         1,2,3 Theoretical Physics Section, Department of Physics, University of Ilorin, P.M.B. 1515, Ilorin, Nigeria..  Sept. 2010 E-mails: wazzy4real@yahoo.com , mjpysics@yahoo.com 4 Department of Physics, University of Stellenbosch, Matieland, Private Bag XI. Stellenbosch, South Africa. Email: ttibrahimng@yahoo.com Bound States of the Relativistic Dirac Equation with Equal Scalar and Vector Eckart Potentials Using the Nikiforov-Uvarov Method     ,               ABSTRACT: The relativistic Dirac equation with equal Eckart scalar and vector potentials is solved using the Nikiforov-Uvarov method. The exact energy equation and the spinor wave function for the s- wave bound states are obtained. A straightforward extension to shifted Hulthén potential is also presented. KEYWORDS: I. INTRODUCTION It is well known that when a particle is in a strong potential field, the relativistic effect must be considered, leading to the relativistic quantum mechanical description of such particle [1-4]. In the relativistic limit the particle motion are commonly described using either the Klein-Gordon equation or the Dirac equation [2, 3] depending on the spin character of the particle. The spin zero particles for example, the mesons, are described by the Klein-Gordon equation. On the other hand, however, the spin-half particles, such as an electron, are satisfactorily described by the Dirac equation. Because the exact solutions of these equations with physically relevant potentials play an important role in our understanding of the relativistic phenomena, attentions have therefore been devoted to obtaining their analytical solutions using different potential models [5-14]. In recent time, a number of articles have studied the bound states of the Klein-Gordon equation and the Dirac equation with mixed potentials by assuming equal scalar and vector potentials. For example, these investigations have employed anharmonic oscillator potential [4], Kratzer potential [8, 15], Hulthén potential [16], Woods-Saxon potential [17], the Pöschl-Teller potential and Rosen- Morse potential [18], and the harmonic oscillator plus a novel angle-dependent potential [19]. Different methods such as the supersymmetry, pseudospin symmetry operator technique, and the Nikiforov-Uvarov technique, etc, have been used to solve the differential equations arising from these considerations. In this paper, our focus is on the bound state solution of the Dirac equation with equal scalar and vector potentials for the Eckart potential using the elegant Nikiforov-Uvarov (NU) method [20]. The Eckart potential, introduced by Eckart [21] in 1930, is a diatomic molecular potential model widely used in applied Physics [22] and chemical Physics [23, 24].