l-state Solutions of the Relativistic and Non-Relativistic Wave Equations for Modified Hylleraas-Hulthen Potential Using the Nikiforov-Uvarov Quantum Formalism

An exact analytical and approximate solution of the relativistic and nonrelativistic wave equations for central potentials has attracted enormous interest in recent years. By using the basic Nikiforov-Uvarov quantum mechanical concepts and formalism, the energy eigenvalue equations and the corresponding wave functions of the Klein–Gordon and Schrodinger equations with the interaction of Modified Hylleraas-Hulthen Potentials (MHHP) were obtained using the conventional Pekeris-type approximation scheme to the orbital centrifugal term.The corresponding unnormalized eigen functions are evaluated in terms of Jacobi polynomials. article History Received: 4 June 2018 Accepted: 28 June 2018


introduction
Quantum mechanical Wavefunctions and their corresponding eigenvalues give significant information in describing various quantum systems 1-3 .Bound state solutions of relativistic and nonrelativistic wave equation arouse a lot of interest for decades.
Schrodinger wave equations constitute nonrelativistic wave equation while Klein-Gordon and Dirac equations constitute the relativistic wave equations 1-5 .The quantum mechanical interacting potentials (MHHP) can be used to compute and predict the bound state energies for both homonuclear and heteronuclear diatomic molecules.Other potentials that have been used to investigate bound state solutions are as follows: Coulomb, Poschl-Teller, Yukawa, Hulthen, Hylleraas, pseudoharmonic, Eckart and many other potential combinations 6-13 .The aforementioned potentials are studied with some specific quantum mechanical methods and concepts like the following:Wentzel, Kramers, and Brillouin known as the WKB approximation, asymptotic iteration method, Nikiforov-Uvarov method, formular method, supersymmetric quantum mechanics approach, exact quantization, and many more 14-24 .
In theoretical physics, the shape form of a potential plays a significant role, particularly when investigating the structure and natureof the interaction between systems.Therefore, our aim, in this present work, is to investigate approximate bound state solutions of the Klein-Gordon and Schrodinger equations with newly proposed Modified Hylleraas-Hulthen potential (MHHP) using the conventional parametric Nikiforov-Uvarov (NU) method.The solutions of this equation will definitely give us a wider and deeper knowledge of the properties of molecules moving under the influence of the mixed interacting potentials which is the goal of this paper.The parametric NU method is very convenient and does not require the truncation of a series like the series solution method which is more difficult to useThis article is divided into five sections.Section 1 is the introduction; Section 2 is the brief introduction of Nikiforov-Uvarov quantum mechanical concept.In Section 3, we presented the angular solutions to Klein-Gordon and Schrodinger wave equations using the proposed potential and obtained both the energy eigenvalue and their corresponding normalized.We gave a brief discussion and conclusion in sections 4 and 5 respectively.

theory of Parametric Nikiforov-Uvarov Method
The parametric form is simply using parameters to obtain explicitly energy eigenvalues and it is still based on the solutions of a generalized second order linear differential equation with special orthogonal functions.The NU is based on solving the second order linear differential equation by reducing to a generalized equation of hyper-geometric type.This method has been used to solve the Schrödinger, Klein-Gordon and Dirac equation for different kind of potentials 24-31 .The second-order differential equation of the NU method has the form.
The parameters obtainable from equation ( 4) serve as important tools to finding the energy eigenvalue and eigenfunctions.They satisfy the following sets of equation respectively .... (7) and P n is the orthogonal polynomials.

Solutions of the klein-gordon Equation
The Klein-Gordon Equation 29 with vector V(r), potential in atomic units (ħ = c = 1) is given as ....( 8) Where E,M,l and V(r The Hulthen potential is one of the important shortrange potentials, which behaves like a Coulomb potential for small values of r and decreases exponentially for large values ofr 33 .The Hulthen potential in it simplest form is given as: .... (10) Where V o and S are the potential depth and the transformation parameter respectively.

Conclusion
In this paper, we solved explicitly the Klein-Gordon and Schrodinger equations for the modified Hylleraas plus Hulthen potential for arbitrary states by using the parametric form of the Nikiforov-Uvarov method.By using the Pekeris-type approximation for the centrifugal term, we obtained approximately the energy eigenvalues and the unnormalized wave function expressed in terms of the Jacobi polynomials for arbitrary wave states.It is hope that the results we obtained in this research work could enlarge and enhance the application of the Hylleraas-Hulthen potentials (which is our newly proposed potentials) in the relevant fields of physics and atomic spectroscopy.