New Bound States for Modified Vibrational-Rotational Structure of Supersingular Plus Coulomb Potential of the Schrödinger Equation in One-Electron Atoms

. In this study, three-dimensional modified time-independent Schrödinger equation of modified vibrational-rotational structure of supersingular plus Coulomb (v.r.s.c) potential was solved using Boopp’s shift method instead to apply star product, in the framework of both noncommutativity three dimensional real space and phase (NC: 3D-RSP). We have obtained the explicit energy eigenvalues for ground state and first excited state for interactions in one-electron atoms. Furthermore, the obtained corrections of energies are depended on infinitesimal parameters    ,  and


Introduction
The exact solutions of the wave equations in non-relativistic and relativistic quantum mechanics (non-relativistic and relativistic spinless particles) in the case of ordinary commutative space with central and non-central potentials are very important for describing atoms, nuclei,…. various methods have been applied to solve the ordinary Schrödinger equation by means of asymptotic iteration method, improved (AIM), Laplace integral transform, factorization method, proper quantization rule, exact quantization rule, Nikiforov-Uvarov method, supersymmetry quantum mechanics (SUSYQM), In the recent years, authors have analytically derived exact solutions for diverse potentials . It is well-known, that, the ordinary quantum mechanics based to the ordinary canonical commutations relations (CCRs) in both Schrödinger (time-independent operators) and Heisenberg pictures (time dependent operators) respectively, as: Here vrsc H denote to the ordinary quantum Hamiltonian operator for studied potential. Furthermore, much considerable effort has been expanded on the solutions of Schrödinger, Dirac and Klein-Gordon equations to noncommutative quantum mechanics, to search a profound interpretation in microscopic scales, which based to new noncommutative canonical commutations relations (NNCCRs) in both Schrödinger and Heisenberg pictures, respectively, as follows : and: where the two new operators       in Heisenberg picture are related to the corresponding two in Schrödinger picture from the two projections relations: in ordinary three dimensional spaces [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57]: are the new function in NC: 2D-RSP) and (NC: 3D-RSP), the two covariant derivatives , respectively, the two following terms and (phase-phase) noncommutativity properties, respectively, and stands for the second and higher order terms of  and  , a Boopp's shift method can be used, instead of solving any quantum systems by using directly star product procedure : The non-vanish 9 commutators in (NC-3D: RSP) can be determined, as follows: which allow us to getting the two operators ( 2 r and 2 p ) in (NC-3D: RSP), respectively, as follows [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]: (12) where the two couplings  L and  and It is-well known, that, in quantum mechanics, the three components ( The study of vibrational-rotational structure of supersingular plus Coulomb potential has now become a very interest field due to their applications in different fields [32], the bound state solutions of the non-relativistic Schrödinger equation, with the modified vibrational-rotational structure of supersingular plus Coulomb potential has not been obtained yet. This is the priority for this work. The modified vibrational-rotational structure of supersingular plus Coulomb potential used in this frame work takes the form: The crucial purpose of this paper is to determine the energy levels of above potential in (NC: 3D-RSP) symmetries using the generalization Boopp's shift method based on mentioned formalisms on above equations to discover the new symmetries and a possibility to obtain another applications to this potential in different fields. It is worth to mention that, the noncommutative idea was introduced firstly by H. Snyder [33]. In the recent years, the problem of finding exact solutions of the non-relativistic modified Schrödinger equation in noncommutative spaces and phases for a number of special potential has been a line of great interest . The organization scheme of the study is given as follows: In next section, we briefly review the Schrödinger equation with vibrational-rotational structure of supersingular plus Coulomb potential on based to Ref.
[32]. The Section 3, devoted to studying the three deformed Schrödinger equation by applying both Boopp's shift method to the vibrational-rotational structure of supersingular plus Coulomb potential. In the fourth section and by applying standard perturbation theory we find the quantum spectrum of the excited states in (NC-3D: RSP) for spin-orbital interaction corresponding the ground states and first International Letters of Chemistry, Physics and Astronomy Vol. 73 excited states. In the next section, we derive the magnetic spectrum for studied potential. In the sixth section, we resume the global spectrum and corresponding noncommutative Hamiltonian for vibrational-rotational structure of supersingular plus Coulomb potential. Conclusions are drawn in Sect 6.

Review the eignenfunctions and the energy eigenvalues for vibrational-rotational structure of supersingular plus Coulomb potential in ordinary three dimensional spaces
Our goal in this section is to review the essential steps, which give the solutions of time independent Schrödinger equation for a fermionic particle like electron of rest mass 0 m and its energy E moving in vibrational-rotational structure of supersingular plus Coulomb potential [32]: where A play the role of positive constant coefficient and Z is the nuclear charge. The (v.r.s.c) potential plays a basic role in chemical and molecular physics since it can be used to calculate the molecular vibration-rotation energy spectrum of linear and non-linear systems. The above potential is the sum of Colombian if we insert this potential into the non-relativistic Schrödinger equation; we obtain the following equation in three dimensional spaces as follows [32]: is the radial function. As it is mentioned in ref.
[32], the solution of above second order differential equation in the coordinate basis turns out that there is a family of solutions: Here is the modified Bessel function and the first term corresponds to the atomic hydrogen atom and the successive terms is the fraction involves the expectation values of inverse power of r ,

Theoretical framework
This section is devoted to constructing of non relativistic modified Schrödinger equation (m.s.e) in (NC-3D: RSP) for (v.r.s.c) potential; to achieve this subject, we apply the essentials following steps :  And the last step corresponds to replace the ordinary old product by new star product    , which allow us to constructing the modified three dimensional Schrödinger equation in (NC-3D: RSP) symmetries for modified (v.r.s.c) potential: It is worth to emphasize that the Boopp's shift method permits to use the ordinary product without star product for modified Schrödinger equation, in (NC-3D: RSP) symmetries for modified (v.r.s.c) potential as follows: In recently work, we are interest with the first variety which presented by eq. (23), after straightforward calculations, we can obtain the five important terms, which will be used to determine the modified (v.r.s.c) potential in (NC: 3D-RSP), as:     for modified (v.r.s.c) potential, which we are subject of discussion in next sub-section.

The exact spin-orbital spectrum for modified (v.r.s.c) potential for ground states and first excited states for one-electron atoms in (NC: 3D-RSP) symmetries:
In this sub section, we are going to study the modifications to the energy levels ( ) for spin up and spin down, respectively, at first order of parameters (  , ), for excited states th n , obtained by applying the standard perturbation theory, using eqs. (19) and (33) in (NC-3D: RSP) symmetries: and where, the 6-terms: and

International Letters of Chemistry, Physics and Astronomy Vol. 73
It is convenient to apply the following integral [68]: (40) where       , we now consider another interested physically meaningful phenomena, which produced from the perturbative terms of inverse-square potential related to the influence of an external uniform magnetic field, it's sufficient to apply the following three replacements to describing these phenomena: