Solutions to the Schr？dinger Equation with Inversely Quadratic Yukawa Plus Inversely Quadratic Hellmann Potential Using Nikiforov-Uvarov Method

The solutions to the Schr？dinger equation with inversely quadratic Yukawa and inversely quadratic Hellmann (IQYIQH) potential for any angular momentum quantum number have been presented using the Nikiforov-Uvarov method. The bound state energy eigenvalues and the corresponding unnormalized eigenfunctions are obtained in terms of the Laguerre polynomials. The NU method is related to the solutions in terms of generalized Jacobi polynomials. In the NU method, the Schr？dinger equation is reduced to a generalized equation of hypergeometric type using the coordinate transformation . The equation then yields a form whose polynomial solutions are given by the well-known Rodrigues relation. With the introduction of the IQYIQH potential into the Schr？dinger equation, the resultant equation is further transformed in such a way that certain polynomials with four different possible forms are obtained. Out of these forms, only one form is suitable for use in obtaining the energy eigenvalues and the corresponding eigenfunctions of the Schr？dinger equation. 1. Introduction The bound state solutions to the Schr？dinger equation (SE) are only possible for some potentials of physical interest [1–5]. Quite recently, several authors have tried to solve the problem of obtaining exact or approximate solutions to the Schr？dinger equation for a number of special potentials [6–10]. Some of these potentials are known to play very important roles in many fields of physics such as molecular physics, solid state, and chemical physics [8]. The purpose of the present work is to present the solution to the Schr？dinger equation with the inversely quadratic Yukawa potential [11] plus inversely quadratic Hellmann potential [12] of the forms The sum of these potentials can be written as where represents the internuclear distance, and are the strengths of the Coulomb and Yukawa potentials, respectively, is the screening parameter, and is the dissociation energy. Equation (2) is then amenable to Nikiforov-Uvarov method. Ita [13] has solved the Schr？dinger equation for the Hellman potential and obtained the energy eigenvalues and their corresponding wave functions using expansion method and Nikiforov-Uvarov method. Also, Hamzavi and Rajabi [14] have used the parametric Nikiforov-Uvarov method to obtain tensor coupling and relativistic spin and pseudospin symmetries of the Dirac equation with the Hellmann potential. Kocak et al. [15] solved the Schr？dinger equation with the Hellmann potential using asymptotic iteration method and obtained energy eigenvalues and the wave functions. However, not