Bound State Solutions of Three-Dimensional Klein-Gordon Equation for Two Model Potentials by NU Method

In this study, we investigate the relativistic Klein-Gordon equation analytically for the Deng-Fan potential and Hulthen plus Eckart potential under the equal vector and scalar potential conditions. Accordingly, we obtain the energy eigenvalues of the molecular systems in di ﬀ erent states as well as the normalized wave function in terms of the generalized Laguerre polynomials function through the NU method, which is an e ﬀ ective method for the exact solution of second-order linear di ﬀ erential equations.


Introduction
The exact solution is of paramount importance in quantum mechanics as it carries essential information on the quantum systems under investigation. It is possible only for quantum systems such as H 2 and harmonic oscillator. As for the majority of quantum systems, the approximation method needs to be used. In most quantum systems, for the analytical solution, methods such as the Nikiforov-Uvarov method [1], quantization rules [2], ansatz method [3], supersymmetry (SUSY) method [4], and series expansion [5] have been used for any arbitrary l state.
Recently, the bound state of the Schrödinger equation has been solved by the Deng-Fan potential [6], modified Morse potential [7], and Eckart potential [8] by approximation to the centrifugal term, and the wave function and energy level for bound states in any arbitrary l state have been identified. The bound state solutions of the Klein-Gordon equation with the Deng-Fan molecular potential are solved by Dong [9]. Wei et al. investigated the relativistic scattering states of the Hulthen potential by taking the same approximation [10]. Wei and Dong examined the approximate solution of the bound state of the Dirac equation with the second Pöschl-Teller potential under spin symmetry conditions and with scalar and vector modified potentials under pseudospin symmetry conditions [11,12]. They also solved the Dirac equation with the scalar and vector Manning-Rosen potentials under pseudospin symmetry conditions by using the function analysis method and algebraic formalism [13].
In our previous works, we solved the Schrödinger equation for different potentials for few-quark systems [14][15][16][17]. However, in the present work, we make use of the NU method to solve the Klein-Gordon equation for a diatomic molecule analytically. The NU method has recently been exploited in a variety of physical fields, including the Schrödinger equation with a spherically harmonic oscillatory ring-shaped potential [18] or the second Pöschl-Teller-like potential by the Nikiforov-Uvarov method [19].
In this paper, first we describe the Nikiforov-Uvarov method. In Review of Nikiforov-Uvarov (NU) Method, we consider the Deng-Fan potential and calculate the energy eigenvalue for different diatomic molecules. Next, we solve the Klein-Gordon equation analytically for the Eckart plus Hulthen potential through the NU method and obtain the energy eigenvalue. And finally, we present the results, discussion, and conclusion.

Review of Nikiforov-Uvarov (NU) Method
The Schrödinger equation can be converted into a secondorder differential equation as follows: where σðsÞ andσðsÞ denote polynomials at most of the second degree andτðsÞ is a first-degree polynomial. We use the following form to find the solution: By introducing Equation (3) into Equation (2), we arrive at where φðsÞ in terms of the Rodriguez formula appears as The weight function ρðsÞ holds in the following formula: The other solution factor is defined as In this method, the polynomial πðsÞ and parameter k are defined as where ψðsÞ is defined as By substituting k into Equation (7): and λ is defined as: The general form of the Schrödinger equation including any potential is Comparing Equation (12) with Equation (2), we get the parametersτ Based on the equations, the constant parameters are defined as The energy equation is obtained from Now we consider the eigenfunctions of the problem with any potential. We obtain the second part of the solution from Equation (4).
From the explicit form of the weight function obtained from Equation (5), we arrive at

Solving the Klein-Gordon Equation for the Ground State
The potential that is selected for molecular spectroscopy and molecular dynamics is of paramount importance. The Deng-Fan oscillator potential [20] is a simple potential model for diatomic molecules. It has the correct physical boundary conditions at r = 0 and ∞. It is defined by where D stands for the dissociation energy, r e for the equilibrium bond length, and α for the potential range. The shifted Deng-Fan potential is of the following form ( Figure 1) The radial part of the Klein-Gordon equation for a particle with a mass m and potential V ðrÞ is Because of the term 1/r 2 in Equation (21), it cannot be analytically solved except for l = 0. Therefore, a suitable approximation to the centrifugal term is required, as used in [9,21,22]: in which αr ≪ 1 and d 0 = 1/12. As illustrated in Figure 2, this approximation is very close to the term 1/r 2 . By introducing the potential and the approximation 1/r 2 and ℏc = 1, Equation (21) becomes By using the variable change s = e −αr which maps the halfline ð0, ∞Þ into the interval ð0, 1Þ, Equation (23) becomes By comparing Equations (11) and (24), the following coefficients are obtained: By combining Equations (23) and (11), the following quantities are obtained: 3 Advances in High Energy Physics Finally, we find the energy eigenvalue as in which In Table 1, we present potential parameters adopted from [23][24][25]. Furthermore, by the following data, we obtain the energy eigenvalue for diatomic molecules mentioned in Table 1.  In Table 2, we calculate energy levels for different n and l states and compare them with other findings in [23,24,26,27]. The radial wave function is of the following form: where 2 F 1 −n, n + ν + μ + 1 ; ν + 1 ; N nl is the normalization constant. P ðν,μÞ n denotes the Jacobi polynomial and 2 F 1 stands for the hypergeometric function. The N nl constant is defined as By using the following formula [28,29]: the normalization constant is obtained as and for the ground state n = 0 We plot wavefunction of the Deng-Fan potential (eV) as a function of rðA 0 Þ for the H 2 diatomic molecule in n = l = 0 in Figure 3.

Eckart plus Hulthen Potential
The Eckart plus Hulthen potential has been used for the analytical solution of the Schrödinger equation. This potential as a diatomic molecular potential model has been utilized in applied physics and chemical physics. The NU method has been exploited to solve the Schrödinger equation for the Eckart plus Hulthen potential [30]. However, in the present work, we make use of the NU method to solve the Klein-Gordon equation for the Eckart plus Hulthen potential. The Eckart plus Hulthen potential runs as shown in Figure 4: where V 0 and V 1 stand for the depths of potential well and α for the inverse of the potential range.
In this way, the potential is obtained as The radial Klein-Gordon equation by using the Eckart plus Hulthen potential in l = 0 is of the following form: With the variable change s = e −2αr , Equation (40) is transformed to With a simple comparison, the following quantities are obtained: 6 Advances in High Energy Physics Accordingly, the α i parameters are obtained through Equation (24). Moreover, by using Equation (14) and the energy eigenvalue is identified as The radial wave function is of the following form: The normalization constant is obtained as We plot wave function of the Eckart plus Hulthen potential (eV) as a function of rðA 0 Þ for the H 2 diatomic molecule in n = l = 0 in Figure 5.

Results and Discussion
In this study, we examined the solution of the Klein-Gordon equation for two different potentials. Accordingly, we calculated the energy eigenvalues and normalized wave function for the diatomic molecules for different n and l states through the NU method. In order to solve the Klein-Gordon equation, we utilized the NU method, which can also be used to identify the wave function and energy eigenvalue for any particular potential. However, with certain potentials, the wave equation fails to furnish the boundary conditions of the method.
The parameters related to the spectroscopic constants of these molecules, taken from [19][20][21] appear in Table 1. Table 2 shows the energy levels for diatomic molecules by using the Morse oscillator potential and the shifted Deng-Fan (sDF) potential. The findings comply well with refs. [23,24] and also with the energies calculated from AP (amplitude phase) [26,27].
Based on the results, as the quantum number n increases, the energy value decreases. Furthermore, an increase in l makes the particle less bound. It can further be inferred that in higher dimensions, the energy value decreases. To show the accuracy of our results, we have calculated the eigenvalues numerically for arbitrary n with l = 0, α = 0:05, and α = 0:1. Tables 3 and 4 show the energy levels of the Eckart plus Hulthen potential in atomic units ℏ = μ = 1 with l = 0, α = 0:05, and α = 0:1, respectively. It is observed that as α increases, the energy eigenvalue decreases.

Conclusion
Considering the importance of the molecular Deng-Fan potential and the Eckart plus Hulthen potential in molecular physics, chemical physics, molecular spectroscopy, and other related areas, we investigated the bound state solution of the relativistic wave equation. We provided exact solutions of the Klein-Gordon equation for these potentials by means of the Nikiforov-Uvarov (NU) method. We formulated the eigenvalues equation and the corresponding wave function in terms of hypergeometric functions via the NU method within an approximation to the centrifugal potential term.
As we know, there is no analytical solution for the radial equation for l ≠ 0. Therefore, the Klein-Gordon equation is transformed into a differential Schrödinger-like equation through a suitable coordinate transformation. The obtained energies are very close to the energies reported in other studies [19][20][21][22]. We preferred to calculate the energy eigenvalue of H 2 , CO, LiH, and HCl as diatomic molecules. The main advantage of these molecules is that their spectroscopic values are already known [31]. This feature has made them suitable candidates for working with in other studies, too, e.g., [32]. They also serve different purposes in various aspects of both physics and chemistry [22,33,34].
This method of approximation is simple and practical. It can be applied to different quantum models to enhance the accuracy of the energy eigenvalues for some potential models of exponential-type, such as the hyperbolical potential and the Manning-Rosen potential [35,36]. Our findings in this section are significant not only in theoretical and chemical physics but also in experimental physics since we obtained general results which are useful for studying nuclear charge radius, spin, and nuclear scattering. In the future, we plan to improve the approximation to solve the Bethe-Salpeter equation with different potentials.

Data Availability
We have not included a data availability statement in our manuscript.