Non-Relativistic Bound State Solutions of Modified Quadratic Yukawa plus q -Deformed Eckart Potential

We have obtained approximate bound state solutions of Schrödinger wave equation with modified quadratic Yukawa plus q-deformed Eckart potential Using Parametric Nikiforov-Uvarov (NU) method. However, we obtained numerical energy eigenvalues and un-normalized wave function using con-fluent hypergeometric function (Jacobi polynomial). With some modifica-tions, our potential reduces to a well-known potential such as Poschl-Teller and exponential inversely quadratic potential. Numerical bound state energies were carried out using a well-designed Matlab algorithm while the plots were obtained using origin software. The result obtained is in agreement with that of the existing literature.


Introduction
Researchers have put on their interest over the years with the aim of investigating the bound state solutions of relativistic and nonrelativistic wave equations for different potentials. A few of these potentials have been solved exactly [1], while others can only be solved approximately [2] [3], with the use of different approximation schemes [4] [5]. Subsequently, various methods have been applied to obtain the solutions of the nonrelativistic wave equations with a chosen potential model. Methods include the factorization method [6], functional analysis approach [7] [8] [9], supersymmetry quantum mechanics (SUSYQM) [10], The Eckart potential is one of the most important exponential-type potentials in the literature and it has been widely used in physics and chemical physics [13] [14] [15]. Recently, [16] has employed the asymptotic iteration method to calculate any ℓ-state solutions of the Schrödinger equation with the Eckart potential by proper approximation of the centrifugal term. Energy eigenvalues and corresponding eigen functions are obtained explicitly. The energy eigenvalues are calculated numerically for some values of ℓ and n.
The Yukawa potential, which is also known as the screened Coulomb potential, is greatly important with applications cutting across nuclear physics and condensed matter physics [17] [18]. The screened Coulomb potential is used mostly in short-range interactions [19] [20]. The Yukawa potential is known to be the potential of a charged particle in a weakly non-ideal plasma. It also describes the charged particle effects in a sea of conduction electrons in solid-state physics.
With approximate analytical solution of the Yukawa potential with arbitrary momenta using the Nikiforov-Uvarov method, they obtained approximate analytical solutions of the radial Schrödinger equation for the Yukawa potential, and the energy eigenvalues and the corresponding eigen functions are calculated in closed forms [21]. Some numerical results are presented and show that these results are in good agreement with those obtained previously by other methods. As the screening parameter tends to zero, the energy levels of Yukawa potential are equivalent to that of the familiar pure Coulomb potential. Eckart potential is one of the most important exponential type potentials for the study of nuclear forces [22] [23]. Basically, many properties of nuclear forces are explained using Yukawa potential and to predict the existence of pi-meson. This is the main reason why the study of the combination of the two potential models is very significant and highly essential. However, after the above mentioned studies on these different potentials and their lofty importance, we seek to investigate the bound state solutions of the Schrodinger equation with the modified quadratic Yukawa plus q-deformed Eckart potential of the form: where 1 2 , , o v v v are the potential depth, q is the deformation parameter, r is the inter-nuclear distance and α is the screening parameter. It can be deduced that when

Nikiforov-Uvarov Method: Parametric Method
The parametric form is simply using parameters to obtain explicitly energy eigenvalues and it is based on the solutions of a generalized second order linear differential equation with special orthogonal functions [2]. The hypergeometric NU method has shown high utility in calculating the exact energy levels of all bound states for some solvable quantum systems.
Given a second order differential equation of the form where σ and σ are polynomials at most degree and τ is first degree polynomials. The parametric generalization of the N-U method is given by the generalized hypergeometric-type equation Thus Equation (2) can be solved by comparing it with Equation (3) and the following polynomials are obtained The parameters obtainable from Equation (3) serve as an important tool for finding the energy eigenvalue and eigenfunctions.
Where 1 2 , c c and 3 c are parametric constants, other parametric constants are given below The energy equation is given as The total wave function is given by where µ is the reduced mass, nl E is the rotational vibrational energy spectra of the diatomic molecules,  is the reduced Planck's constant, n and l are the radial and orbital angular momentum quantum numbers, respectively [7] [14].

Energy of the Proposed Potential
The energy can be calculated by substituting Equation (14) and Equation (15) into (5)

Deductions from the Energy Equation
Case 1: If we replace parameter The energy equation for Poschl-Teller potential becomes: Case 2: If we also replace parameter 1 we obtain exponential inversely quadratic potential which is expressed as Also, the energy equation becomes:

The Wave Function of the Proposed Potential
The wave function for the proposed potential can be calculated using equation (6), expressed in terms of Jacobi polynomial and substituting the necessary parameters. Thus, where n N is the normalization constant.

The Numerical Computation for Energy Equation of the Proposed Potential Using Schrödinger Equation
The numerical computation was carried out using MATLAB 8.0 software while graphical plot was carried out using origin software. The computation parameters have been incorporated in the numerical table. The energy equation for the proposed potential is given in Equation (18

Discussion
The results of the numerical computation that were carried out for the screening parameter α = (0.1 to 0.5) with the potential depths V 0 = 0.3, V 1 = 0.1, V 2 = 0.2. Tables 1-5 show that for α = (0.1 to 0.5), the numerical bound state energy decreases with an increase in quantum state. It is also observed that the energy increases with a decrease in screening parameter α. However, the energy spectral diagram as shown in Figures 1-5 shows unique quantization of different energy levels with respect to the quantum state. The numerical bound state energies obtained for the various values of α are predominantly negative which the sufficient and necessary condition for bound state solutions. Also the negative bound state eigenvalues obtained show that the potential is suitable for describing particle state and their bound state properties.

Conclusion
We used the Nikiforov-Uvarov method to compute bound state solution of non-relativistic wave equation. We developed a new potential model which is the     A. D. Antia et al.     modified quadratic Yukawa plus q-deformed Eckart potential. This potential reduces to a well-known potential called the Poschl-Teller potential and the exponential inversely quadratic potential. We also implement some algorithms to carry out numerical computation for the resulting energy equations for various values of screening parameters. Our analytic and computational results for the proposed potential are in agreement with the existing literature.