New bound-state energy of double magic number plus one nucleon nuclei with a relativistic and non-relativistic mean-field approach in 3D-RNCQS and 3D-NRNCQS symmetries

Relativistic and non-relativistic quantum mechanics formulated in a non-commutative space-space have recently become the object of renewed interest. In the context of three-dimensional relativistic non-commutative quantum space ( 3D-RNCQS ) symmetries with arbitrary spin–orbit coupling quantum number k , we approximate the solution of the deformed Dirac equation for a new modified Eckart plus Hulthen potentials ( NMEHPs ) for interaction between the core and single nucleon of ( 17 O, 41 Ca, 49 Ca, and 57 Ni) isotopes with one additional nucleon (valence) in the (1 d 5/2 , 2 s 1/2 ), (1 f 7/2 , 2 p 3/2 ), (2 p 3/2 , 1 f 5/2 ) and (2 p 3/2 , 1 f 5/2 ) levels that correspond the ground-state and the first excited energy, respectively. In the framework of the spin and pseudospin ( p -spin) symmetry, we obtain the global new energy eigenvalue, which equals the energy eigenvalue in the usual relativistic QM as the main part plus three corrected parts produced from the effect of the spin–orbit interaction, the new modified Zeeman, and the rotational Fermi term. The new values that we get appear to be sensitive to the quantum numbers j , k , l , l p , s , s p , m , m p , the mixed potential depths v 0 , v 1 , α , and non-commutativity parameters (NP) Θ , τ , χ . We recovered several potentials, including the NMEHPs . We were able to define intervals for the NP. In addition, we were able to reduce the difference between our new theoretical values and their experimental values by comparing this difference with what is found in the literature.