New Bound State Energies for Spherical Quantum Dots in Presence of a Confining Potential Model at Nano and Plank’s Scales

Abdelmadjid Maireche*

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


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(2l +1)sub states.


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


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 m0 is the isotropic effective mass and En,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 nth 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 operatorsVpert−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 Eu-cmph and Ed-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 Eu and Ed 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 Eul and Edl 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):


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 ( 2l +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(2l +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 (θkk )→(0,0) we can obtain all the results of ordinary quantum mechanics.


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.


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


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*Correspondence to:

Abdelmadjid MAIRECHE
Laboratory of physical and chemical materials
physics department, Sciences faculty
University of M’sila M’sila- Algeria
Tel: 00213664438317

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) ( which permits commercial use, including reproduction, adaptation, and distribution of the article provided the original author and source are credited.

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