Applied Physics Research; Vol. 4, No. 3; 2012 ISSN 1916-9639 E-ISSN 1916-9647 Published by Canadian Center of Science and Education 94 Heavy Quarkonium Mass Spectra in A Coulomb Field Plus Quadratic Potential Using Nikiforov-Uvarov Method Ahmed Al-Jamel 1 & Hatem Widyan 1 1 Physics Department, Al Al-Bayt University, Mafraq, Jordan Correspondence: Ahmed Al-Jamel, Department of Physics, Al Al-Bayt University, Mafraq 25113, Jordan. Tel: 962-2-629-7000-3592. E-mail: aaljamel@gmail.com, aaljamel@aabu.edu.jo Received: May 30, 2012 Accepted: June 16, 2012 Online Published: July 18, 2012 doi:10.5539/apr.v4n3p94 URL: http://dx.doi.org/10.5539/apr.v4n3p94 Abstract In this work, the spin-averaged mass spectra of heavy quarkonia (ҧ and  ത ) in a Coulomb plus quadratic potential is studied within the framework of nonrelativistic Schrodinger equation. The energy eigenvalues and eigenfunctions are obtained in compact forms for any -value using Nikiforov-Uvarov method. The obtained results are used to produce potential parameters (, , and ߜ) for the charmonium and bottomonium systems, from which then their full mass spectra are determined. The obtained values are compared with the available experimental results. The predictions from our model are found to be in good agreement with the experimental results. As a side result, the Hydrogen atom known spectrum is recovered. Keywords: heavy quarkonium, mass spectra, Nikiforov-Uvarov 1. Introduction Since their discoveries, investigation of heavy quarkonium systems ( ,  ,  , ҧ) provides us with a crucial role for quantitative tests of QCD and the standard model. For detailed review of heavy quarkonium physics recent progress, see e.g. (Brambilla & Vario, 2007; Brambilla et al., 2005; Zalewski, 1998) and the references therein. Because of the heavy masses of the constituent quarks ( ௤ ൐൐ Λ ொ஼஽ , i.e. than about 200 MeV, where Λ ொ஼஽ is the hadronization scale), many features of these systems can be studied within the framework of nonrelativistic Schrodinger equation, where one assumes that the quark-antiquark strong interaction is described by a phenomenological potential. There are many potential models that are commonly used to study heavy quarkonium spectra; for instance, Martin, logarithmic, and Cornell potentials (Al-Jamel, 2011; Patel & Vinodkumar, 2009; Rai, Patel, & Vinodkumar, 2008; Reyes, Rigol, & Soneira, 2003; Zalewski, 1998). Any of these potential should take into account the two distinctive features of the strong interaction, namely, asymptotic freedom and confinement. A successful potential model for such systems is the one that produces its mass spectra in agreeing with the experimental data within about 20 MeV and leptonic decay widths within a factor of two (Bhanot & Rudaz, 1978). The main obstacle in such studies arises due to the abscence of the exact solutions of Schrödinger equation for such systems, particulalry when the centrifugal potential ԰ మ ௟ሺ௟ାଵሻ ଶఓ௥ మ is included. For 0 , there are some approximation techniques, analytical and numerical, were developed, such as supersymmetry (Morales, 2004), 1/ expansion (Bag, Panja, & Dutt, 1992), Pekeris approximation (Pekeris, 1934), variational methods (Montgomery, 2001, 2011), and asymptotic iteration methods (Ciftci, Hall, & Saad, 2009). In the present paper, we consider the mass spectra of heavy quarkonium systems in a Coulomb plus quardatic potential using Nikiforov-Uvarov method. In section 2, we review the main formalism of the conventional Nikiforov-Uvarov (NU) method used in our analysis. In section 3, we present our main problem and its analytic solution. In section 4, results and discussion are given. In the last section, summary and conclusions are presented. 2. Nikiforov-Uvarov Method The Nikiforov-Uvarov method (hereafter, NU) is a method that provides us an exact solution of nonെrelativistic Schrödinger equation, or Schrödinger-like equation, for certain shape of potentials. It is based on the solutions of general second order linear differential equation with special orthogonal functions (Nikiforov & Uvarov, 1988; Szego, 1975). With an appropriate ݏൌ ݏሺݔሻ coordinate transformation, the Schrödinger equation in one