Diatomic molecules and fermionic particles with improved Hellmann-generalized Morse potential through the solutions of the deformed Klein-Gordon, Dirac and Schrödinger equations in extended relativistic quantum mechanics and extended nonrelativistic quantum mechanics symmetries

Abstract

In this paper, we investigate the new approximate bound state solution of deformed Klein--Gordon, Dirac and Schr\"{o}dinger equations in the symmetries of extended relativistic quantum mechanics ERQM and extended nonrelativistic quantum mechanics ENRQM have been obtained with a newly proposed potential called improved Hellmann-generalized Morse potential (IHGMP, for short). To the best of our knowledge, this problem is examined in literature in the usual RQM and NRQM with Hellmann-generalized Morse potential. The potential is a superposition of Hellmann potential, generalized Morse or Deng-Fan potential, and some other exponential terms. By employing the improved approximation to deal with the centrifugal term, Bopp's shift and standard perturbation theory method. The new approximate analytical energy shift and the corrections of bound state energyeigenvalues in ERQM and ENRQM are obtained for some selected diatomic molecules such as (HCl, LiH, H2, ScH, TiH, VH, CrH, CuLi, TiC, NiC, ScN and ScF). The new values that we get appeared sensitive to the quantum numbers $% \left( j,l,s,m\right) $, the potential depths of the improved Hellmann-generalized Morse potential ($a,b$), the range of the potential $%\alpha $, the dissociation energy $D_{e}$, the equilibrium bond length $%r_{e} $, and noncommutativity parameters$\left( \Theta ,\sigma ,\chi \right)$ . We have highlighted three physical phenomena that automatically generate a result of the topological properties of noncommutativity, the first physical phenomena are the perturbative spin-orbit coupling, the second the magnetic induction while the third corresponds to the rotational proper phenomena. In both relativistic and nonrelativistic problems, we show thatthe corrections on the spectrum energy are smaller than the main energy in the ordinary cases of quantum field theory and quantum mechanics. In the new symmetries of NCQM, it is not possible to get the exact analytical solutions for $l=0$ and $l\neq 0$, the approximate solutions are available. Four special cases, i.e., l wave are investigated in the context of deformed Klien-Gordon and Schr\"{o}dinger theories. The relativistic energy equations and the new nonrelativistic energy for some potentials such as improved Hellmann potential and improved generalized Morse potential have also been obtained by varying some potential parameters. We have clearly shown that the Schr\"{o}dinger and Klein Gordon equations in the new symmetries canphysically describe each of the two Dirac equations and the Duffin--Kemmer equation under the effect of IHGMP.

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

“…In the search for solutions of the NR deformed Schrödinger equation DSE under the influence of several different potentials [56][57][58][59][60][61][62][63][64]. The success of this method was not limited to the DSE, but was extended to the study of various relativistic physics problems, for example, for the DKGE [65][66][67][68][69][70][71], for the DDE [72][73][74][75][76][77][78][79][80] and the deformed DKP equation [81,82]. Thus, the BS method allows us to simplify second-order linear differential equations of the DSE, DKG, DDE, and DDKPE with the Weyl-Moyal star product to the second-order linear differential SE, KGE, DE, and DKP equations with the ordinary product with simultaneous translation in the spacespace.…”

“…This allows us to find ̂︀ 𝑟 2 equal (︀ 𝑟 2 − LΘ )︀ and (𝑟 2 − − ̃︀ LΘ) for the spin and p-spin symmetries, respectively [72][73][74][75][76][77][78][79][80] which we will use in the next subsection.…”

The influence of noncommutativity on the energy spectra of bosonic particles in the framework of the DKGE with improved spatially-dependent mass including mixed scalar-vector Coulomb potentials in the ERQM symmetries

“…As for the deformed Dirac equation DDE, many typical potentials were successfully processed, despite the complexity of the calculations (See some refs. [81][82][83][84]) while the relativistic deformed Duffin-Kemmer-Petiau equation DDKPE for particles with spin-(1,2,...) [85,86]. Thus, Bopp's shift method BS method is based on reducing second-order linear differential equations of the DSE, DKGE, DDE, and DDKPE with Weyl-Moyal star product to second-order linear differential equations of SE, KGE, DE, and DKPE without Weyl-Moyal star product with simultaneous translation in the space-space.…”

Deformed Dirac and Shrödinger Equations with Improved Mie-Type Potential for Diatomic Molecules and Fermionic Particles in the Framework of Extended Quantum Mechanics Symmetries

“…[68][69][70][71][72][73][74][75]), for the DDE (see Refs. [12,13,[76][77][78]) for deformed Duffin-Kemmer-Petiau equation (DD-KPE) [79,80]. Thus, the Bopp's shift method is based on reducing second-order linear differential equations such as DSE, DKG, DDE, and DDKPE with Weyl-Moyal star product to second-order linear differential equations, namely, SE, KGE, DE, and DKPE without Weyl-Moyal star product with a simultaneous translation in the space-space.…”