The Influence of Deformation Phase-Space on Spectra of Heavy Quarkonia in Improved Energy Potential at Finite Temperature Model of Shrodinger Equation Via the Generalized Boob’s Shift Method and Standard Perturbation Theory

Abstract

In this work, we obtain solutions of the deformed Schrödinger equation (DSE) with improved internal energy potential at a finite temperature model in a 3-dimensional nonrelativistic noncommutative phase-space (3D-NRNCPS) symmetries framework, using the generalized Bopp’s shift method in the case of perturbed nonrelativistic quantum chromodynamics (pNRQCD). The modified bound state energy spectra are obtained for the heavy quarkonium system such as charmonium cc- and bottomonium bb- at finite temperature. It is found that the perturbative solutions of the discrete spectrum are sensible to the discreet atomic quantum numbers (j,l,s,m) of the ( QQ- (Q=c,b)) state, the parameters of internal energy potential (T,αs(T), mD (T),β,c), which are the Debye screening mass mD (T), the running coupling constant αs(T) the critical temperature β, the free parameter c in addition to noncommutativity parameters (Θ,θ-). The new Hamiltonian operator in 3D-NRNCPS symmetries is composed of the corresponding operator in commutative phase-space and three additive parts for spin-orbit interaction, the new magnetic interaction, and the rotational Fermi-term. The obtained energy eigenvalues are applied to obtain the mass spectra of heavy quarkonium systems (cc- and bb-). The total complete degeneracy of the new energy levels of the improved internal energy potential changed to become equal to the new value 3n2 in 3D-NRNCPS symmetries instead of the value n2 in the symmetries of 3D-NRQM. Our non-relativistic results obtained from DSE will possibly be compared with the Dirac equation in high-energy physics.

New Eigensolution of the Klein--Gordon and Shrödinger equations for improved modified Yukawa-Kratzer potential and its applications using Boop's Shift method and standard perturbation theory in the 3D-ERQM and 3D-ENRQM symmetries

“…This procedure is called Bopp's shifts method (BSM) [72][73][74][75][76][77]. This quantization procedure is called Bopp quantization [76].…”

“…The Bopp's shift method has achieved great success when applied by specialized researchers to the four basic equations correspond to the relativistic deformed Schrödinger equation (see, e.g., [32][33][34][77][78][79][80][81][82]) and the other three relativistic equations represented by the deformed Klein-Gordon equation (see, e.g., [83][84][85][86][87][88]), deformed Dirac equation (see, e.g., [47,93,94]) and the deformed Duffin-Kemmer-Petiau equation (see, e.g., [86][87][88][89]). It is worth motioning that that Bopp's shift method permutes us to rewrite Eq.…”

“…The modified algebraic structure of covariant canonical commutation relations with the notion of the Weyl-Moyal star product in Eqs. ( 3) and ( 4), which become new METNCCCRs with ordinary known products in literature, are as follows (see, e.g., [72][73][74][75][76][77]):…”

“…) employing of linear transformations. This can be generally done by using the Seiberg-Witten map, given by [72][73][74][75][76][77]:…”