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 energy
eigenvalues 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 that
the 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 can
physically describe each of the two Dirac equations and the Duffin--Kemmer equation under the effect of IHGMP.