A New Model to Describe Quarkonium Systems under Modified Cornell Potential at Finite Temperature in pNRQCD

Abstract

In the present work, the three-dimensional modified radial Schrödinger equation is analytically solved. The nonrelativistic interactions under new modified Cornell potential (NMCP, in short) at finite temperature, are extended to the symmetries of nonrelativistic noncommutative space phase (NRNSP, in short), using the generalized Bopp’s shift method in the case of perturbed nonrelativistic quantum chromodynamics (pNRQCD). W generalize this process by adding multi-variable coupling potentials , and together with the modified Cornell potential model in three-dimensional nonrelativistic quantum mechanics noncommutative phase space (3D-NCSP, in short). The new energy eigenvalues and the corresponding Hamiltonian operator are calculated in 3D-NCSP symmetries instead of solving the modified Schrödinger equation with the Weyl Moyal star product. The present results, in (3D-NCSP), are applied to the charmonium and bottomonium masses at finite temperature. The present approach successfully generalizes the energy eigenvalues at finite temperature in 3D-NCSP symmetries. It is found that the perturbative solutions of the discrete spectrum and quarkonium mass can be expressed by the Gamma function, the discreet atomic quantum numbers of the state and the potential parameters ( ), in addition to noncommutativity parameters ( and ). The total complete degeneracy of new energy levels of NMCP changed to become equals to the value instead the values in ordinary quantum mechanics. Our obtained results are in good agreement with the already existing literature in NCSP. Keywords: Schrödinger Equation, Heavy Quarkonium System, Cornell Potential, Noncommutative Space Phase, Bopp’s Shift Method. Subject Classification Numbers: 03.65.-w; 03.65.Ge; 03.65. Fd; 03.65.Ca