parameters are given as(19)3+2-E2v1+v1+v2+8v1=0,ψ2z=1-z-E2/2ev1/2zz2-z1+z2z+z1z2,-v1-v2-v3-E2+4v1z1+z2+9-E2-9v1+20=0,where the two distinct roots z1 and z2 are obtainable from the Bethe ansatz equations(20)2z1-z2+12z1+1+-E2z1-1+v1=0,2z2-z1+12z2+1+-E2z2-1+v1=0.In 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 v1=1, v2=-12, and v3=15.255.
Solutions of the first three states for the generalized Manning DWP with v1=1 and the possible values of v2, where z=tanh2(x).
The generalized Manning DW potential with v1=1, v2=-12, and v3=15.255.
2.2. The Razavy Bistable Potential
Here, we consider the hyperbolic Razavy potential (also called the double sinh-Gordon (DSHG) potential) defined by [41](21)Vx=ξcosh2x-M2,where ξ is a real parameter. While the value of M is not restricted generally, but according to [8], the solutions of the first M states can be found exactly if M is a positive integer. This potential exhibits a double-well structure for M>ξ with the two minima lying at cosh(2x0)=M/ξ. 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 [44]. The Schrödinger equation with potential (21) is(22)-d2dx2+ξcosh2x-M2ψx=Eψx.Exact and approximate solutions of the first M states for M=1,2,3,4,5,6,7 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 M and obtain general exact expressions for the energies and wave functions using the BAM and QES methods. Using the change of variable z=e2x and the gauge transformation(23)ψz=z1-M/2e-ξ/4z+1/zϕz,we obtain(24)Hϕz=0,H=-4z2d2dz2+2ξz2+4M-8z-2ξddz+-2ξM-1z+ξ2+2M-1-E.Now, we consider the polynomial solutions for (24) as(25)ϕMz=∏k=2Mz-zkM≠11M=1,where zk are unknown parameters to be determined by the Bethe ansatz equations. In this case, the operators AM and AM+ are defined as(26)AM=ddz-∑k=2M1z-zk,AM+=-ddz+ξ2+M-2z-ξ2z2-∑k=2M1z-zk.As a result, we have(27)AM+AM=-d2dz2+2ξz2+4M-2z-2ξ4z2ddz+-ξ2-M-2z+ξ2z2+∑j≠kM2zj-zk∑k=2M1z-zk.Now, evaluating the residues at the two simple poles z=zk and z=0, and comparing the results with (24), we obtain the following relations:(28)EM=ξ2+2M-1+2ξ∑k=2Mzk,(29)∑j≠kM2zk-zj+ξ2zk2-M-2zk-ξ2=0,for the energy eigenvalues and the roots zk, respectively. For example, for the ground state M=1, from (28) and (23), we have the following relations for energy and wave function:(30)E1=ξ2+1,ψ1z=e-ξ/4z+1/z.For the first excited state M=2, from (28) and (23), we have(31)E2=ξ2+3+2ξz2,ψ2z=z-1/2e-ξ/4z+1/zz-z2,where z2=±1. Similarly, the second excited state solution corresponding to M=3 is given by(32)E3=ξ2+5+2ξz2+z3,ψ3z=z-1e-ξ/4z+1/zz2-z2+z3z+z2z3,where the distinct roots z2 and z3 are obtained from the Bethe ansatz equation (29) as follows:(33)z2z3=±1∓1,±21+4ξ2+1±4ξ2+1±121+4ξ2+1-1+4ξ2+1224ξ2+1+31+4ξ2+1±44ξ2+1±4ξ±21+4ξ2+1+4ξ2+1+12ξ,--1+4ξ2+1∓2-24ξ2+12-24ξ2+11+4ξ2+12ξ4ξ2+1-32-24ξ2+1±44ξ2+1∓41-4ξ2+1±2-24ξ2+12ξ.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 M, the energy differences between the two adjacent levels satisfy the inequality E2-E1<E4-E3<⋯ and therefore the energy levels are paired together.
Solutions of the first three states for the Razavy bistable potential with ξ=2, where z=e2x.
The Razavy bistable potential and the energy levels splitting with ξ=2 and M=12.
2.3. The Hyperbolic Shifman Potential
Now, we consider a hyperbolic potential introduced by Shifman as [9](34)Vx=a22sinh2x-an+12coshx,where the parameter a is a real constant. The Schrödinger equation for potential (34) is given by(35)-d2dx2+a22sinh2x-an+12coshx-Eψx=0.According to the asymptotic behaviours of the wave function at the origin and infinity, we consider the following transformations:(36)ψx=e-acoshxϕx,z=coshx.Therefore, the differential equation for φ(x) reads(37)Hϕz=0,H=-12z2+12d2dz2+az2-12z-addz-naz+E.Now, by assuming(38)ϕnz=∏k=1nz-zkn≠01n=0,and defining the operators An and An+ as(39)An=ddz-∑k=1n1z-zk,An+=-ddz-2az2-z-2a1-z2-∑k=1n1z-zk,we obtain(40)An+An=-d2dz2-2az2-z-2a1-z2ddz+2az2-z-2a1-z2+∑j≠kn2zj-zk∑k=1n1z-zk.Comparing the residues at the simple poles z=±1 and z=zk with (37), we obtain the following set of equations for the energy and the zeros zk:(41)En=a∑k=1nzk-n22,(42)∑j≠kn2zj-zk+2azk2-zk-2a1-zk2=0,respectively. Here, we obtain exact solutions of the first three levels. For n=0, from (41) and (36), we get (43)E0=0,ψ0z=e-az,and for the first excited state n=1, (44)E1=az1-12,ψ1z=e-azz-z1,where the root z1 is obtained from Bethe ansatz equation (42) as(45)z1=1±1+16a24a.Solutions of the second excited state corresponding to n=2 are given as(46)E2=az1+z2-2,ψ2z=e-azz2-z1+z2z+z1z2,where the roots z1 and z2 are obtained from (42) as (47)z1,z2=-0.7378701975,0.6708350007,17.31284254,3.232902445,-0.3019410394,14.82323126.Here, we have taken the parameter a=0.1. Our numerical results obtained for the first three levels are displayed and compared in Table 4. Also, the Shifman DWP for the parameter values a=0.1 and n=1 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.
Solutions of the first three states for the Shifman DWP with a=0.1, where z=cosh(x).
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 sl(2) 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 sl(2) and thereby we obtain the exact solutions of the systems using the representation theory of sl(2). 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 form(48)Hϕz=0,H=-Jn+Jn--Jn0Jn--v1Jn++32+n+-E-v1Jn0-n+12Jn-+n22+32+-E-v1n2+ε4,if the following condition (constraint of quasi-exact solvability) is fulfilled, (49)3+2-Env1+v1+v2=-4nv1,which is the same result as (12). As a result, the operator H preserves the finite-dimensional invariant subspace ϕ(z)=∑m=0namzm spanned by the basis 1,z,z2,…,zn and therefore the n+1 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),(50)ε4-12000-nv1-E-v1+6+ε4-3000-n-1v1⋱⋱00⋱⋱⋱⋮00⋱⋱n-2n22000-v1n12+-E-v1+n+ε4=0,which provides important constraints on the potential parameters. Also, from (3), the wave function is as(51)ψnz=1-z-En/2ev1/2z∑m=0namzm,where the expansion coefficients am obey the following three-term recurrence relation:(52)m2+32m+12pm+1+2v1pm-1-mχ+m-1+ε4pm=0,with boundary conditions a-1=0 and an+1=0. Now, for comparison with the results of BAM obtained in the previous section, we study the first three states. For n=0, from (49), the energy equation is as (53)3+2-E0v1+v1+v2=0,where the constraints on the potential parameters is obtained from (50) as(54)-v1-v2-v3-E0+-E0-v1=0.For the first excited state n=1, the energy equation is given by(55)7+2-E1v1+v1+v2=0,where the potential parameters satisfy the constraint(56)-v1-v2-v3-E1+-E1-v12+2-v1-v2-v3-E1+-E1-v13+2-E1-2v1-8v1=0.Solutions of the second excited state corresponding to n=2 are given as(57)11+2-E2v1+v1+v2=0,where the constraints on the potential parameters are obtained from (50) as follows:(58)-v1-v2-v3-E2+-E2-v13+8-v1-v2-v3-E2+-E2-v12134+32-E2-32v1+16-v1-v2-v3-E2+-E2-v13+2-E2-2v122+3+2-E2-6v1+64v1-3-2-E2+2v1-2v1=0.Some numerical results are reported and compared with BAM results in Table 1. As can be seen the results achieved by the two methods are identical.
3.2. The Razavy Bistable Potential
In this case, using the results of the Appendix, the operator H of (24) is expressed as an element of the universal enveloping algebra of sl(2) as(59)Hϕz=0,H=-JM+JM-+ξ2JM++JM0+ξ2JM-+M-12+E+1-2M-ξ24,where M=n+1=1,2,3,…. As a result of the above algebraization, we can use the representation theory of sl(2) which results in a general matrix equation for arbitrary M whose nontrivial solution condition gives the exact solutions of the system as(60)E+1-2M-ξ24ξ20000M-1ξ2E+9-6M-ξ24ξ0000M-2ξ2⋱⋱0000⋱⋱⋱0000⋱⋱M-1ξ20000ξ2E+1-2M-ξ24=0.Also, from (23), the wave function of the system is as(61)ψMz=z1-M/2e-ξ/4z+1/z∑m=0M-1amzm,where the coefficients am satisfy the following recursion relation:(62)mξ2am+E-ξ2-6m+34am-1+ξam-2=0,with initial conditions aM=0 and a-1=0. As examples of the general formula (60) and also, for comparison purpose, we study the first three levels. For the ground state M=1, from (60), we have (63)E0=1+ξ2.For the first excited state, from (60), we obtain(64)E2=ξ2+3±2ξ.In a similar way, for the second excited state M=3, we have (65)164ξ4+-2E3-2ξ2+E32-14E3+45-ξ2+E3-5=0,which yields the energy as (66)E3=ξ2+5ξ2+7±24ξ2+1.The numerical results are reported and compared with BAM results in Table 2.
3.3. The Hyperbolic Shifman Potential
Comparing (37) with the results of the Appendix, it is seen that the operator H can be expressed in the Lie algebraic form(67)Hϕz=0,H=12Jn+Jn-+Jn-Jn--aJn++Jn--n+12Jn0-E+nn+14.Then, using the representation theory of sl(2), the general condition for existence of a nontrivial solution is obtained as (68)-E-a1000-na-E-12-2a30⋮0-n-1a⋱⋱⋱000⋱⋱⋱nn-12⋮00⋱⋱-na0⋯⋯0-a-E-n22=0,which gives the same results as those obtained by BAM in the previous section. For example, for n=0, from (68), the ground state energy of the system is E0=0. For n=1, the energy equation is as follows:(69)2E12+E1-2a2=0,which yields(70)E1=-1±16a2+14.Likewise, for n=2, the second excited state energy of the system is obtained from the following relation:(71)-E23-52E22+4a2-1E2+6a2=0.The numerical results are reported and compared with BAM results in Table 4. The wave function of the system from (36) is given as(72)ψnz=e-az∑m=0namzm,where the expansion coefficients am’s satisfy the recursion relation(73)-m+1aam-2-E+m-12am-1-maam+mm+12am+1=0,with boundary conditions a-2=0, a-1=0, and an+1=0.
4. Conclusions
Using the Bethe ansatz method, we have solved the Schrödinger equation for a class of QES DWPs and obtained the general expressions for the energies and the wave functions in terms of the roots of the Bethe ansatz equations. In addition, we have solved the same problems using the Lie algebraic approach within the framework of quasi-exact solvability and obtained the exact solutions using the representation theory of sl(2) Lie algebra. It was found that the results of the two methods are consistent. Also, we have provided some numerical results for the Razavy potential to display the energy splitting explicitly. The main advantage of the methods we have used is that we determine the general exact solutions of the systems and thus, we can quickly calculate the solution of any arbitrary state, without cumbersome procedures and without difficulties in obtaining the solutions of the higher states.
AppendixThe Lie Algebraic Approach of Quasi-Exact Solvability
In this Appendix, we outline the connection between sl(2) Lie algebra and the second-order QES differential equations. A differential equation is said to be QES if it lies in the universal enveloping algebra of a QES Lie algebra of differential operators [12]. In the case of one-dimensional systems, sl(2) Lie algebra is the only algebra of first-order differential operators which possesses finite-dimensional representations, whose generators [12],(A.1)Jn+=-z2ddz+nz,Jn0=zddz-n2,Jn-=ddz,obey the sl(2) commutation relations as(A.2)Jn+,Jn-=2Jn0,Jn0,Jn±=±Jn±and leave invariant the finite-dimensional space (A.3)Pn+1=1,z,z2,…,zn.Hence, the most general one-dimensional second-order differential equation H can be expressed as a quadratic combination of the sl(2) generators as(A.4)H=∑a,b=0,±CabJaJb+∑a=0,±CaJa+C.On the other hand, the operator (A.4) as an ordinary differential equation has the following differential form:(A.5)Hϕz=0,H=-P4zd2dz2+P3zddz+P2z,where Pl are polynomials of at most degree l. Generally, this operator does not have the form of Schrödinger operator but can always be turned into a Schrödinger-like operator (A.6)H~=e-AxHeAx=-12d2dx2+A′2-A′′+P2zx,using the following transformations:(A.7)x=±∫dzP4,ϕz=e-∫P3/P4dz+logx′ψx.The interested reader is referred to [9–14] for further details.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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