Abstract
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 different states as well as the normalized wave function in terms of the generalized Laguerre polynomials function through the NU method, which is an effective method for the exact solution of second-order linear differential equations.
1. 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 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 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 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 [14–17]. 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.
2. Review of Nikiforov-Uvarov (NU) Method
The Schrödinger equation can be converted into a second-order differential equation as follows: where and denote polynomials at most of the second degree and 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 in terms of the Rodriguez formula appears as
The weight function holds in the following formula:
The other solution factor is defined as
In this method, the polynomial and parameter are defined as
where is defined as
By substituting into Equation (7): and is defined as:
The general form of the Schrödinger equation including any potential is
Comparing Equation (12) with Equation (2), we get the parameters
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
is the Jacobi polynomial. From Equation (6), we arrive at
Then the general solution becomes
3. 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 and ∞. It is defined by where stands for the dissociation energy, for the equilibrium bond length, and 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 and potential is
Because of the term in Equation (21), it cannot be analytically solved except for . Therefore, a suitable approximation to the centrifugal term is required, as used in [9, 21, 22]: in which and . As illustrated in Figure 2, this approximation is very close to the term . By introducing the potential and the approximation and , Equation (21) becomes
By using the variable change which maps the half-line into the interval , 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:
Finally, we find the energy eigenvalue as in which
In Table 1, we present potential parameters adopted from [23–25]. 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
is the normalization constant. denotes the Jacobi polynomial and stands for the hypergeometric function. The constant is defined as
By using the following formula [28, 29]: the normalization constant is obtained as and for the ground state
We plot wavefunction of the Deng-Fan potential (eV) as a function of for the diatomic molecule in in Figure 3.
4. 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 and stand for the depths of potential well and for the inverse of the potential range. The hyperbolic functions are defined as
In this way, the potential is obtained as
The radial Klein-Gordon equation by using the Eckart plus Hulthen potential in is of the following form:
With the variable change , Equation (40) is transformed to
With a simple comparison, the following quantities are obtained:
Accordingly, the 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:
With the variable change and the parameterrs of Equation (42), we obtain
The normalization constant is obtained as
We plot wave function of the Eckart plus Hulthen potential (eV) as a function of for the diatomic molecule in in Figure 5.
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 and 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 [19–21] 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 increases, the energy value decreases. Furthermore, an increase in 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 with , , and . Tables 3 and 4 show the energy levels of the Eckart plus Hulthen potential in atomic units with , , and , respectively. It is observed that as increases, the energy eigenvalue decreases.
6. 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 . 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 [19–22]. We preferred to calculate the energy eigenvalue of , 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.
Conflicts of Interest
The authors have not declared any conflict of interest.