# Abdelmadjid Maireche^{*}

Laboratory of Physical and Chemical Materials, Physics Department, Sciences Faculty, University of M’sila, M’sila- Algeria

# Abstract

In present work, the exact analytical bound-state solutions of modified
Schrödinger equation with Modified central potential consisting of a Cornellmodified
plus pseudoharmonic harmonic potential (MCMpH) have been
presented using both Boopp’s shift method and standard perturbation theory, we
have also constructed the corresponding noncommutative Hamiltonian which
containing two new terms, the first one is modified Zeeman effect and the second
is new spin-orbital interaction. The theoretical results show that the automatically
appearance for both spin-orbital interaction and modified Zeeman Effect leads
to the degenerate to energy levels to 2(2*l* +1)sub states.

# Keywords

Schrödinger equation, Star product, Boopp’s shift method, Pseudoharmonic, linear, Coulomb potentials, Noncommutative space and noncommutative phase

**Pacs numbers:** 11.10.Nx, 32.30-r, 03.65-w

# Introduction

It is well known that to study any quantum chemical-physical model, in different fields of sciences like atomic, nuclear, molecular, harmonic and harmonic spectroscopy, we need to solve the non relativistic Schrödinger equation and relativistic two equations: Klein-Gordon and Dirac [1-22]. To obtain profound interpretation in Nano and plank’s scales, much work in case of the noncommutative space-phase at two, three and N generalized dimensions has been done for solving the three fundamental previously equations [23-47]. The notions of noncommutativity of space and phase developed on based to the Seiberg-Witten map, Boopp’s shift method and the star product, defined on the first order of two infinitesimal parameters antisymmetric as [23-47]:

Which allow us to obtaining the two new non nulls commutators and , respectively as:

It’s important to notice, that the Boopp’s shift method will be applying in this paper instead of solving the (NC-3D: RSP) with star product, the Schrödinger equation will be treated by using directly usual commutators on quantum mechanics, in addition to the two commutators [29-43]:

The main goal of this work is to extend our study in reference [41] for the potential (MCMpH) including new term into noncommutative three dimensional spaces and phases on based to the principal reference [48] to discover the new spectrum and possibility to obtain new applications for the modified potential in different fields. The rest of present search is organized as follows: In next section, we give briefly review to the Schrödinger equation with (CMpH) in three dimensional spaces. In section 3, we shall briefly introduce the fundamental concepts of Boopp’s shift method and then we apply this method to derive the deformed potential and noncommutative spin-orbital Hamiltonian. In section 4, we apply perturbation theory to find the spectrum for ground stat and first excited states and then we deduce the spectrum produced automatically by the external magnetic field. In section 5, we conclude the global noncommutative Hamiltonian and we resume the global spectrum for (MCMpH) in first order of two infinitesimal parameters’ Θ and θ in (NC-3D: RSP). Finally, the important found results and the conclusions are discussed in the last section.

### Review of the Eigenvalues and Eigenfunctions for (CMpH) in Three Dimensional

In this section, we shall review the eigenvalues and eignenfunctions for spherically symmetric for the potential known by Cornell-modified plus pseudoharmonic potential (CMpH) in three dimensional (3D) spaces [48]:

The four parameters: *, b , c* and *d* are constants, the
above confining interaction potential consisting of a sum of
harmonic, linear, Coulombic and pseudoharmonic potential
terms, the last term is incorporated into the quarkouniom
potential for the sake of coherence while the rest terms
represents the Cornell potential [48, 49], the complex
eigenfunctions Ψ(r,θ ,ϕ ) in 3-dimensional space for above
potential satisfied the Schrödinger equation (SE) in spherical
coordinates is [48]:

Where m_{0} is the isotropic effective mass and E_{n,l} is
the total energy of the particle and Ψ(r,θ ,ϕ ) can be written
as [48]:

Here (θ ,ϕ )
is the spherical harmonic and the radial
wave function ψ _{n,l} (r) is the solution of the equation [48]:

To eliminate the first order of derivation, it’s covariant to
rewritten the radial wave function ψ _{n,l} (r) to the form [48]:

Then, the equation (7) reduces to following form [48]:

Where, and then, the
complete normalized wave functions and corresponding
energies for the ground state, the first existed states, and * _{n}^{th}*
excited state, respectively [48]:

Where and the two normalized constants are given by [48]:

### Noncommutative Phase-space Hamiltonian for (MCMpH)

**Formalism of Boopp’s shift**

Based on the previous works [31-43], we give a brief review to the fundamental principles of modified Schrödinger equation in (NC-3D: RSP), to achieve this goal we apply the important 4-steps on the ordinary (SE):

On based to our references [37-40], we can write the two operators and in noncommutative three dimensional spaces and phases as follows:

Which allow us to writing the modified three dimensional studied potential (MCMpH) in (NC-3D: RSP) as follows:

It is clear that, the first 4-terms in eq. (19) represent
the ordinary potential while the rest term is produced by
the deformation of space and phase. The global perturbative
potential operatorsV_{pert−mcph} (r,Θ,θ ) for studied potential
(MCMpH) in both (NC-3D: RSP) will be written as:

### The Spin-orbital Noncommutative Hamiltonian for (CMpH) in (NC: 3D- RSP)

In order, to discover the new contribution of (MCMpH), we replace the two couplings and by and , respectively, then the above perturbed operator becomes as:

### The Exact Spectrum of Ground States Produced by Noncommutative Spinorbital Hamiltonian for (MCMpH) in (NC: 3D- RSP)

Now, the aim of this subsection is to obtain the
modifications to the energy levels for ground states E_{u-cmph}
and E_{d-cmph} for spin up and spin down, respectively, at first
order of two parameters Θ and θ . In order to achieve this
goal, we apply the standard perturbation theory:

In order to obtain the above integrals, we applying the following special integration [50]:

Inserting the above expressions into equations (26.1) and
(26.2), one obtains the following results for exact modifications
of ground states E_{u} and E_{d} produced by new spin-orbital
effect for (MCMPH):

It’s important to notice that the above two terms and are represent the noncommutative geometry of space and phase, respectively.

### The Exact Spectrum of First Excited States Produced by Noncommutative Spinorbital Hamiltonian for (MCMpH) in (NC: 3D- RSP)

The aim of this subsection is to obtain the new
modifications to the energy levels for first excited states *E*_{ul}
and *E*_{dl} corresponding spin up and spin down, respectively at
first order of two parameters Θ and θ for (MCMpH) which
are obtained by applying the standard perturbation theory as:

### The Exact Spectrum Produced by Noncommutative Magnetic Hamiltonian for (MCMpH) in (NC: 3D- RSP)

On the other hand, it is possible to found another automatically symmetry for (CMpH) related to the influence of an external uniform magnetic field, generated from the effect of the new geometry of space and phase, it is deduced by the two following two replacements:

Here χ and σ are infinitesimal real two proportional’s constants and to simplify the calculations we choose the magnetic field B = Bk and then we can make the following translation:

Which allow us to introduce the modified new magnetic Hamiltonian in (NC-3D: RSP) for (MCMpH) as:

Here denote to the ordinary operator of Hamiltonian for of Zeeman Effect in quantum mechanics. To obtain the exact noncommutative magnetic modifications of energy for (MCMpH) we just replaced the

Where and are the exact magnetic modifications of spectrum corresponding the ground states and first excited states, respectively. The new global exact spectrum of lowest excited states for (MCMpH) in (NC-3D: RSP) produced by the diagonal elements of noncommutative Hamiltonian operator . It is clearly, that the obtained previous results which are presented by Eqs. (30.1), (30.2), (38.1), (38.2), (43.1) and (43.2) of eigenvalues of energies are reels and then the noncommutative diagonal Hamiltonian operator will be Hermitian operator. Furthermore, we can obtain the explicit physical form of this operator based on the results (23) and (42) for (CMpH), its represent by diagonal noncommutative matrix of order (3×3), with elements and in both (NC-3D: RSP):

and

Now we find the corresponding global modified energies for ground and first excited states of a particle fermionic with spin up and spin moving in the (MCMpH), referring to Eqs. (10.1), (10.2), (30.1), (30.2), (38.1), (38.2), (43.1) and (43.2), we fond the results as follows:

As it’s mentioned in our previous works [31-43], the
atomic quantum number *m* can be taken ( 2*l* +1) values
and we have also two values for , thus every state in
usually three dimensional space of (MCMpH) will be in (NC-
3D: RSP): 2(2*l* +1) sub-states.

It is important to notice, that the appearance of the
polarization states of a fermionic particle indicates the validity
of the results in the field of high energy where Dirac equation
is applied, which allowing to the validity to results of present
search on the Plank’s and Nano scales level. If we make the
limits (θ_{k} ,θ_{k} )→(0,0) we can obtain all the results of ordinary
quantum mechanics.

# Conclusion

In this article, we have investigated the solutions of the
Schrödinger equation for modified (MCMpH) potential.
We showed that the obtained degenerated spectrum for the
modified studied potential depended by new discrete atomic
quantum numbers: and *m* of electron, our obtained
results could be extended to applied at and Plank’s and Nano
scales.

# Acknowledgements

This work was supported with search laboratory of: Physique et Chimie des matériaux, in university of M’sila, Algeria.

# References

2. Bose SK. 1994. Exact bound states for the central fraction power singular potential V(r)=αr^{2/3}+βr^{−2/3}+γr^{−4/3}. Il Nuovo Cimento B (1971-1996) 109(11): 1217-1220. doi: 10.1007/BF02726685

4. Ahmed SAS, Borah BC, Sarma D. 2001. Generation of exact analytic bound state solutions from solvable non-powerlaw potentials by a transformation method. *Eur Phys J D* 17(1): 5-11. doi: 10.1007/s100530170032

5. Ikhdair SM, Sever R. 2007. Exact solutions of the radial Schrödinger equation for some physical potentials. *CEJP* 5(4): 516- 527. doi: 10.2478/s11534-007-0022-9

6. Nieto MM. 1979. Hydrogen atom and relativistic pi-mesic atom in N-space dimension. *Am J Phys* 47: 1067-1072. doi: 10.1119/1.11976

7. Ikhdair SM, Sever R. 2007. Exact polynomial eigensolutions of the Schrödinger equation for the pseudoharmonic potential. *J Mol Struc-Theochem* 806: 155-158. doi: 10.1016/j.theochem.2006.11.019

10. Flesses GP, Watt A. 1981. An exact solution of the Schrödinger equation for a multiterm potential. *J Phys A: Math Gen* 14(9): L315-L318. doi: 10.1088/0305-4470/14/9/001

11. Ikhdair SM, Sever R. 2007. Exact solution of the Klein-Gordon equation for the PT symmetri generalized Woods-Saxon potential by the Nikiforov-Uvarov method. *Ann Phys (Leipzig) *16(3): 218-232. doi: 10.1002/andp.200610232

12. Dong SH. 2001. Schrödinger equation with the potential V(r) = Ar-4 + Br-3 + Cr-2 + Dr-1. *Phys scr* 64(4): 273-276. doi: 10.1238/Physica.Regular.064a00273

13. Dong SH, Ma ZQ. 1998. Exact solutions to the Schrödinger equation for the potential V(r) = ar2 + br-4 + Cr-6 in two dimensions. J Phys A 31(49): 9855-9859. doi: 10.1088/0305-4470/31/49/009

14. Dong SH. 2001. A new approach to the relativistic Schrödinger equation with central potential: Ansatz method. *Int J Theor Phys *40(2): 559-567. doi: 10.1023/A:1004119928867

15. Rajabi AA, Hamzavi M. 2013. A new Coulomb ring-shaped potential via generalized parametric Nikiforov-Uvarov method. *Journal of Theoretical and Applied Physics* 7: 17. doi: 10.1186/2251-7235-7-17

16. Ikhdair SM, Sever R. 2013. Relativistic Two-Dimensional Harmonic Oscillator Plus Cornell Potentials in External Magnetic and AB Fields. *Adv High Energy Phys* 2013: 562959. doi: 10.1155/2013/562959

17. Dong SH, San GH. 2003. Quantum Spectrum of Some Anharmonic Central Potentials: Wave Functions Ansatz. Foundations of Physics Letters 16(4): 357-367. doi: 10.1023/A:1025313809478

19. Ikhdair SM. 2012. Exact solution of Dirac equation with charged harmonic oscillator in electric field: bound states. *J Mod Phys* 3(2): 170-179. doi: 10.4236/jmp.2012.32023

20. Hassanabadi H, Maghsoodi E, Oudi R, Zarrinkamar S, Rahimov H. 2012. Exact solution Dirac equation for an energy-depended potential. *Tur Phys J Plus* 127: 120. doi: 10.1140/epjp/i2012-12120-1

21. Hassanabadi H, Hamzavi M, Zarrinkamar S, Rajabi AA. 2011. Exact solutions of N-Dimensional Schrödinger equation for a potential containing coulomb and quadratic terms. *International Journal of the Physical Sciences* 6(3): 583-586. doi: 10.5897/IJPS10.555

22. Dong SH, Ma ZQ, Esposito G. 1999. Exact solutions of the Schrödinger equation with inverse-power potential. *Found Phys Lett* 12(5): 465-474. doi: 10.1023/A:1021633411616

23. Connes A. 1994. Noncommutative geometry. Academic Press, Paris, France.

24. Snyder H. 1947. The quantization of space time. *Phys Rev* 71: 38-41. doi: 10.1103/PhysRev.71.38

25. Dossa AF, Avossevou GYH. 2013. Noncommutative Phase Space and the Two Dimensional Quantum Dipole in Background Electric and Magnetic Fields. *J Mod Phys* 4(10): 1400-1411. doi: 10.4236/jmp.2013.410168

26. Jacobus DT. 2010. Department of Physics, Stellenbosch University, South Africa.

27. Smailagic A, Spallucci E. 2002. New isotropic versus anisotropic phase of noncommutative 2-D Harmonic oscillator. *Phys Rev D* 65: 107701. doi: 10.1103/PhysRevD.65.107701

28. Yang ZH, Long CY, Qin SJ, Long ZW. 2010. DKP Oscillator with spin-o in Three dimensional Noncommutaive Phase-Space. *Int J Theor Phys* 49(3): 644-657. doi: 10.1007/s10773-010-0244-2

29. Yuan Y, Kang L, Jian-Hua W, Chi-Yi C. 2010. Spin ½ relativistic particle in a magnetic field in NC Ph. *Chin Phys C* 34(5): 543. doi: 10.1088/1674-1137/34/5/005

30. Mirza B, Narimani R, Zare S. 2011. Relativistic Oscillators in a Noncommutative space and in a Magnetic field. *Commun Theor Phys* 55: 405-409. doi: 10.1088/0253-6102/55/3/06

36. Maireche A. 2015. Nonrelativistic Atomic Spectrum for Companied Harmonic Oscillator Potential and its Inverse in both NC-2D: RSP. *International Letters of Chemistry, Physics and Astronomy* 56: 1-9. doi: 10.18052/www.scipress.com/ILCPA.56.1

37. Maireche A. 2015. A New Approach to the Non Relativistic Schrödinger equation for an Energy-Depended Potential V(r,En,l)=V0(1+ηEn,l)r2 in Both Noncommutative three Dimensional spaces and phases.*International Letters of Chemistry, Physics and Astronomy* 60: 11-19. doi: 10.18052/www.scipress.com/ILCPA.60.11

39. Maireche A. 2015. New Exact Solution of the Bound States for the Potential Family V(r) =A/r^{2}-B/r+Cr^{k} (k=0,-1,-2) in both Noncommutative Three Dimensional Spaces and Phases: Non Relativistic Quantum Mechanics. *International Letters of Chemistry, Physics and Astronomy* 58: 164-176. doi: 10.18052/www.scipress.com/ILCPA.58.164

41. Maireche A. 2015. A new study to the Schrödinger equation for modified potential V(r)=ar2+br-4+cr-6 in nonrelativistic three dimensional real spaces and phases. *International Letters of Chemistry, Physics and Astronomy* 61: 38-48. doi: 10.18052/www.scipress.com/ILCPA.61.38

45. Cai S, Jing T, Guo G, Zhang R. 2010. Dirac Oscillator in Noncommutative Phase Space, *International Journal of Theoretical Physics* 49(8): 1699-1705. doi: 10.1007/s10773-010-0349-7

46. Lee J. 2005. Star Products and the Landau Problem. Journal of the Korean Physical Society 47(4): 571-576. doi: 10.3938/jkps.47.571

^{*}Correspondence to:

Abdelmadjid MAIRECHE

Laboratory of physical and chemical materials

physics department, Sciences faculty

University of M’sila M’sila- Algeria

Tel: 00213664438317

E-mail: abmaireche@gmail.com

**Received:** December 28, 2015

**Accepted:** January 20, 2016

**Published:** January 22, 2016

**Citation:** Maireche A. 2016. New Bound State Energies for Spherical Quantum Dots in Presence of a Confining Potential Model at Nano and
Plank’s Scales. *NanoWorld J* 1(4): 122-129.

**Copyright:** © 2016 Maireche. This is an Open Access article distributed under the terms of the Creative Commons Attribution 4.0 International License (CC-BY) (http://creativecommons.org/licenses/by/4.0/) which permits commercial use, including reproduction, adaptation, and distribution of the article provided the original author and source are credited.