The Influence of Deformation Space-Space on High and Low Energy Spectra of Fermionic Particles and Spectra of Heavy Quarkonia with Improved Hulthén Plus Hyperbolic Exponential Inversely Quadratic Potential

Abstract

In this work, the modified approximation to the centrifugal barrier term is applied to find approximate bound-state solutions of the deformed Dirac equation for the spin and pseudospin symmetries in a model with the improved hyperbolic Hulth´en and hyperbolic exponential inversely quadratic potentials (IHHEIQPs) using the parametric method of Bopp’s shift and the standard perturbation theory in the extended relativistic quantum mechanics (ERQM). Our results indicate that the new energy eigenvalues are highly sensitive to the potential parameters (ν1, A) and to the values of quantum atomicnumbers (j, k, l, m,̃︀ l, ̃︀ m, s, ̃︀s), range of the potential ν, and noncommutativity parameters (θ, σ, β). We found that the effect of a space-space deformation gives a correction in the energy spectrum, where the main energy term remains due to the effect of the hyperbolic Hulth´en and hyperbolic exponential inversely quadratic potentials known in the literature. The new nonrelativistic energies are obtained by applying the nonrelativistic limit to the relativistic spin-energy equation in the extended nonrelativistic quantum mechanics (ENRQM). The proposed potential model reduces to the improved Hulth´en and exponential inversely quadratic potentials as special cases in ERQM. The present results are applied for calculating the new mass spectra Mhiqpnc−nl of heavy mesons such as cc, bb, bc, bs, cs, and bq,q = (u, d) in ENRQM. It turns out that the values of masses come from the contribution of the mass spectra Mhiqpnl in NRQM, while the effect of a space-space deformation δMhiqpnc−nl is an infinitesimal correction as compared with Mhiqpnl. Our results seem to be significant and agree perfectly with the ones in the literature.