1. Introduction

The interactions of quantum systems with the spherically symmetric potentials have been studied by researchers since the discovery of quantum mechanics [¹-³]. From the early days of quantum mechanics, the study of exactly solvable problems has attracted a considerable attention in many areas of physics, particularly in atomic physics, information theory, nuclear physics, particle physics, molecular physics and its importance cannot be overemphasized [⁴,⁵]. The exact solution of the relativistic and non-relativistic equations for most of these types of potentials is not attainable and approximations become indispensable [⁶]. For instance, in the case of the Schrödinger equation, when the angular momentum quantum number is present, one can resort to solve the non-relativistic equation approximately via a suitable approximation scheme [⁷]. Some of such approximations, yielding good results, consist of the conventional approximation scheme proposed by Greene and Aldrich [⁸], the improved approximation scheme by Jia et al. [⁹], the elegant approximation scheme [¹⁰], the Pekeris approximation [¹¹], the improved approximation scheme by Yazarloo et al. [¹²], improved approximation scheme in Refs. [¹³-¹⁵] and in Refs. [¹⁶,¹⁷].

Over the past decades, problems involving the multidimensional Schrödinger equation have been addressed by many researchers using different analytical procedures. For examples, Ntibi et al. [¹⁸] investigated the analytical solution of the D-Dimensional radial Schrödinger equation with Yukawa potential. Oyewumi et al. [¹⁹] studied the N-dimensional Pseudoharmonic oscillator. Gönül and Koc¸ak [²⁰] investigated explicit solutions for N-dimensional Schrödinger equations with position-dependent mass. The N-dimensional Kratzer-Fues potential was discussed by Oyewumi [²¹], Ikhdair and Sever [²²], studied the modified Kratzer-Fues potential plus the ring shape potential in D-dimensions while Dong [²³], reviewed the wave equations in higher dimensions.

The Varshni potential is greatly important with applications, cutting across nuclear physics, particle physics and molecular physics. The Varshni potential model takes the form [²⁴,²⁵]:

where η ₀ and η ₁ are potential strength parameters, δ is the screening parameter and r the inter-nuclear separation. The Varshni potential is a short-range repulsive potential energy.

The Hulthén potential is one of the important short-range potentials in physics. Its relevance to diverse areas of physics including nuclear and particle physics, atomic physics, molecular physics, condensed matter and chemical physics has been of great interest and concern to researchers in recent times [²⁶,²⁷]. The Hulthén potential model takes the form [²⁸]

where δ is the screening parameter and η ₂ is the potential strength constant which is sometimes identified with the atomic number when the potential is used for atomic phenomena. Many authors have obtained bound state solutions of SE with this potential. For example, Tazimi and Ghasempour [²⁹] used the Nikiforov Uvarov (UN) method to obtain bound state solutions of the three-Dimensional Klein-Gordon equation for two model potentials. Quantization rule was employed by Ikhdair and Abu-Hasna [³⁰] to obtain the solution to the Hulthén potential in arbitrary dimension with a new approximate scheme for the centrifugal term. Okorie et al. [³¹] obtain the solution of Schrödinger equation (SE) with energydependent screened Coulomb potential with the new form of Greene-Aldrich approximation using the NU method.

With the experimental proof of the Schrödinger wave equation, researchers have made great effort to solve the SE by the combination of two or more potentials, which can be used for a wider range of applications [³²]. For example, Edet et al. [³³] obtained bound state solutions of the SE for the modified Kratzer potential plus screened Coulomb potential including a centrifugal term. Also, Edet et al. [³⁴] obtained any-state solutions of the SE interacting with Hellmann-Kratzer potential model. William et al. [³⁵] obtained bound state solutions of the radial SE by the combination of Hulthén and Hellmann potential within the frame-work of Nikiforov-Uvarov (NU) method for any arbitrarystate, with the Greene-Aldrish approximation in the centrifugal term. Inyang et al. [³⁶] studied any-state solutions of the SE interacting with class of Yukawa-Eckart potentials within the framework of NU method. Hence, motivated by the success of the combination of exponential-type potentials, we seek to investigate the bound state solutions of the SE by the combination of Varshni potential, of Eq. (1), and the Hulthén potential of Eq. (2) using the NU method. The resulting potential, so-called the Varshni-Hulthén potential (VHP), takes the form

Considering the achievements in the previous studies, we combine the potentials to allow for more physical application and comparative analysis to existing studies of molecular physics. Also in molecular physics, it is well-known that the potential energy functions with more parameters have a tendency to fit experimental data better than those with fewer parameters [³⁷]. The plot of the combined potential for different values of the screening parameter δ is presented in Fig. 1.

However, it must be noted that the exact solution of the SE with the combined potentials in Eq. (3) is not possible due to the presence of the centrifugal term. Therefore, to obtain approximate solutions, we employ a suitable approximation scheme. It is found that such approximation proposed by Greene and Aldrich [⁸],

is a good approximation to the centrifugal or inverse square term for a short range potential which is valid for δࠏ1.

The paper is organized as follows: In Sec. 2, we derive the bound state solutions of the SE with VHP using NU method and also derive the corresponding normalized wave function. In Sec. 3, we present the results and discussion and finally, we make a concluding remark in Sec. 4.

2. Bound state solutions of the Schrödinger equation with Varshni plus Hulthén potentials

In this study, we adopt the Nikiforov-Uvarov method which is based on solving the second-order differential equation of the hypergeometric type. The details can be found in Ref. [³⁸]. The SE takes the form Ref. [³⁹]

where l, µ, r, N, and ℏ and are the angular momentum, the reduced mass of the particle, inter-particle distance, the number of space, and reduced Plank constant, respectively. Substituting Eqs. (3) and (4) into Eq. (5) we obtain

By using coordinate transformation

we obtain the differential equation as

where

Now that Eq. (8) and Eq. (1) of Ref. [³⁸] are in the same shape, we have the following parameters:

Substituting Eq. (10) into Eq. (11) of Ref. [³⁷], we obtain π(x) as

where

To find the constant k, the discriminant of the expression under the square root of Eq. (11) must be equal to zero. As such we have that

Substituting Eq. (13) into Eq. (11) yields π(x) as

and τ(x) can be written as

Taking the derivative of Eq. (15) with respect to x, we have

Referring to Eq. (10) of Ref. [³⁸], we define the constant γ as,

and taking the derivative of σ(x) with respect to x from Eq. (10), we have

Substituting Eqs. (15) and (18) into Eq. (13) of Ref. [³⁸], we obtain

By comparing Eqs. (17) and (19), using Eq. (9) yields the energy eigenvalues equation of the VHP as a function of n and l as

where

To obtain the corresponding wavefunction, we consider Eq. (3) of Ref. [³⁸], and upon substituting Eqs. (10) and (14) and integrating, we get

To get the hypergeometric function, we first determine the weight function ρ(x). By substituting Eqs. (10) and (14) into Eq. (3) of Ref. [³⁸] and after integrating, we obtain

Hence, by substituting Eqs. (10) and (23) into Eq. (2) yields the Rodrigues equation given as

where N _nl is the normalization constant. Equation (24) is equivalent to

where Pnα,β is the Jacobi Polynomials. Hence, the wave function is given as

Using the normalization condition, we obtain the normalization constant as follows:

From our coordinate transformation of Eq. (7), we have that

By letting, y = 1 − 2x, we have

Let

and

By substituting Eq. (29) into Eq. (28), using Eq. (30), we have

According to Onate and Ojonubah [⁴⁰], integral of the form in Eq. (31) can be expressed as

Hence, by comparing Eq. (31) with the standard integral of Eq. (32), we obtain the normalization constant as

3. Discussion

Solution of the radial N-dimensional SE for the newly proposed potential obtained by the superposition of Varshni and Hulthén potentials otherwise known as Varshni-Hulthén potential are obtain within the framework of NU method by approximation to the centrifugal barrier. In Table I, we reported the numerical energy eigenvalues (eV) with ℏ= 2, µ = 1 for 1S, 2S, 2P, 3S, 3P, 3d, 4S and 4P states with the potential strength: η ₀ = η ₁ = −1, η ₂ = 0.025 and for N = 3, 4, and 5, respectively. We observed that the energy increases as the screening parameters and N increases. In Table II, we have presented the energy eigenvalues (eV) with ℏ= 2µ = 1 of the VHP for N = 3 with three different values of the potential range: η ₀ = −1; η ₁ = −1, η ₂ = 0.050: η ₀ = −2, η ₁ = −1, η ₂ = 0.075 and η ₀ = η ₁ = −2, η ₂ = 0.10, respectively, and observed that the energy increases as the screening parameters increases for constant N. We have plotted the graph of energy eigenvalues against the number of dimensional space, potential strength, screening parameter, and the reduced mass in the ground and excited states. In Fig. 2, we show the variation of energy as a function of and observed that the energy increases as both and increases. From Figs. 3a) and b) - Figs. 5a) and b) respectively, we plotted the ground and excited states energy eigenvalues of the different quantum states as a function of the VHP strengths. We observed that there is a decrease in energy in both the ground and excited states as the potential strength, η ₀, η ₁, and η ₂ respectively, increases.

The variation of energy eigenvalues of VHP as a function of the screening parameter is as shown in Fig. 6a) and 6b). We observed that the energy increases as the screening parameter increases in the ground and excited states. In Fig. 7a) and 7b), we show the variation of energy eigenvalues as a as a function of the reduced mass. It is observed that there is a decrease in the ground and excited states energy for different quantum states as the reduced mass, µ increases.

4. Conclusion

The analytical solutions of the N-dimensional Schrödinger equation for the newly proposed Varshni-Hulthén potential are obtained via NU method by using Greene-Aldrich approximation scheme to the centrifugal barrier. The numerical energy eigenvalues and the corresponding normalized eigenfunctions are obtained for various values of orbital angular momentum quantum number and space dimension. We have also obtained the numerical energy eigenvalues for two special cases of the newly proposed potential and their results were found to be in agreement with the existing literature. However, the behavior of the energy for the ground state and several excited states is illustrated graphically. Therefore, studying the analytical solution of the N-dimensional Schrödinger equation for the newly proposed Varshni-Hulthen potential could provide valuable information on the quantum mechanics dynamics at atomic and molecular physics and opens new window.