A New Approach to the Approximate Analytic Solution of the Three-Dimensional Schrӧdinger Equation for Hydrogenic and Neutral Atoms in the Generalized Hellmann Potential Model

Abstract

Within the framework of nonrelativistic noncommutative quantum mechanics using the improved approximation scheme to the centrifugal term for any l-states via the generalized Bopp’s shift method and standard perturbation theory, we have obtained the energy eigenvalues of a newly proposed generalized Hellmann potential model (the GHP model) for the hydrogenic atoms and neutral atoms. The potential is a superposition of the attractive Coulomb potential plus Yukawa one, and new central terms appear as a result of the effects of noncommutativity properties of space and phase in the Hellmann potential model. The obtained energy eigen-values appear as a function of the generalized gamma function, the discrete atomic quantum numbers (j, n, l, s and m), infinitesimal parameters (a, b, б) which are induced by the position-position and phase-phase noncommutativity, and, the dimensional parameters (Θ, 0) of the GHP model, in the nonrelativistic noncommutative three-dimensional real space phase (NC: 3D-RSP). Furthermore, we have shown that the corresponding Hamiltonian operator with (NC: 3D-RSP) symmetries is the sum of the Hamiltonian operator of the Hellmann potential model and two operators, the first one is the modified spin-orbit interaction, while the second is the modified Zeeman operator for the hydrogenic and neutral atoms.

Analytical Expressions to Energy Eigenvalues of the Hydrogenic Atoms and the Heavy Light Mesons in the Framework of 3D-NCPS Symmetries Using the Generalized Bopp's Shift Method

“…are the potential parameters, r is the distance between the two particles. The algebraic structure of nonrelativistic 3D-NCPS based to NCcanonical commutations relations in (Schrödinger, Heisenberg and interactions) pictures (SP, HP and IP), respectively, as follows (Throughout this paper, the natural units c = = 1 will be used) (see, e.g., [40][41][42][43][44][45][46][47][48]):…”

“…To perform this task the physical form DSE, it is necessary to replace ordinary three-dimensional Hamiltonian operators H trm (p, x), ordinary energy E trm nl and corresponding complex wave function Ψ ( r) in the symmetries of NRQM with new three-dimensional Hamiltonian operators H trm nc (p nc , x nc ), new unknown values E trm nc of energy and corresponding new complex wave function Ψ ( r nc ), respectively in 3D-NCPS symmetries. In addition, to replace the ordinary product by star product * , this allows us to construct the deformed Schrödinger DSE in 3D-NCPS symmetries as (see, e.g., [41][42][43][44][45][46][47][48][49][50][51]):…”

“…The above equation allows us to obtain the two operators r 2 nc and p 2 nc in 3D-NCPS (see, e.g., [44,51,63,64]):…”

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

“…Concerning the combination of Hellmann and generalized Morse potentials, the subject of the current study, I have previously dealt with the non-specific study of each of them separately, but I have not dealt with them in a combined way, and neither has any other researcher done so far. We have treated the generalized Hellmann potential in the symmetries of NERQM [60]. Moreover, we have applied the Hellmann potential on the Mirror Nuclei 17 O and 17 F in the symmetries of NERQM [61].…”

A theoretical study of the modified equal scalar and vector Manning-Rosen potential within the deformed Klein-Gordon and Schrödinger in RNCQM and NRNCQM symmetries

“…=i θ µν , and θ µν = ε µν θ (θ is the non-commutative parameter), which is an infinitesimals parameter if compared to the energy values and elements of antisymmetric 3 × 3 real matrix and δ µν is the identity matrix. The symbol ( * ) denotes the Weyl Moyal star product, which is generalized between two ordinary functions f (x)g(x) to the new modified formf (x)ĝ(x) ≡ f (x) * g(x) in the symmetries of RNCQM as follows [47][48][49][50][51][52][53][54][55]:…”