Heavy quarkonium systems for the deformed unequal scalar and vector Coulomb–Hulthén potential within the deformed effective mass Klein–Gordon equation using the improved approximation of the centrifugal term and Bopp’s shift method in RNCQM symmetries

Abstract

In this study, the analytical solutions of the Klein–Gordon equation for any [Formula: see text] states of the modified effective mass potential under the modified unequal scalar and vector Coulomb–Hulthén potential (MUSVCH-P) are derived by using an approximation method to the centrifugal potential term in the symmetries of relativistic noncommutative three-dimensional real space (RNC: 3D-RS). The new analytical expressions for eigenvalues of the energy spectrum and the new mass of mesons, such as charmonium and bottomonium that have the quark and antiquark flavor, have been estimated by using Bopp’s shift method, and perturbation theory. The energy state equation depends on the global parameters characterizing the noncommutativity space and the potential parameter [Formula: see text] in addition to the Gamma function and the discreet atomic quantum numbers [Formula: see text]. The expression for the new energy spectra is applied to obtain the new mass spectra of heavy quarkonium systems (charmonium and bottomonium) in the symmetries of (RNC: 3D-RS). The comparisons show that our theoretical results are in very good agreement with the reported works.

The Effects of Noncommutativity on the Energy Spectra for Mass-Dependent Klein-Gordon and Shrödinger Equations with Vector Quark-Antiquark Interaction and Harmonic Oscillator Potential: Application to HLM Systems

“…In the physics literature, this is known as Bopp's shift, and this quantization procedure is called Bopp quantization. It is known to the specialists that Bopp's shift method [47][48][49][50] has been applied effectively and has succeeded in simplifying the three basic equations deformed Schrödinger equation DSE [51][52][53][54][55][56][57][58][59][60][61][62], the DKGE [63][64][65][66][67][68][69][70][71][72][73][74], the deformed Dirac equation DDE [75][76][77][78][79][80] and the deformed Duffin-Kemmer equation [81][82][83][84] with the notion of star product to the SE, the KGE, the DE and the Duffin-Kemmer equation with the notion of ordinary product, respectively. The results of the application of this method were very useful and yielded promising results in many physical and chemical fields.…”

The Impact of deformed space-phase parameters into HAs and HLM systems with the improved Hulthen plus Hellmann potentials model in the presence of temperature-dependent confined Coulomb potential within the framework of DSE

“…, respectively. It is known to specialists that BSM has been applied effectively and has succeeded in simplifying the known four fundamental equations: the NRDSE [67][68][69][70], the relativistic deformed Klein-Gordon equation RDKGE [60][61][62][63][64][65][66][67][68], the relativistic deformed Dirac equation RDDE [20,41,69], and the deformed relativistic Duffin-Kemmer-Petiau equation (DRDKPE) [70] with the notion of star product to the NRSE, RKGE, RDE and RDKPE with the notion of ordinary product, respectively. Thus, BSM is based on reducing second-order linear differential equations of DNRSE, DRKGE, DRDE, and DRDKPE with star product to second-order linear differential equations of NRSE, RKGE, RDE, and RDKPE without star products with simultaneous translation in the 3D-NCSP.…”

Analytical Expressions to Energy Eigenvalues of the Hydrogenic Atoms and the Heavy Light Mesons in the Framework of 3D-NCPS Symmetries Using the Generalized Bopp's Shift Method

“…The new coupling Lη is L x η 12 + L y η 23 + L z η 13 while LΘ is defined in the first section. It should be noted that GBSM has been successfully applied at the relativistic level in the field of high energies using both the Klein-Gordon equation [65][66][67][68][69][70][71][72][73][74][75][76][77][78], Dirac equation [79][80][81] and Duffin-Kemmer-Petiau equation [82]. Thus, the reduced SE (without star product) can be written as…”