# Semiexact Solutions of the Razavy Potential

Authors: Qian Dong; F. A. Serrano; Guo-Hua Sun; Jian Jing; Shi-Hai Dong
Journal: Advances in High Energy Physics
 (2018)

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## 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 onm are illustrated graphically for a given potential parameter ξ. We find that the even and odd wave functions with definite parity are changed to odd and even wave functions when the potential parameter m increases. This arises from the fact that the parity, which is a defined symmetry for very small m, is completely violated for large m. We also notice that the energy levels ϵi decrease with the increasing potential parameter m.


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## Body

## 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]:(1)Vx=ħ2β22μ18ξ2cosh⁡4βx-m+1ξcosh⁡2βx-18ξ2,which depends on three potential parameters β, ξ, and a positive integer m. In Figure 1 we plot it as the function of the variables x with various m, in which we take β=1 and ξ=3. Choose atomic units ħ=μ=1 and also take V(x)=2V(x). Using series expansion around the origin, we have(2)Vx=-mξ-ξ+x2-2mξ+ξ2-2ξ+23x4-mξ+2ξ2-ξ+445x6-mξ+8ξ2-ξ+Ox7,which shows that V(x) is symmetric to variable x. We find that the minimum value of the potential Vmin(x)=-(m+1)2-ξ2/4 at two minimum values x=±1/2cosh-1[2(m+1)/ξ]. For a given value ξ=3, we find that the potential has a flat bottom for n=0, but for n>1 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 η of the confluent Heun functions Hc(α,β,γ,δ,η,z). 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.Figure 1
(Color online) A plot of potential as function of the variablesx and m.This paper is organized as follows. In Section2, we present the solutions of the Schrödinger equation with the Razavy potential. It should be recognized that the Razavy potential is single or double-well depends on the potential parameter m. In Section 3 some fundamental properties of the solutions are studied. The energy levels for different m are calculated numerically. Some concluding remarks are given in Section 4.


## 2. Semiexact Solutions

Let us consider the one-dimensional Schrödinger equation:(3)-ħ22μd2dx2ψx+Vxψx=Eψx.Substituting potential (1) into (3), we have(4)d2dx2ψx+ε-18ξ2cosh⁡4x-m+1ξcosh⁡2x-18ξ2ψx=0,ϵ=2E.Take the wave functions of the form(5)ψx=eξcosh2x/2yx.Substituting this into (4) allows us to obtain(6)y′′x+ξsinh⁡2xy′x+m+2ξcosh⁡2x+ϵyx=0.Take a new variablez=cosh2(x). The above equation becomes(7)4z-1zy′′z+4zξz-1+1-2y′z+m+2ξ2z-1+ϵyz=0which can be rearranged as(8)y′′z+ξ+121z+1z-1y′z+m+2ξ2z-1+ϵ4z-1zyz=0.When comparing this with the confluent Heun differential equation in the simplest uniform form [13](9)d2Hzdz2+α+1+βz+1+γz-1dHzdz+μz+νz-1Hz=0,we find the solution to (8) is given by the acceptable confluent Heun function Hc(α,β,γ,δ,η;z) with(10)α=ξ,β=-12,γ=-12,μ=ξm+2-ε4,ν=ξm+2+ε4,from which we are able to calculate the parameters δ and η involved in Hc(α,β,γ,δ,η;z) as(11)δ=μ+ν-12αβ+γ+2=12m+1ξ,η=12αβ+1-μ-12β+γ+βγ=18-2m+1ξ+2ϵ+3.It is found that the parameterη related to energy levels is involved in the confluent Heun function. The wave function given by this function seems to be analytical, but the key issue is how to first get the energy levels. Otherwise, the solution becomes unsolvable. Generally, the confluent Heun function can be expressed as a series of expansions:(12)HCα,β,γ,δ,η,z=∑n=0∞vnα,β,γ,δ,η,ξzn,z<1.The coefficients vn are given by a three-term recurrence relation:(13)Anvn-Bnvn-1-Cnvn-2=0,v-1=0,v0=1,with(14)An=1+βn,Bn=1+1nβ+γ-α-1+1n2η-12β+γ-α-αβ2+βγ2,Cn=αn2δα+β+γ2+n-1.To make the confluent Heun functions reduce to polynomials, two termination conditions have to be satisfied [13, 14]:(15)μ+ν+Nα=0,ΔN+1μ=0,where(16)μ-p11+β0…000Nαμ-p2+α22+β…0000N-1αμ-p3+2α…000⋮⋮⋮⋱⋮⋮⋮000…μ-pN-1+N-2αN-1N-1+β0000…2αμ-pN+N-1αNN+β000…0αμ-pN+1+Nα=0with(17)pN=N-1N+β+γ.For present problem, it is not difficult to see that the first condition is violated. That is,μ+ν+α=0 when N=1. From this we have m=-4. This is contrary to the fact that m is a positive integer. Therefore, we cannot use this method to obtain the eigenvalues. On the other hand, we know that z∈[1,∞). Thus, the series expansion method is invalid. This is unlike previous study [22, 24], in which the quasiexact wave functions and eigenvalues can be obtained by studying those two constraints. The present case is similar to our previous study [20, 21], in which some constraint is violated. We have to choose other approach to study the eigenvalues as used in [20, 21].


## 3. Fundamental Properties

In this section we are going to study some basic properties of the wave functions as shown in Figures2–4. We first consider the positive integer m. Since the energy spectrum cannot be given explicitly we have to solve the second-order differential equation (4) numerically. We denote the energy levels as ϵi(i∈[1,6]) in Table 1. We find that the energy levels ϵi decrease with the increasing m. Originally, we wanted to calculate the energy levels numerically by using powerful MAPLE, which includes some special functions such as the confluent Heun function that cannot be found in MATHEMATICA. As we know, the wave function is given by ψ(z)=exp(zξ/2)Hc(α,β,γ,δ,η,z). Generally speaking, the wave function requires ψ(z)→0 when z→∞; i.e., x→∞. Unfortunately, the present study is unlike our previous study [20, 21], in which z→1 when x goes to infinity. The energy spectra can be calculated by series expansions through taking z→1. On the other hand, the wave functions have a definite parity; e.g., for m=0 some wave functions are symmetric. It is found that such properties are violated when the potential parameter m becomes larger as shown in Figure 4. That is, the wave functions for m=12 are nonsymmetric. In addition, on the contrary to the case discussed by Razavy [18], in which he supposed the m is taken as positive integers, we are going to show what happens to the negative m case. We display the graphics in Figures 5 and 6 for this case. We find that the wave functions are shrunk towards the origin. This makes the amplitude of the wave function increase.Table 1
Energy levels of the Schrödinger equation with potential (1).
ν ϵ 1 ϵ 2 ϵ 3 ϵ 4 ϵ 5 ϵ 6 m=-6 21.6608 35.7557 51.3448 68.3341 86.6500 106.233 m=-5 18.1891 31.3844 46.1503 62.3746 79.9715 98.8740 m=-4 14.6806 26.9167 40.8214 56.2549 73.1150 91.3249 m=-3 11.1259 22.3314 35.3346 49.9525 66.0599 83.5680 m=-2 7.51110 17.5996 29.6610 43.4412 58.7838 75.5860 m=-1 3.81463 12.6800 23.7644 36.6914 51.2639 67.3635 m=0 0.00007 7.51170 17.6027 29.6729 43.4799 58.8919 m=1 -3.99968 2.00200 11.1343 22.3606 35.4208 50.1750 m=2 -8.32288 -3.99300 4.34771 14.7494 27.0959 41.2385 m=3 -13.2815 -10.6927 -2.64788 6.87526 18.5501 32.1389 m=4 -19.5196 -9.46859 -1.17161 9.87916 22.9677 38.0537 m=5 -27.7547 -15.7094 -9.29612 1.24110 13.8439 28.5940 m=6 -38.0314 -21.6913 -17.5131 -7.12621 4.89289 19.3065 m=7 -49.9928 -28.2027 -25.9897 -14.8827 -3.78434 10.2625 m=8 -63.3335 -35.8866 -21.7455 -12.1464 1.51447 17.5661 m=9 -77.8339 -44.5255 -27.8571 -20.2355 -6.89162 8.76577 m=10 -93.3024 -54.9017 -33.6970 -28.1690 -14.8944 0.229704 m=11 -109.592 -65.743 -39.7373 -36.1005 -22.4007 -8.04337 m=12 -126.580 -77.2416 -46.3335 -29.3139 -16.0647 1.06475Figure 2
(Color online) The characteristics of the potentialV(z) as a function of the position z. We take m=0,1 and ξ=3.Figure 3
(Color online) The characteristics of the potentialV(z) as a function of the position z. We take m=6,8 and ξ=3.Figure 4
(Color online) The same as the above case butm=10,12.Figure 5
(Color online) The characteristics of the potentialV(z) as a function of the position z. We take m=-1,-6 and ξ=3.Figure 6
(Color online) The same as the above case butm=-10,-90.


## 4. Conclusions

In this work we have studied the quantum system with the Razavy potential, which is symmetric with respect to the variablex and showed how its exact solutions are found by transforming the original differential equation into a confluent type Heun differential equation. It is found that the solutions can be expressed by the confluent Heun functions Hc(α,β,γ,δ,η), in which the energy levels are involved inside the parameter η. This makes us calculate the eigenvalues numerically. The properties of the wave functions depending on m are illustrated graphically for a given potential parameter ξ. We have found that the even and odd wave functions with definite parity are changed to odd and even wave functions when the potential parameter m increases. This arises from the fact that the parity, which is a defined symmetry for very small m, is completely violated for large m. We have also noticed that the energy levels ϵi decrease with the increasing potential parameter m.



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