Abstract

We solve the -dimensional Schrödinger equation with hyperbolic Pöschl-Teller potential plus a generalized ring-shaped potential. After the separation of variable in the hyperspherical coordinate, we used Nikiforov-Uvarov (NU) method to solve the resulting radial equation and obtain explicitly the energy level and the corresponding wave function in closed form. The solutions to the energy eigenvalues and the corresponding wave functions are obtained using the NU method as well.

1. Introduction

The noncentral potentials in recent times have been an active field of research in physics and quantum chemistry [1–3]. For instance, the occurrence of accidental degeneracy and hidden symmetry in the noncentral potentials and their application in quantum chemistry and nuclear physics are used to describe ring-shaped molecules like benzene and the interaction between deformed pair of nuclei [4, 5]. It is known that this accidental degeneracy occurring in the ring-shaped potential was explained by constructing an SU (2) algebra [6]. Owing to these applications, many authors have investigated a number of real physical problems on nonspherical oscillator [7], ring-shaped oscillator (RSO) [8], and ring-shaped nonspherical oscillator [9]. Berkdemir [10] had shown that either Coulomb or harmonic oscillator will give a better approximation for understanding the spectroscopy and structure of diatomic molecules in the ground electronic state. Other applications of the ring-shaped potential can be found in ring-shaped organic molecules like cyclic polyenes and benzene [11, 12].

On the other hand, Chen and Dong studied the Schrödinger equation with a new ring-shaped potential [3]. Cheng and Dai investigated modified Kratzer potential plus the new ring-shaped potential using Nikiforov-Uvarov method [13]. Recently, Ikot et al. [14–16] investigated the Schrödinger equation with Hulthen potential plus a new ring-shaped potential [3], nonspherical harmonic and Coulomb potential [15], and pseudo-Coulomb potential in the cosmic string space-time [16]. Many authors have used different methods to obtain exact solutions of the wave equation such as the methods of Supersymmetric Quantum Mechanics (SUSY-QM) [17–19], the Tridiagonal Representation Approach (TRA) [20–23], and Nikiforov-Uvarov (NU) method [24–28].

Motivated by the recent studies of the ring-shaped potential [29–32], we proposed a novel hyperbolical Pöschl-Teller potential plus generalized ring-shaped potential of the form (see Figure 1)where is the screening parameter and , , , , and are real potential parameters. As a special case when with , , and ; the potential of (1) turns to nonspherical harmonic oscillator plus generalized ring-shaped potential.

Figure 1 

The plot of the novel Pöschl-Teller plus ring-shaped potential as a function of and for , , ., , , and .

2. -Dimensional Schrödinger Equation in Hyperspherical Coordinates

The D-dimensional Schrödinger equation is given as follows [33, 34]:where is the effective mass of two interacting particles, is Planck’s constant, is the energy eigenvalue, is the potential energy function, is the position vector in D-dimensions, where is the angular position vector written in terms of hyperspherical coordinates [35, 36], and is the D-dimensional Laplacian operator given in Appendix B.

The solvable potentials that allow separation of variable in (3) must be of the formThe separable wave function takes the following form:Applying (5) to (3) with the use of (4), we obtain the following radial and angular wave equations:where , (8) holds for , with , and . Solutions of (8) will not be affected by the presence of the proposed potential and thus are common to different systems and they were done before using different approaches [37]. Consequently, we will only solve (6) and (8) using the Nikiforov-Uvarov method [24, 25].

3. Nikiforov-Uvarov Method

Many problems in physics lead to the following second-order linear differential equation [24]:where and are polynomials of degree 2 at most and is at most linear in . Equation (9) is sometimes called of hypergeometric type. Let us consider ; this will transform (9) to the following differential equation for:where we assumed the following conditions:where and are constants chosen such that is polynomial which is at most linear in and . This will transform (10) to the following:where and are polynomials of degrees 2 and 1, respectively. In this case, solutions to (12) are polynomials of degree ; , where is given as follows:Equation (13) will be used to obtain the energy spectrum formula of the quantum mechanical system. We should point out here that the polynomial solutions to (12) for and on the boundaries of the finite space (the latter case is omitted for infinite space) are the classical orthogonal polynomials. It is well known that each set of polynomials is associated with a weight function . For the polynomial solutions to (12), this function must be bounded on the domain of the system and must satisfy . This weight function will be used to construct the Rodrigues formula for these polynomials, which readswhere is just a constant obtained by the normalization conditions and .

4. The Solutions of the -Dimensional Radial Equation

We use the NU method to solve (6); in the presence of our potential,where . Equation (15) cannot be solved analytically due to the centrifugal term . Different authors used different approximation techniques to allow an approximate analytical solution of (15) and these methods rely on Taylor expansion of the centrifugal potential in terms of the other components of the potential of interest [38]. In this work, we use the following approximation obtained by Taylor expansion [39]:The advantage of this approximation is that it is valid not only for but also for with high accuracy. Also, it satisfies the limits on at zero and infinity; that is, and , where RHS denotes the right-hand side of (16). Now, Using (16) back in (15), we getwhere , , and . Making change of variable and by writing , this transforms (17) to (9) with the polynomials being , , and . We now use (11d) to calculate , which readsThe choice of that makes (18) a polynomial of first degree must satisfy , where , , and . This giveswhere we will pick the negative part in (19) which makes . The function can be easily calculated using (11b):Using (19) and (20) in (13) and (11e), we getSolutions of (21) for areThe next step is to use the value of in (21) with the constraint on mentioned in (18), which is ; we obtainThe conditions for bound states are and , . Using (22) in (23), we write the bound states formula as follows:In terms of the original parameters , , and , the spectrum formula in -dimensions readswhere conditions for bound states become and . We will only take the (−) sign in (25) as explained below. The s-wave spectrum formula in three dimensions is the only exact solution that is obtained by setting in (25). However, for other higher states, the above solution is acceptable with high accuracy as far as the condition is satisfied.

The transformation makes the domain of the function be . This suggests a change of variable to bring the domain to that of Jacobi polynomials which are well-known classical orthogonal polynomials. By using (18) in (11a), we obtain , where and . The weight function can be easily calculated using (20) in , which gives , where and . The solution of (12) in our case is written in the following Rodrigues formula:By comparison to Jacobi polynomials, we conclude that , where is the Jacobi polynomial of order in . Thus, the bound state solution of the radial wave equation now readswhere is just a normalization constant. We must clarify here that, for Jacobi polynomials, we have to have . Thus, the parameters and are chosen to satisfy and . Moreover, since those parameters depend on the energy as we mentioned previously, this yields us to reject the (+) sign in (24) and (25). Consequently, bound states occur for (which does not violate the old restriction ) and . The latter condition on is already satisfied as we can see in (25), so we do not have to worry about it.

To calculate the normalization constant , we first use the following identity of Jacobi polynomials [40]:Next, we use the normalization constraint , where ; this givesTo calculate the integral in (29), we will use the following very useful integral formula [41]:where is the generalized hypergeometric function [41]. By direct comparison between (29) and (30) we get , where is given as follows:where and .

The only issue that is left for discussion in this section is that the solutions of the radial wave equation , where denotes the maximum number in which we get bound states, are not orthogonal! But they are normalized as we discussed above. We know that Hermitian operators with distinct eigenvalues must have orthogonal eigenvectors [42]. To solve this problem, one must use the Gram-Schmidt (GS) method to obtain an orthonormal set by linear combinations [43]. The latter set will be the solutions of the radial wave equation. The process is a bit lengthy and we will not be able to do it here. However, we encourage the interested reader to do these calculations by referring to the process of GS.

5. Solutions of the Angular Equations

It is well known from literature that solutions of (7) are written in terms of Jacobi polynomials as follows [41]:where is just a constant factor, , and . Moreover, the latter parameters are written in terms of the quantum numbers as , which yields and . The above solution was obtained using different methods including the NU technique [24]. Hence, solutions of (7) are written below:To solve (8), we introduce coordinate transformation as , which giveswhere for real parameters are related to by a factor of . Equation (34) is of hypergeometric type with the polynomials being , , and , where , , and . The solutions of (34) are written as , where satisfies (11a). Next, we need to find the function using (11d); we find that this function takes the following form:where and the parameter defined in (11e) must satisfy . The latter constraint will be used later to obtain the eigenvalues of (34). Now, we use (35) in (11b); we get , which satisfies . Using (11a), we can obtain to be , where and . We can also calculate the weight function by solving , which gives . The Rodrigues formula of the polynomials readswhere is just a constant. By direct comparison with the Rodrigues formula of Jacobi polynomials , we conclude that . As required by Jacobi polynomials, we must impose that . Now, we use (35) and (13) in (11e) to obtain the following quadratic formula for :The solutions of (37) are given below:Moreover, we use to obtain another solution for k:Direct comparison between (38) and (39) givesIn the next section, we will consider different examples and try to obtain the unknown parameters for each case.