Investigations on the Relativistic Interactions in One-Electron Atoms with Modified Anharmonic Oscillator

One of the interesting problems of the relativistic quantum mechanics is to find exact solutions to the Klein-Gordon (to the treatment of a zero-spin particle) and Dirac (spin 1⁄2 particles and anti-particles) equations for certain potentials of the physical interest, in recent years, considerable efforts have been done to obtain the analytical solution of central and non-central physics problems for different areas of atoms, nuclei, and hadrons, numerous papers of the physicist have discussed in details all the necessary information for the quantum system and in particularly the bound states solutions [1-21]. Some of these potentials are known to play important roles in many fields, one of such potential is the anharmonic oscillator has been a subject of many studies, it is a central potential of nuclear shell model, etc [20,21]. The ordinary quantum structures obey the standard Weyl-Heisenberg algebra in both Schrödinger and Heisenberg (the operators are depended on time) pictures, respectively, as (Throughout this paper the natural unit 1 = =  c are employed):