Journal of ISSN: 2377-4282JNMR

Nanomedicine Research
Research Article
Volume 4 Issue 4 - 2016
Investigations on the Relativistic Interactions in One-Electron Atoms with Modified Anharmonic Oscillator
Abdelmadjid Maireche*
Department of Physics, University of M’sila M’sila, Algeria
Received: November 30, 2016 | Published: December 16, 2016
*Corresponding author: Abdelmadjid Maireche, Laboratory of Physics and Material Chemistry, Physics department, University of M’sila-M’sila Algeria, Tel: +213664834317; Email:
Citation: Maireche A (2016) Investigations on the Relativistic Interactions in One-Electron Atoms with Modified Anharmonic Oscillator. J Nanomed Res 4(4): 00097. DOI: 10.15406/jnmr.2016.04.00097

Abstract

The bound-state solutions of the modified Dirac equation (m.d.e.) for the modified anharmonic oscillator (m.a.o.) are presented exactly for arbitrary spin-orbit quantum number k( k ˜ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgada qadaqaaiqadUgagaacaaGaayjkaiaawMcaaaaa@39F1@  by means Bopp’s shift method instead to solving (m.d.e.) with star product, in the framework of noncommutativity three dimensional real space (NC: 3D-RS). The exact corrections for n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gada ahaaqcfasabeaacaWG0bGaamiAaaaaaaa@39A2@ excited states are found straightforwardly for interactions in one-electron atoms by applying the standard perturbation theory. Furthermore, the obtained corrections of energies are depended on two infinitesimal parameters ( Θ ij , χ ij ) ε ij k ( Θ k , χ k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaeuiMde1aaSbaaKqbGeaacaWGPbGaamOAaaqabaqcfaOaaiilaiab eE8aJnaaBaaajuaibaGaamyAaiaadQgaaKqbagqaaaGaayjkaiaawM caaiabggMi6kabew7aLnaaDaaajuaibaGaamyAaiaadQgaaeaacaWG RbaaaKqbaoaabmaabaGaeuiMde1aaSbaaKqbGeaacaWGRbaabeaaju aGcaGGSaGaeq4Xdm2aaSbaaKqbGeaacaWGRbaajuaGbeaaaiaawIca caGLPaaaaaa@5170@  which induced by position-position noncommutativity, in addition to the non-relativistic quantum mechanics ( n,j,l=l±1/2,m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamOBaiaacYcacaWGQbGaaiilaiaadYgacqGH9aqpcaWGSbGaeyyS aeRaaGymaiaac+cacaaIYaGaaiilaiaad2gaaiaawIcacaGLPaaaaa a@43E6@  and ( n,j= l ˜ ± s ˜ , m ˜ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamOBaiaacYcacaWGQbGaeyypa0JabmiBayaaiaGaeyySaeRabm4C ayaaiaGaaiilaiqad2gagaacaaGaayjkaiaawMcaaaaa@4140@  under spin-symmetry and p-spin symmetry in (NC: 3D-RS), respectively. In limit of parameters ( Θ k , χ k )( 0,0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaeuiMde1aaSbaaKqbGeaacaWGRbaajuaGbeaacaGGSaGaeq4Xdm2a aSbaaKqbGeaacaWGRbaajuaGbeaaaiaawIcacaGLPaaacqGHsgIRda qadaqaaiaaicdacaGGSaGaaGimaaGaayjkaiaawMcaaaaa@4514@ , the energy equation is consistent with the results of ordinary relativistic quantum mechanics.

Keywords: Anharmonic oscillator; Noncommutative space; Star product; Bopp’s shift method; Dirac equation

Abbreviations

MOA: Modified Anharmonic Oscillator; (NC: 3D-RS): Noncommutativity Three Dimensional Real Space; MDE: Modified Dirac Equation; (NCCRs): NC Canonical Commutations Relations

Introduction

One of the interesting problems of the relativistic quantum mechanics is to find exact solutions to the Klein-Gordon (to the treatment of a zero-spin particle) and Dirac (spin ½ particles and anti-particles) equations for certain potentials of the physical interest, in recent years, considerable efforts have been done to obtain the analytical solution of central and non-central physics problems for different areas of atoms, nuclei, and hadrons, numerous papers of the physicist have discussed in details all the necessary information for the quantum system and in particularly the bound states solutions [1-21]. Some of these potentials are known to play important roles in many fields, one of such potential is the anharmonic oscillator has been a subject of many studies, it is a central potential of nuclear shell model, etc [20,21]. The ordinary quantum structures obey the standard Weyl-Heisenberg algebra in both Schrödinger and Heisenberg (the operators are depended on time) pictures, respectively, as (Throughout this paper the natural unit c==1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabg2 da9iabl+qiOjabg2da9iaaigdaaaa@3ACE@ are employed):

[ x i , p j ]=[ x i ( t ), p j ( t ) ]=i δ ij [ x i , x j ]=[ p i , p j ]=[ x i ( t ), x j ( t ) ]=[ p i ( t ), p j ( t ) ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aam WaaeaacaWG4bWaaSbaaKqbGeaacaWGPbaabeaajuaGcaGGSaGaamiC amaaBaaajuaibaGaamOAaaqcfayabaaacaGLBbGaayzxaaGaeyypa0 ZaamWaaeaacaWG4bWaaSbaaKqbGeaacaWGPbaabeaajuaGdaqadaqa aiaadshaaiaawIcacaGLPaaacaGGSaGaamiCamaaBaaajuaibaGaam OAaaqcfayabaWaaeWaaeaacaWG0baacaGLOaGaayzkaaaacaGLBbGa ayzxaaGaeyypa0JaamyAaiabes7aKnaaBaaajuaibaGaamyAaiaadQ gaaeqaaaGcbaqcfa4aamWaaeaacaWG4bWaaSbaaKqbGeaacaWGPbaa beaajuaGcaGGSaGaamiEamaaBaaajuaibaGaamOAaaqabaaajuaGca GLBbGaayzxaaGaeyypa0ZaamWaaeaacaWGWbWaaSbaaKqbGeaacaWG PbaabeaajuaGcaGGSaGaamiCamaaBaaajuaibaGaamOAaaqcfayaba aacaGLBbGaayzxaaGaeyypa0ZaamWaaeaacaWG4bWaaSbaaKqbGeaa caWGPbaabeaajuaGdaqadaqaaiaadshaaiaawIcacaGLPaaacaGGSa GaamiEamaaBaaajuaibaGaamOAaaqabaqcfa4aaeWaaeaacaWG0baa caGLOaGaayzkaaaacaGLBbGaayzxaaGaeyypa0ZaamWaaeaacaWGWb WaaSbaaKqbGeaacaWGPbaabeaajuaGdaqadaqaaiaadshaaiaawIca caGLPaaacaGGSaGaamiCamaaBaaajuaibaGaamOAaaqabaqcfa4aae WaaeaacaWG0baacaGLOaGaayzkaaaacaGLBbGaayzxaaGaeyypa0Ja aGimaaaaaa@82C3@ ……( 1 )

where the two operators ( x i ( t ), p i ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamiEamaaBaaajuaibaGaamyAaaqabaqcfa4aaeWaaeaacaWG0baa caGLOaGaayzkaaGaaiilaiaadchadaWgaaqcfasaaiaadMgaaeqaaK qbaoaabmaabaGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa @433E@ in Heisenberg picture are related to the corresponding operators ( x i , p i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamiEamaaBaaajuaibaGaamyAaaqabaqcfaOaaiilaiaadchadaWg aaqcfasaaiaadMgaaeqaaaqcfaOaayjkaiaawMcaaaaa@3E3A@  in Schrödinger picture from the following projections relations:

x i ( t )=exp(i H ^ ( t t 0 )) x i exp(i H ^ ( t t 0 )) p i ( t )=exp(i H ^ ( t t 0 )) p i exp(i H ^ ( t t 0 )) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam iEamaaBaaajuaibaGaamyAaaqabaqcfa4aaeWaaeaacaWG0baacaGL OaGaayzkaaGaeyypa0JaciyzaiaacIhacaGGWbGaaiikaiaadMgace WGibGbaKaadaqadaqaaiaadshacqGHsislcaWG0bWaaSbaaKqbGeaa caaIWaaajuaGbeaaaiaawIcacaGLPaaacaGGPaGaamiEamaaBaaaju aibaGaamyAaaqabaqcfaOaciyzaiaacIhacaGGWbGaaiikaiabgkHi TiaadMgaceWGibGbaKaadaqadaqaaiaadshacqGHsislcaWG0bWaaS baaKqbGeaacaaIWaaajuaGbeaaaiaawIcacaGLPaaacaGGPaaakeaa juaGcaWGWbWaaSbaaKqbGeaacaWGPbaabeaajuaGdaqadaqaaiaads haaiaawIcacaGLPaaacqGH9aqpciGGLbGaaiiEaiaacchacaGGOaGa amyAaiqadIeagaqcamaabmaabaGaamiDaiabgkHiTiaadshadaWgaa qcfasaaiaaicdaaeqaaaqcfaOaayjkaiaawMcaaiaacMcacaWGWbWa aSbaaKqbGeaacaWGPbaabeaajuaGciGGLbGaaiiEaiaacchacaGGOa GaeyOeI0IaamyAaiqadIeagaqcamaabmaabaGaamiDaiabgkHiTiaa dshadaWgaaqcfasaaiaaicdaaKqbagqaaaGaayjkaiaawMcaaiaacM caaaaa@7B0A@ …… ……. .. ……..( 2 )

here H ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcaaaa@3756@  denote to the ordinary quantum Hamiltonian operator. In addition, for spin ½ particles described by the Dirac equation, experiment tells us that must satisfy Fermi Dirac statistics obey the restriction of Pauli, which imply to gives the only non-null equal-time anti-commutator for field operators as follows:

{ Ψ α ( t,r ), Ψ ¯ β ( t,r' ) }=i ( γ 0 ) αβ δ 3 ( r-r' )   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaacmaaba GaeuiQdK1aaSbaaKqbGeaacqaHXoqyaKqbagqaamaabmaabaGaamiD aiaacYcacaqGYbaacaGLOaGaayzkaaGaaiilaiqbfI6azzaaraWaaS baaKqbGeaacqaHYoGyaKqbagqaamaabmaabaGaamiDaiaacYcacaqG YbGaae4jaaGaayjkaiaawMcaaaGaay5Eaiaaw2haaiabg2da9iaadM gadaqadaqaaiabeo7aNnaaCaaabeqcfasaaiaaicdaaaaajuaGcaGL OaGaayzkaaWaaSbaaKqbGeaacqaHXoqycqaHYoGyaKqbagqaaiabes 7aKnaaCaaabeqcfasaaiaabodaaaqcfa4aaeWaaeaacaqGYbGaaeyl aiaabkhacaqGNaaacaGLOaGaayzkaaGaaeiiaiaabccaaaa@5EA7@ ……… ... ... . …….( 3 )

with Ψ ¯ β ( t,r' )= Ψ + β ( t,r' ) γ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuiQdKLbae badaWgaaWcbaGaeqOSdigabeaakmaabmaabaGaamiDaiaacYcacaqG YbGaae4jaaGaayjkaiaawMcaaiabg2da9iabfI6aznaaCaaaleqaba Gaey4kaScaaOWaaSbaaSqaaiabek7aIbqabaGcdaqadaqaaiaadsha caGGSaGaaeOCaiaabEcaaiaawIcacaGLPaaacqaHZoWzdaahaaWcbe qaaiaaicdaaaaaaa@4B29@ . Very recently, many authors have worked on solving these equations with physical potential in the new structure of quantum mechanics, known by NC quantum mechanics, which known firstly H Snyder [22], to obtaining profound and new applications for different areas of matter sciences in the microscopic and nano scales [23-68]. It is important to noticing that, the new quantum structure of NC space based on the following NC canonical commutations relations (NCCRs) in both Schrödinger and Heisenberg pictures, respectively, as follows [23-60]:

[ x ^ i , p ^ j ]=[ x ^ i ( t ) , p ^ j ( t ) ]=i δ ij ,[ x ^ i , x ^ j ]=[ x ^ i ( t ) , x ^ j ( t ) ]=i θ ij    [ p ^ i , p ^ j ]= [ p ^ i ( t ) , p ^ j ( t ) ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aam WaaeaaceWG4bGbaKaadaWgaaqcfasaaiaadMgaaeqaaKqbaoaaxaca baGaaiilaaqabeaacqGHxiIkaaGabmiCayaajaWaaSbaaKqbGeaaca WGQbaabeaaaKqbakaawUfacaGLDbaacqGH9aqpdaWadaqaaiqadIha gaqcamaaBaaajuaibaGaamyAaaqabaqcfa4aaeWaaeaacaWG0baaca GLOaGaayzkaaWaaCbiaeaacaGGSaaabeqaaiabgEHiQaaaceWGWbGb aKaadaWgaaqaamaaBaaajuaibaGaamOAaaqabaaajuaGbeaadaqada qaaiaadshaaiaawIcacaGLPaaaaiaawUfacaGLDbaacqGH9aqpcaWG PbGaeqiTdq2aaSbaaKqbGeaacaWGPbGaamOAaaqabaqcfaOaaiilam aadmaabaGabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGdaWf GaqaaiaacYcaaeqabaGaey4fIOcaaiqadIhagaqcamaaBaaajuaiba GaamOAaaqabaaajuaGcaGLBbGaayzxaaGaeyypa0ZaamWaaeaaceWG 4bGbaKaadaWgaaqcfasaaiaadMgaaeqaaKqbaoaabmaabaGaamiDaa GaayjkaiaawMcaamaaxacabaGaaiilaaqabeaacqGHxiIkaaGabmiE ayaajaWaaSbaaKqbGeaacaWGQbaabeaajuaGdaqadaqaaiaadshaai aawIcacaGLPaaaaiaawUfacaGLDbaacqGH9aqpcaWGPbGaeqiUde3a aSbaaKqbGeaacaWGPbGaamOAaaqabaqcfaOaaeiiaaGcbaqcfaOaae iiamaadmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaG daWfGaqaaiaacYcaaeqabaGaey4fIOcaaiqadchagaqcamaaBaaaju aibaGaamOAaaqabaaajuaGcaGLBbGaayzxaaGaeyypa0Jaaeiiamaa dmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGdaqada qaaiaadshaaiaawIcacaGLPaaadaWfGaqaaiaacYcaaeqabaGaey4f IOcaaiqadchagaqcamaaBaaajuaibaGaamOAaaqabaqcfa4aaeWaae aacaWG0baacaGLOaGaayzkaaaacaGLBbGaayzxaaGaeyypa0JaaGim aaaaaa@9337@  …………( 4 )

Where the two new operators ( x ^ i ( t ), p ^ i ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmiEayaajaWaaSbaaKqbGeaacaWGPbaajuaGbeaadaqadaqaaiaa dshaaiaawIcacaGLPaaacaGGSaGabmiCayaajaWaaSbaaKqbGeaaca WGPbaabeaajuaGdaqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIca caGLPaaaaaa@435E@ in Heisenberg picture are related to the corresponding new operators ( x ^ i , p ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGcaGGSaGabmiC ayaajaWaaSbaaKqbGeaacaWGPbaabeaaaKqbakaawIcacaGLPaaaaa a@3E5A@  in Schrödinger picture from the new projections relations:

x ^ i ( t )=exp(i H ^ nc ( t t 0 ))* x ^ i *exp(i H ^ nc ( t t 0 )) p ^ i ( t )=exp(i H ^ nc ( t t 0 ))* p ^ i *exp(i H ^ nc ( t t 0 )) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOabm iEayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGdaqadaqaaiaadsha aiaawIcacaGLPaaacqGH9aqpciGGLbGaaiiEaiaacchacaGGOaGaam yAaiqadIeagaqcamaaBaaajuaibaGaamOBaiaadogaaeqaaKqbaoaa bmaabaGaamiDaiabgkHiTiaadshadaWgaaqaaiaaicdaaeqaaaGaay jkaiaawMcaaiaacMcacaGGQaGabmiEayaajaWaaSbaaKqbGeaacaWG PbaabeaajuaGcaGGQaGaciyzaiaacIhacaGGWbGaaiikaiabgkHiTi aadMgaceWGibGbaKaadaWgaaqaamaaBaaabaGaamOBaiaadogaaeqa aaqabaWaaeWaaeaacaWG0bGaeyOeI0IaamiDamaaBaaajuaibaGaaG imaaqcfayabaaacaGLOaGaayzkaaGaaiykaaGcbaqcfaOabmiCayaa jaWaaSbaaKqbGeaacaWGPbaabeaajuaGdaqadaqaaiaadshaaiaawI cacaGLPaaacqGH9aqpciGGLbGaaiiEaiaacchacaGGOaGaamyAaiqa dIeagaqcamaaBaaajuaibaGaamOBaiaadogaaKqbagqaamaabmaaba GaamiDaiabgkHiTiaadshadaWgaaqaaiaaicdaaeqaaaGaayjkaiaa wMcaaiaacMcacaGGQaGabmiCayaajaWaaSbaaKqbGeaacaWGPbaabe aajuaGcaGGQaGaciyzaiaacIhacaGGWbGaaiikaiabgkHiTiaadMga ceWGibGbaKaadaWgaaqcfasaaiaad6gacaWGJbaabeaajuaGdaqada qaaiaadshacqGHsislcaWG0bWaaSbaaKqbGeaacaaIWaaajuaGbeaa aiaawIcacaGLPaaacaGGPaaaaaa@86CF@ ... .... .. …( 5 )

with H ^ nc MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamOBaiaadogaaeqaaaaa@3980@ being the Hamiltonian operator of the quantum system described on (NC: 3D-RS) symmetries. The very small parameters θ μν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXn aaCaaabeqcfasaaiabeY7aTjabe27aUbaaaaa@3BED@  (compared to the energy) are elements of anti symmetric real matrix of dimension ( length ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba WaaeWaaeaacaqGSbGaaeyzaiaab6gacaqGNbGaaeiDaiaabIgaaiaa wIcacaGLPaaadaahaaqcfasabeaacaaIYaaaaaqcfayaaiabl+qiOb aaaaa@4069@  and ( ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba Gaey4fIOcacaGLOaGaayzkaaaaaa@38F1@  denote to the new star product (the Moyal-Weyl product), which is generalized between two arbitrary functions f( x )  f ^ ( x ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada qadaqaaiaadIhaaiaawIcacaGLPaaacaqGGaGaeyOKH4QabmOzayaa jaWaaeWaaeaaceWG4bGbaKaaaiaawIcacaGLPaaaaaa@400B@ and g( x ) g ^ ( x ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEgada qadaqaaiaadIhaaiaawIcacaGLPaaacqGHsgIRceWGNbGbaKaadaqa daqaaiqadIhagaqcaaGaayjkaiaawMcaaaaa@3F6A@ to f ^ ( x ^ ) g ^ ( x ^ )( fg )( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadAgaga qcamaabmaabaGabmiEayaajaaacaGLOaGaayzkaaGabm4zayaajaWa aeWaaeaaceWG4bGbaKaaaiaawIcacaGLPaaacqGHHjIUdaqadaqaai aadAgacqGHxiIkcaWGNbaacaGLOaGaayzkaaWaaeWaaeaacaWG4baa caGLOaGaayzkaaaaaa@463A@  instead of the usual product ( fg )( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamOzaiaadEgaaiaawIcacaGLPaaadaqadaqaaiaadIhaaiaawIca caGLPaaaaaa@3C5F@  in ordinary three dimensional spaces [23-68]:

f ^ ( x ^ ) g ^ ( x ^ )( fg )( x )exp( i 2 θ μν μ x ν x ( fg )( x,p ) (fg i 2 θ μν μ x f ν x g| ( x μ = x ν ν ) +O( θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadAgaga qcamaabmaabaGabmiEayaajaaacaGLOaGaayzkaaGabm4zayaajaWa aeWaaeaaceWG4bGbaKaaaiaawIcacaGLPaaacqGHHjIUdaqadaqaai aadAgacqGHxiIkcaWGNbaacaGLOaGaayzkaaWaaeWaaeaacaWG4baa caGLOaGaayzkaaGaeyyyIORaciyzaiaacIhacaGGWbGaaiikamaala aabaGaamyAaaqaaiaaikdaaaGaeqiUde3aaWbaaKqbGeqabaGaeqiV d0MaeqyVd4gaaKqbakabgkGi2oaaDaaajuaibaGaeqiVd0gabaGaam iEaaaajuaGcqGHciITdaqhaaqcfasaaiabe27aUbqaaiaadIhaaaqc fa4aaeWaaeaacaWGMbGaam4zaaGaayjkaiaawMcaamaabmaabaGaam iEaiaacYcacaWGWbaacaGLOaGaayzkaaGaeyyyIO7aaqGaaeaacaGG OaGaamOzaiaadEgacqGHsisldaWcaaqaaiaadMgaaeaacaaIYaaaai abeI7aXnaaCaaajuaibeqaaiabeY7aTjabe27aUbaajuaGcqGHciIT daqhaaqcfasaaiabeY7aTbqaaiaadIhaaaqcfaOaamOzaiabgkGi2o aaDaaajuaibaGaeqyVd4gabaGaamiEaaaajuaGcaWGNbaacaGLiWoa daWgaaqaamaabmaabaGaamiEamaaCaaabeqcfasaaiabeY7aTbaaju aGcqGH9aqpcaWG4bWaaWbaaKqbGeqabaGaeqyVd4gaaKqbaoaaCaaa juaibeqaaiabe27aUbaaaKqbakaawIcacaGLPaaaaeqaaiabgUcaRi aad+eadaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaaajuaG caGLOaGaayzkaaaaaa@920C@ ..…… ...... ...... ( 6 )

where f ^ ( x ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadAgaga qcamaabmaabaGabmiEayaajaaacaGLOaGaayzkaaaaaa@3A0A@  and g ^ ( x ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadEgaga qcamaabmaabaGabmiEayaajaaacaGLOaGaayzkaaaaaa@3A0B@  are the new function in (NC: 3D-RS), the following term ( i 2 θ mn μ x f( x ) ν x g( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTm aalaaabaGaamyAaaqaaiaaikdaaaGaeqiUde3aaWbaaeqajuaibaGa amyBaiaad6gaaaqcfaOaeyOaIy7aa0baaKqbGeaacqaH8oqBaeaaca WG4baaaKqbakaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH ciITdaqhaaqcfasaaiabe27aUbqaaiaadIhaaaqcfaOaam4zamaabm aabaGaamiEaaGaayjkaiaawMcaaaaa@4E6C@ ) is induced by (space-space) noncommutativity properties and O( θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad+eada qadaqaaiabeI7aXnaaCaaajuaibeqaaiaaikdaaaaajuaGcaGLOaGa ayzkaaaaaa@3C26@  stands for the second and higher order terms of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ , a Bopp’s shift method can be used, instead of solving any quantum systems by using directly star product procedure [23-55]:

[ x ^ i , x ^ j ]=[ x ^ i ( t ), x ^ j ( t ) ]=i θ ij and [ p ^ i , p ^ j ]=[ p ^ i ( t ), p ^ j ( t ) ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba GabmiEayaajaWaaSbaaKqbGeaacaWGPbaajuaGbeaacaGGSaGabmiE ayaajaWaaSbaaKqbGeaacaWGQbaabeaaaKqbakaawUfacaGLDbaacq GH9aqpdaWadaqaaiqadIhagaqcamaaBaaajuaibaGaamyAaaqcfaya baWaaeWaaeaacaWG0baacaGLOaGaayzkaaGaaiilaiqadIhagaqcam aaBaaajuaibaGaamOAaaqabaqcfa4aaeWaaeaacaWG0baacaGLOaGa ayzkaaaacaGLBbGaayzxaaGaeyypa0JaamyAaiabeI7aXnaaBaaaju aibaGaamyAaiaadQgaaKqbagqaauaabeqabiaaaeaacaqGHbGaaeOB aiaabsgaaeaadaWadaqaaiqadchagaqcamaaBaaajuaibaGaamyAaa qcfayabaGaaiilaiqadchagaqcamaaBaaajuaibaGaamOAaaqcfaya baaacaGLBbGaayzxaaGaeyypa0ZaamWaaeaaceWGWbGbaKaadaWgaa qcfasaaiaadMgaaKqbagqaamaabmaabaGaamiDaaGaayjkaiaawMca aiaacYcaceWGWbGbaKaadaWgaaqcfasaaiaadQgaaKqbagqaamaabm aabaGaamiDaaGaayjkaiaawMcaaaGaay5waiaaw2faaiabg2da9iaa icdaaaaaaa@6EA6@ ………………………………………..…………………. ( 7 )

The three-generalized coordinates ( x ^ = x ^ 1 , y ^ = x ^ 2 , z ^ = x ^ 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmiEayaajaGaeyypa0JabmiEayaajaWaaSbaaKqbGeaacaaIXaaa beaajuaGcaGGSaGabmyEayaajaGaeyypa0JabmiEayaajaWaaSbaaK qbGeaacaaIYaaabeaajuaGcaGGSaGabmOEayaajaGaeyypa0JabmiE ayaajaWaaSbaaKqbGeaacaaIZaaabeaaaKqbakaawIcacaGLPaaaaa a@4790@  in the NC space are depended with corresponding three-usual generalized positions ( x,y,z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamiEaiaacYcacaWG5bGaaiilaiaadQhaaiaawIcacaGLPaaaaaa@3C5C@  and momentum coordinates ( p x , p y , p z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamiCamaaBaaajuaibaGaamiEaaqabaqcfaOaaiilaiaadchadaWg aaqcfasaaiaadMhaaeqaaKqbakaacYcacaWGWbWaaSbaaKqbGeaaca WG6baajuaGbeaaaiaawIcacaGLPaaaaaa@41D2@  by the following relations, as follows [25,28,29,32-34,37-47]:

x ^ =x θ 12 2 p y θ 13 2 p z ,      y ^ =y θ 21 2 p x θ 23 2 p z z ^ =z θ 31 2 p x θ 32 2 p y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOabm iEayaajaGaeyypa0JaamiEaiabgkHiTmaalaaabaGaeqiUde3aaSba aKqbGeaacaaIXaGaaGOmaaqabaaajuaGbaGaaGOmaaaacaWGWbWaaS baaKqbGeaacaWG5baabeaajuaGcqGHsisldaWcaaqaaiabeI7aXnaa BaaajuaibaGaaGymaiaaiodaaKqbagqaaaqaaiaaikdaaaGaamiCam aaBaaajuaibaGaamOEaaqabaqcfaOaaiilaiaabccacaqGGaGaaeii aiaabccacaqGGaGabmyEayaajaGaeyypa0JaamyEaiabgkHiTmaala aabaGaeqiUde3aaSbaaKqbGeaacaaIYaGaaGymaaqcfayabaaabaGa aGOmaaaacaWGWbWaaSbaaKqbGeaacaWG4baabeaajuaGcqGHsislda WcaaqaaiabeI7aXnaaBaaajuaibaGaaGOmaiaaiodaaKqbagqaaaqa aiaaikdaaaGaamiCamaaBaaajuaibaGaamOEaaqabaaakeaajuaGce WG6bGbaKaacqGH9aqpcaWG6bGaeyOeI0YaaSaaaeaacqaH4oqCdaWg aaqcfasaaiaaiodacaaIXaaajuaGbeaaaeaacaaIYaaaaiaadchada WgaaqcfasaaiaadIhaaeqaaKqbakabgkHiTmaalaaabaGaeqiUde3a aSbaaKqbGeaacaaIZaGaaGOmaaqcfayabaaabaGaaGOmaaaacaWGWb WaaSbaaKqbGeaacaWG5baajuaGbeaaaaaa@7717@ ……….……………..( 8 )

The non-vanish-commutators in (NC-3D: RS) can be determined as follows:

[ x ^ , p ^ x ]=[ y ^ , p ^ y ]=[ z ^ , p ^ z ]=i, [ x ^ , y ^ ]=i θ 12 ,[ x ^ , z ^ ]=i θ 13 ,[ y ^ , z ^ ]=i θ 23 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aam WaaeaaceWG4bGbaKaacaGGSaGabmiCayaajaWaaSbaaKqbGeaacaWG 4baabeaaaKqbakaawUfacaGLDbaacqGH9aqpdaWadaqaaiqadMhaga qcaiaacYcaceWGWbGbaKaadaWgaaqcfasaaiaadMhaaKqbagqaaaGa ay5waiaaw2faaiabg2da9maadmaabaGabmOEayaajaGaaiilaiqadc hagaqcamaaBaaajuaibaGaamOEaaqcfayabaaacaGLBbGaayzxaaGa eyypa0JaamyAaiaacYcaaOqaaKqbaoaadmaabaGabmiEayaajaGaai ilaiqadMhagaqcaaGaay5waiaaw2faaiabg2da9iaadMgacqaH4oqC daWgaaqcfasaaiaaigdacaaIYaaajuaGbeaacaGGSaWaamWaaeaace WG4bGbaKaacaGGSaGabmOEayaajaaacaGLBbGaayzxaaGaeyypa0Ja amyAaiabeI7aXnaaBaaajuaibaGaaGymaiaaiodaaeqaaKqbakaacY cadaWadaqaaiqadMhagaqcaiaacYcaceWG6bGbaKaaaiaawUfacaGL DbaacqGH9aqpcaWGPbGaeqiUde3aaSbaaKqbGeaacaaIYaGaaG4maa qabaaaaaa@7081@ ……………….………………….……….( 9 )

which allow us to getting the operator r ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaaja WaaWbaaeqabaGaaGOmaaaaaaa@37DB@  on NC three dimensional spaces as follows [25,28,29,32,33,34,37-48]:

r ^ 2 = r 2 L Θ         MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadkhaga qcamaaCaaajuaibeqaaiaaikdaaaqcfaOaeyypa0JaamOCamaaCaaa juaibeqaaiaaikdaaaqcfaOaeyOeI0ccbeGab8htayaalaGafuiMde LbaSaacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaaa aa@4485@ ……………………..................… ... ...…...……( 10 )

Where the coupling LΘ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqbakaa=X eacqqHyoquaaa@38C7@  is given by ( Θ ij = θ ij /2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaeuiMde1aaSbaaeaacaWGPbqcfaIaamOAaaqcfayabaGaeyypa0Ja eqiUde3aaSbaaKqbGeaacaWGPbGaamOAaaqabaqcfaOaai4laiaaik daaiaawIcacaGLPaaaaaa@4318@ :

LΘ L x Θ 12 + L y Θ 23 + L z Θ 13   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqbakaa=X eacqqHyoqucqGHHjIUcaWGmbWaaSbaaKqbGeaacaWG4baajuaGbeaa cqqHyoqudaWgaaqcfasaaiaaigdacaaIYaaajuaGbeaacqGHRaWkca WGmbWaaSbaaKqbGeaacaWG5baajuaGbeaacqqHyoqudaWgaaqcfasa aiaaikdacaaIZaaabeaajuaGcqGHRaWkcaWGmbWaaSbaaKqbGeaaca WG6baabeaajuaGcqqHyoqudaWgaaqcfasaaiaaigdacaaIZaaabeaa juaGcaqGGaaaaa@505F@ ……..…… .. ………… ( 11 )

with L x =y p z z p y   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeada WgaaqcfasaaiaadIhaaeqaaKqbakabg2da9iaadMhacaWGWbWaaSba aKqbGeaacaWG6baajuaGbeaacqGHsislcaWG6bGaamiCamaaBaaaju aibaGaamyEaaqabaqcfaOaaeiiaaaa@4358@ , L y  = zp x -xp z   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeada WgaaqcfasaaiaadMhaaeqaaKqbakaabccacqGH9aqpcaqG6bGaaeiC amaaBaaajuaibaGaaeiEaaqcfayabaGaaeylaiaabIhacaqGWbWaaS baaKqbGeaacaqG6baabeaajuaGcaqGGaaaaa@43B1@ and L z =x p y y p x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeada WgaaqcfasaaiaadQhaaKqbagqaaiabg2da9iaadIhacaWGWbWaaSba aKqbGeaacaWG5baajuaGbeaacqGHsislcaWG5bGaamiCamaaBaaaju aibaGaamiEaaqabaaaaa@4225@ . Furthermore, the new equal-time anti-commutator for fermionic field operators’ noncommutative spaces can be expressed in the following postulate relations:

{ Ψ ^ α ( t,r ) , * Ψ ¯ ^ β ( t,r' ) }=i ( γ 0 ) αβ δ 3 ( r-r' )  { Ψ ^ α ( t,r ) , * Ψ ^ α ( t,r' ) }={ Ψ ¯ ^ α ( t,r ) , * Ψ ¯ ^ β ( t,r' ) }=i θ αβ δ 3 ( r-r' )   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aai WaaeaacuqHOoqwgaqcamaaBaaajuaibaGaeqySdegabeaajuaGdaqa daqaaiaadshacaGGSaGaaeOCaaGaayjkaiaawMcaamaawagabeqabK qbGeaacaGGQaaajuaGbaGaaiilaaaacuqHOoqwgaqegaqcamaaBaaa juaibaGaeqOSdigajuaGbeaadaqadaqaaiaadshacaGGSaGaaeOCai aabEcaaiaawIcacaGLPaaaaiaawUhacaGL9baacqGH9aqpcaWGPbWa aeWaaeaacqaHZoWzdaahaaqabKqbGeaacaaIWaaaaaqcfaOaayjkai aawMcaamaaBaaabaGaeqySdeMaeqOSdigabeaacqaH0oazdaahaaqa bKqbGeaacaqGZaaaaKqbaoaabmaabaGaaeOCaiaab2cacaqGYbGaae 4jaaGaayjkaiaawMcaaiaabccaaOqaaKqbaoaacmaabaGafuiQdKLb aKaadaWgaaqcfasaaiabeg7aHbqabaqcfa4aaeWaaeaacaWG0bGaai ilaiaabkhaaiaawIcacaGLPaaadaGfGbqabeqajuaibaGaaiOkaaqc fayaaiaacYcaaaGafuiQdKLbaKaadaWgaaqcfasaaiabeg7aHbqcfa yabaWaaeWaaeaacaWG0bGaaiilaiaabkhacaqGNaaacaGLOaGaayzk aaaacaGL7bGaayzFaaGaeyypa0ZaaiWaaeaacuqHOoqwgaqegaqcam aaBaaajuaibaGaeqySdegabeaajuaGdaqadaqaaiaadshacaGGSaGa aeOCaaGaayjkaiaawMcaamaawagabeqabKqbGeaacaGGQaaajuaGba GaaiilaaaacuqHOoqwgaqegaqcamaaBaaabaGaeqOSdigabeaadaqa daqaaiaadshacaGGSaGaaeOCaiaabEcaaiaawIcacaGLPaaaaiaawU hacaGL9baacqGH9aqpcaWGPbGaeqiUde3aaSbaaKqbGeaacqaHXoqy cqaHYoGyaeqaaKqbakabes7aKnaaCaaajuaibeqaaiaabodaaaqcfa 4aaeWaaeaacaqGYbGaaeylaiaabkhacaqGNaaacaGLOaGaayzkaaGa aeiiaiaabccaaaaa@9D6C@  …( 12 )
Here T is the time-ordered product. The purpose of the present work is to extend and present the solution of the Dirac equation with spin-1/2 particle moving in (m.a.o.) potential of the new form:

V ao ( r ^ )= 1 2 M ω 2 r 2 + α 2M r 2 +{ ( α 2M r 4 1 2 M ω 2 ) L Θ   for the    spin symmetric    case  ( α 2M r 4 1 2 M ω 2 ) L ˜ Θ   for the p-spin symmetric    case  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfada WgaaqcfasaaiaadggacaWGVbaabeaajuaGdaqadaqaaiqadkhagaqc aaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiaaikdaaa GaamytaiabeM8a3naaCaaajuaibeqaaiaaikdaaaqcfaOaamOCamaa CaaajuaibeqaaiaaikdaaaqcfaOaey4kaSYaaSaaaeaacqaHXoqyae aacaaIYaGaamytaiaadkhadaahaaqcfasabeaacaaIYaaaaaaajuaG cqGHRaWkdaGabaabaeqabaWaaeWaaeaadaWcaaqaaiabeg7aHbqaai aaikdacaWGnbGaamOCamaaCaaajuaibeqaaiaaisdaaaaaaKqbakab gkHiTmaalaaabaGaaGymaaqaaiaaikdaaaGaamytaiabeM8a3naaCa aabeqcfasaaiaaikdaaaaajuaGcaGLOaGaayzkaaacbeGab8htayaa laGafuiMdeLbaSaafaqabeqacaaabaGaaeiiaiaabccacaqGMbGaae 4BaiaabkhaaeaacaqG0bGaaeiAaiaabwgacaqGGaGaaeiiaiaabcca caqGGaGaae4CaiaabchacaqGPbGaaeOBaiaabccacaqGZbGaaeyEai aab2gacaqGTbGaaeyzaiaabshacaqGYbGaaeyAaiaabogacaqGGaGa aeiiaiaabccacaqGGaGaae4yaiaabggacaqGZbGaaeyzaiaabccaaa aabaWaaeWaaeaadaWcaaqaaiabeg7aHbqaaiaaikdacaWGnbGaamOC amaaCaaajuaibeqaaiaaisdaaaaaaKqbakabgkHiTmaalaaabaGaaG ymaaqaaiaaikdaaaGaamytaiabeM8a3naaCaaajuaibeqaaiaaikda aaaajuaGcaGLOaGaayzkaaGab8htayaalyaaiaGafuiMdeLbaSaafa qabeqacaaabaGaaeiiaiaabccacaqGMbGaae4BaiaabkhaaeaacaqG 0bGaaeiAaiaabwgacaqGGaGaaeiCaiaab2cacaqGZbGaaeiCaiaabM gacaqGUbGaaeiiaiaabohacaqG5bGaaeyBaiaab2gacaqGLbGaaeiD aiaabkhacaqGPbGaae4yaiaabccacaqGGaGaaeiiaiaabccacaqGJb GaaeyyaiaabohacaqGLbGaaeiiaaaaaaGaay5Eaaaaaa@A9B5@ ……….( 13 )

In (NC: 3D-RS) using the generalization Bopp’s shift method to discover the new symmetries and a possibility to obtain another applications to this potential in different fields. This work based essentially on our previously works [23-48]. The outline of our recently article is as follows: In next section, we briefly review the Dirac equation with anharmonic oscillator on based to [18-21]. In section three, we give a description of the Bopp’s shift method for the (m.d.e.) with (m.a.o).Then in section four, we apply standard perturbation theory to establish exact modifications at first order of infinitesimal parameters ( Θ,χ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaeuiMdeLaaiilaiabeE8aJbGaayjkaiaawMcaaaaa@3BE0@  for the perturbed Dirac equation in (NC-3D: RS) for spin-orbital (pseudo-spin orbital) and the relativistic magnetic spectrum for (m.a.o.). In the fifth section, we resume the global spectrum and corresponding NC Hamiltonian for (m.a.o.). Finally, some important concluding remarks are drawn from the present study in last section.

Review the Dirac equation for anharmonic oscillator in ordinary quantum

We start this section by considering a relativistic particle in spherically symmetric for the potential V( r,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabm aabaGaamOCaiaacYcacqaH4oqCaiaawIcacaGLPaaaaaa@3BB7@  which known by anharmonic oscillator, given by in the main reference [21]:

V( r,θ )= 1 2 M ω 2 r 2 + α 2M r 2 + η 2M r 2 sin( θ ) η0 V( r )= 1 2 M ω 2 r 2 + α 2M r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfada qadaqaaiaadkhacaGGSaGaeqiUdehacaGLOaGaayzkaaGaeyypa0Za aSaaaeaacaaIXaaabaGaaGOmaaaacaWGnbGaeqyYdC3aaWbaaKqbGe qabaGaaGOmaaaajuaGcaWGYbWaaWbaaeqajuaibaGaaGOmaaaajuaG cqGHRaWkdaWcaaqaaiabeg7aHbqaaiaaikdacaWGnbGaamOCamaaCa aajuaqbeqaaiaaikdaaaaaaKqbakabgUcaRmaalaaabaGaeq4TdGga baGaaGOmaiaad2eacaWGYbWaaWbaaKqbGeqabaGaaGOmaaaajuaGci GGZbGaaiyAaiaac6gadaqadaqaaiabeI7aXbGaayjkaiaawMcaaaaa daGdKaqaaiabeE7aOjabgkziUkaaicdaaeqacaGLsgcacaWGwbWaae WaaeaacaWGYbaacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaaIXaaa baGaaGOmaaaacaWGnbGaeqyYdC3aaWbaaeqajuaibaGaaGOmaaaaju aGcaWGYbWaaWbaaKqbGeqabaGaaGOmaaaajuaGcqGHRaWkdaWcaaqa aiabeg7aHbqaaiaaikdacaWGnbGaamOCamaaCaaajuaibeqaaiaaik daaaaaaaaa@71DF@ ………… (14)

where M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2eaaa a@374B@ , ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeM8a3b aa@3846@ , ( α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3795@  and η MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdGgaaa@37A2@ )denote the rest mass, frequency of particle and dimensionless parameters. The Dirac equation describing a fermionic particle (spin-1/2 particle) with scalar S(r,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaacI cacaWGYbGaaiilaiabeI7aXjaacMcaaaa@3B84@  and vector V( r,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabm aabaGaamOCaiaacYcacqaH4oqCaiaawIcacaGLPaaaaaa@3BB7@  potentials is given by [18-21]:

( αP+β(M+S(r,θ)) )Ψ( r,θ,ϕ )=( EV( r,θ ) )Ψ( r,θ,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaeqySdeMaaeiuaiabgUcaRiabek7aIjaacIcacaWGnbGaey4kaSIa am4uaiaacIcacaWGYbGaaiilaiabeI7aXjaacMcacaGGPaaacaGLOa GaayzkaaGaeuiQdK1aaeWaaeaacaWGYbGaaiilaiabeI7aXjaacYca cqaHvpGzaiaawIcacaGLPaaacqGH9aqpdaqadaqaaiaadweacqGHsi slcaWGwbWaaeWaaeaacaWGYbGaaiilaiabeI7aXbGaayjkaiaawMca aaGaayjkaiaawMcaaiabfI6aznaabmaabaGaamOCaiaacYcacqaH4o qCcaGGSaGaeqy1dygacaGLOaGaayzkaaaaaa@6173@ … (15)

here M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2eaaa a@374B@  are E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweaaa a@3743@  the fermions’ mass and the relativistic energy while ( α i =( 0 σ i σ i 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg7aHn aaBaaajuaibaGaamyAaaqabaqcfaOaeyypa0ZaaeWaaeaafaqabeGa caaabaGaaGimaaqaaiabeo8aZnaaBaaajuaibaGaamyAaaqabaaaju aGbaGaeq4Wdm3aaSbaaKqbGeaacaWGPbaabeaaaKqbagaacaaIWaaa aaGaayjkaiaawMcaaaaa@4512@ , β=( I 2×2 0 0 I 2×2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIj abg2da9maabmaabaqbaeqabiGaaaqaaiaadMeadaWgaaqcfasaaiaa ikdacqGHxdaTcaaIYaaajuaGbeaaaeaacaaIWaaabaGaaGimaaqaai aadMeadaWgaaqcfasaaiaaikdacqGHxdaTcaaIYaaajuaGbeaaaaaa caGLOaGaayzkaaaaaa@46A1@ ) are the usual Dirac matrices, the spinor Ψ( r,θ,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiQdK1aae WaaeaacaWGYbGaaiilaiabeI7aXjaacYcacqaHvpGzaiaawIcacaGL Paaaaaa@3EE3@  can be expressed as [21]:

Ψ nk ( r,θ,ϕ )=( f nk ( r ) g nk ( r ) )= 1 r ( F nk ( r ) Y jm l ( θ,ϕ ) i G n k ˜ ( r ) Y jm l ˜ ( θ,ϕ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI6azn aaBaaajuaibaGaamOBaiaadUgaaeqaaKqbaoaabmaabaGaamOCaiaa cYcacqaH4oqCcaGGSaGaeqy1dygacaGLOaGaayzkaaGaeyypa0Zaae WaaeaafaqabeGabaaabaGaamOzamaaBaaajuaibaGaamOBaiaadUga aeqaaKqbaoaabmaajuaybaqcfa4aa8HaaKqbGfaacaWGYbaacaGLxd caaiaawIcacaGLPaaaaKqbagaacaWGNbWaaSbaaKqbGeaacaWGUbGa am4Aaaqabaqcfa4aaeWaaeaadaWhcaqaaiaadkhaaiaawEniaaGaay jkaiaawMcaaaaaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaaigda aeaacaWGYbaaamaabmaaeaqabeaacaWGgbWaaSbaaKqbGeaacaWGUb Gaam4AaaqcfayabaWaaeWaaeaacaWGYbaacaGLOaGaayzkaaGaamyw amaaDaaajuaibaGaamOAaiaad2gaaeaacaWGSbaaaKqbaoaabmaaba GaeqiUdeNaaiilaiabew9aMbGaayjkaiaawMcaaaqaaiaadMgacaWG hbWaaSbaaKqbGeaacaWGUbGabm4AayaaiaaabeaajuaGdaqadaqcfa saaiaadkhaaiaawIcacaGLPaaajuaGcaWGzbWaa0baaKqbGeaacaWG QbGaamyBaaqaaiqadYgagaacaaaajuaGdaqadaqaaiabeI7aXjaacY cacqaHvpGzaiaawIcacaGLPaaaaaGaayjkaiaawMcaaaaa@7D3A@ ………(16)

where σ 1 =( 0 1 1 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aaBaaajuaibaGaaGymaaqabaqcfaOaeyypa0ZaaeWaaeaafaqabeGa caaabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGimaaaaaiaawI cacaGLPaaaaaa@3F5D@ , σ 2 =( 0 i i 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aaBaaajuaibaGaaGOmaaqabaqcfaOaeyypa0ZaaeWaaeaafaqabeGa caaabaGaaGimaaqaaiabgkHiTiaadMgaaeaacaWGPbaabaGaaGimaa aaaiaawIcacaGLPaaaaaa@40B1@ and σ 3 =( 1 0 0 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aaBaaajuaibaGaaG4maaqabaqcfaOaeyypa0ZaaeWaaeaafaqabeGa caaabaGaaGymaaqaaiaaicdaaeaacaaIWaaabaGaeyOeI0IaaGymaa aaaiaawIcacaGLPaaaaaa@404C@  and are 2×2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaikdacq GHxdaTcaaIYaaaaa@3A08@  three Pauli matrices while k( k ˜ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgada qadaqaaiqadUgagaacaaGaayjkaiaawMcaaaaa@39F1@  is related to the total angular momentum quantum numbers for spin symmetry l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgaaa a@376A@  and p-spin symmetry l ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadYgaga acaaaa@3779@  as [18-21]:

k={ ( l+1 )  if  -( j+1/2 ),( s 1/2 , p 3/2 ,etc )j=l+ 1 2 , aligned spin ( k0 ) +l    if  j=l+ 1 2 ,( p 1/2 , d 3/2 ,etc )j=l 1 2 , unaligned spin ( k0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacq GH9aqpdaGabaabaeqabaGaeyOeI0YaaeWaaeaacaWGSbGaey4kaSIa aGymaaGaayjkaiaawMcaaiaabccacaqGGaGaaeyAaiaabAgacaqGGa Gaaeiiaiaab2cadaqadaqaaiaabQgacqGHRaWkcaqGXaGaae4laiaa bkdaaiaawIcacaGLPaaacaqGSaWaaeWaaeaacaqGZbWaaSbaaKqbGe aacaqGXaGaae4laiaabkdaaeqaaKqbakaacYcacaWGWbWaaSbaaKqb GeaacaaIZaGaai4laiaaikdaaeqaaKqbakaacYcacaWGLbGaamiDai aadogaaiaawIcacaGLPaaacaqGSaGaaeiiaiaadQgacqGH9aqpcaWG SbGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGSaGaaeiiai aabggacaqGSbGaaeyAaiaabEgacaqGUbGaaeyzaiaabsgacaqGGaGa ae4CaiaabchacaqGPbGaaeOBaiaabccadaqadaqaaiaabUgacqGHPm s4caqGWaaacaGLOaGaayzkaaaabaGaey4kaSIaamiBaiaabccacaqG GaGaaeiiaiaabccacaqGPbGaaeOzaiaabccacaqGGaGaamOAaiabg2 da9iaadYgacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaaiaacYca daqadaqaaiaabchadaWgaaqcfasaaiaabgdacaqGVaGaaeOmaaqcfa yabaGaaiilaiaadsgadaWgaaqcfasaaiaaiodacaGGVaGaaGOmaaqa baqcfaOaaiilaiaadwgacaWG0bGaam4yaaGaayjkaiaawMcaaiaabY cacaqGGaGaamOAaiabg2da9iaadYgacqGHsisldaWcaaqaaiaaigda aeaacaaIYaaaaiaacYcacaqGGaGaaeyDaiaab6gacaqGHbGaaeiBai aabMgacaqGNbGaaeOBaiaabwgacaqGKbGaaeiiaiaabohacaqGWbGa aeyAaiaab6gacaqGGaWaaeWaaeaacaqGRbGaeyOkJeVaaeimaaGaay jkaiaawMcaaaaacaGL7baaaaa@A50E@ …(17)

and

k ˜ ={ l ˜     if  -( j+1/2 ),( s 1/2 , p 3/2 ,etc )j= l ˜ 1 2 , aligned spin ( k0 ) +( l ˜ +1 )    if   j= l ˜ + 1 2 ,( p 1/2 , d 3/2 ,etc )j= l ˜ + 1 2 , unaligned spin ( k0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadUgaga acaiabg2da9maaceaaeaqabeaacqGHsislceWGSbGbaGaacaqGGaGa aeiiaiaabccacaqGGaGaaeyAaiaabAgacaqGGaGaaeiiaiaab2cada qadaqaaiaabQgacqGHRaWkcaqGXaGaae4laiaabkdaaiaawIcacaGL PaaacaqGSaWaaeWaaeaacaqGZbWaaSbaaeaacaqGXaGaae4laiaabk daaeqaaiaacYcacaWGWbWaaSbaaeaacaaIZaGaai4laiaaikdaaeqa aiaacYcacaWGLbGaamiDaiaadogaaiaawIcacaGLPaaacaqGSaGaae iiaiaadQgacqGH9aqpceWGSbGbaGaacqGHsisldaWcaaqaaiaaigda aeaacaaIYaaaaiaacYcacaqGGaGaaeyyaiaabYgacaqGPbGaae4zai aab6gacaqGLbGaaeizaiaabccacaqGZbGaaeiCaiaabMgacaqGUbGa aeiiamaabmaabaGaae4AaiabgMYiHlaabcdaaiaawIcacaGLPaaaae aacqGHRaWkdaqadaqaaiqadYgagaacaiabgUcaRiaaigdaaiaawIca caGLPaaacaqGGaGaaeiiaiaabccacaqGGaGaaeyAaiaabAgacaqGGa GaaeiiaiaabccacaWGQbGaeyypa0JabmiBayaaiaGaey4kaSYaaSaa aeaacaaIXaaabaGaaGOmaaaacaGGSaWaaeWaaeaacaqGWbWaaSbaae aacaqGXaGaae4laiaabkdaaeqaaiaacYcacaWGKbWaaSbaaeaacaaI ZaGaai4laiaaikdaaeqaaiaacYcacaWGLbGaamiDaiaadogaaiaawI cacaGLPaaacaqGSaGaaeiiaiaadQgacqGH9aqpceWGSbGbaGaacqGH RaWkdaWcaaqaaiaaigdaaeaacaaIYaaaaiaacYcacaqGGaGaaeyDai aab6gacaqGHbGaaeiBaiaabMgacaqGNbGaaeOBaiaabwgacaqGKbGa aeiiaiaabohacaqGWbGaaeyAaiaab6gacaqGGaWaaeWaaeaacaqGRb GaeyOkJeVaaeimaaGaayjkaiaawMcaaaaacaGL7baaaaa@A461@ ……(18)

The radial functions ( F nk ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAeada Wgaaqcfasaaiaad6gacaWGRbaabeaajuaGdaqadaqaaiaadkhaaiaa wIcacaGLPaaaaaa@3C84@ , G nk ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEeada Wgaaqcfasaaiaad6gacaWGRbaabeaajuaGdaqadaqaaiaadkhaaiaa wIcacaGLPaaaaaa@3C85@ ) are obtained by solving the following differential equations [18-21]:

[ d 2 d r 2 k(k+1) r 2 ( M+ E nk Δ( r )( M E nk +Σ( r ) )+ dΔ( r ) dr ( d dr + k r ) M E nk +Σ( r ) ) ] F nk ( r )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba WaaSaaaeaacaWGKbWaaWbaaKqbGeqabaGaaGOmaaaaaKqbagaacaWG KbGaamOCamaaCaaabeqcfasaaiaaikdaaaaaaKqbakabgkHiTmaala aabaGaam4AaiaacIcacaWGRbGaey4kaSIaaGymaiaacMcaaeaacaWG YbWaaWbaaKqbGeqabaGaaGOmaaaaaaqcfaOaeyOeI0YaaeWaaeaaca WGnbGaey4kaSIaamyramaaBaaajuaibaGaamOBaiaadUgaaKqbagqa aiabgkHiTiabfs5aenaabmaabaGaamOCaaGaayjkaiaawMcaamaabm aabaGaamytaiabgkHiTiaadweadaWgaaqcfasaaiaad6gacaWGRbaa beaajuaGcqGHRaWkcqqHJoWudaqadaqaaiaadkhaaiaawIcacaGLPa aaaiaawIcacaGLPaaacqGHRaWkdaWcaaqaamaalaaabaGaamizaiab fs5aenaabmaabaGaamOCaaGaayjkaiaawMcaaaqaaiaadsgacaWGYb aaamaabmaabaWaaSaaaeaacaWGKbaabaGaamizaiaadkhaaaGaey4k aSYaaSaaaeaacaWGRbaabaGaamOCaaaaaiaawIcacaGLPaaaaeaaca WGnbGaeyOeI0IaamyramaaBaaajuaibaGaamOBaiaadUgaaeqaaKqb akabgUcaRiabfo6atnaabmaabaGaamOCaaGaayjkaiaawMcaaaaaai aawIcacaGLPaaaaiaawUfacaGLDbaacaWGgbWaaSbaaKqbGeaacaWG UbGaam4AaaqcfayabaWaaeWaaeaacaWGYbaacaGLOaGaayzkaaGaey ypa0JaaGimaaaa@8001@  …(19)

and

[ d 2 d r 2 k(k1) r 2 ( M+ E nk Δ( r )( M E nk +Σ( r ) )+ dΣ( r ) dr ( d dr + k r ) M+ E nk Δ( r ) ) ] G n k ˜ ( r )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba WaaSaaaeaacaWGKbWaaWbaaKqbGeqabaGaaGOmaaaaaKqbagaacaWG KbGaamOCamaaCaaajuaibeqaaiaaikdaaaaaaKqbakabgkHiTmaala aabaGaam4AaiaacIcacaWGRbGaeyOeI0IaaGymaiaacMcaaeaacaWG YbWaaWbaaKqbGeqabaGaaGOmaaaaaaqcfa4aaeWaaeaacaWGnbGaey 4kaSIaamyramaaBaaajuaibaGaamOBaiaadUgaaeqaaKqbakabgkHi Tiabfs5aenaabmaabaGaamOCaaGaayjkaiaawMcaamaabmaabaGaam ytaiabgkHiTiaadweadaWgaaqcfasaaiaad6gacaWGRbaabeaajuaG cqGHRaWkcqqHJoWudaqadaqaaiaadkhaaiaawIcacaGLPaaaaiaawI cacaGLPaaacqGHRaWkdaWcaaqaamaalaaabaGaamizaiabfo6atnaa bmaabaGaamOCaaGaayjkaiaawMcaaaqaaiaadsgacaWGYbaaamaabm aabaWaaSaaaeaacaWGKbaabaGaamizaiaadkhaaaGaey4kaSYaaSaa aeaacaWGRbaabaGaamOCaaaaaiaawIcacaGLPaaaaeaacaWGnbGaey 4kaSIaamyramaaBaaajuaibaGaamOBaiaadUgaaeqaaKqbakabgkHi Tiabfs5aenaabmaabaGaamOCaaGaayjkaiaawMcaaaaaaiaawIcaca GLPaaaaiaawUfacaGLDbaacaWGhbWaaSbaaKqbGeaacaWGUbGabm4A ayaaiaaajuaGbeaadaqadaqaaiaadkhaaiaawIcacaGLPaaacqGH9a qpcaaIWaaaaa@7F2F@  …(20)

The exact spin symmetry corresponding dΔ( r ) dr =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaamizaiabfs5aenaabmaabaGaamOCaaGaayjkaiaawMcaaaqaaiaa dsgacaWGYbaaaiabg2da9iaaicdaaaa@3EF8@ , thus the radial function F nk ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAeada Wgaaqcfasaaiaad6gacaWGRbaajuaGbeaadaqadaqaaiaadkhaaiaa wIcacaGLPaaaaaa@3C84@  satisfying the following like Schrödinger equation [21]:

[ d 2 d r 2 k(k+1) r 2 M 2 E 2 ( ME )( 1 2 M ω 2 r 2 + α 2M r 2 ) ] F nk ( r )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba WaaSaaaeaacaWGKbWaaWbaaKqbGeqabaGaaGOmaaaaaKqbagaacaWG KbGaamOCamaaCaaabeqcfasaaiaaikdaaaaaaKqbakabgkHiTmaala aabaGaam4AaiaacIcacaWGRbGaey4kaSIaaGymaiaacMcaaeaacaWG YbWaaWbaaKqbGeqabaGaaGOmaaaaaaqcfaOaeyOeI0IaamytamaaCa aajuaibeqaaiaaikdaaaqcfaOaeyOeI0IaamyramaaCaaajuaibeqa aiaaikdaaaqcfa4aaeWaaeaacaWGnbGaeyOeI0IaamyraaGaayjkai aawMcaamaabmaabaWaaSaaaeaacaaIXaaabaGaaGOmaaaacaWGnbGa eqyYdC3aaWbaaKqbGeqabaGaaGOmaaaajuaGcaWGYbWaaWbaaKqbGe qabaGaaGOmaaaajuaGcqGHRaWkdaWcaaqaaiabeg7aHbqaaiaaikda caWGnbGaamOCamaaCaaajuaibeqaaiaaikdaaaaaaaqcfaOaayjkai aawMcaaaGaay5waiaaw2faaiaadAeadaWgaaqcfasaaiaad6gacaWG RbaabeaajuaGdaqadaqaaiaadkhaaiaawIcacaGLPaaacqGH9aqpca aIWaaaaa@69ED@ ……(21)

The relativistic energy E n,k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaGGSaGaam4Aaaqabaaaaa@3A25@  and radial upper wave F nk ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAeada Wgaaqcfasaaiaad6gacaWGRbaajuaGbeaadaqadaqaaiaadkhaaiaa wIcacaGLPaaaaaa@3C84@  are given by [21]:

M 2 E n,k 2 M( M+ E n,k ) +4n+2L+3=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaamytamaaCaaabeqcfasaaiaaikdaaaqcfaOaeyOeI0Iaamyramaa DaaajuaibaGaamOBaiaacYcacaWGRbaabaGaaGOmaaaaaKqbagaada Gcaaqaaiaad2eadaqadaqaaiaad2eacqGHRaWkcaWGfbWaaSbaaKqb GeaacaWGUbGaaiilaiaadUgaaKqbagqaaaGaayjkaiaawMcaaaqaba aaaiabgUcaRiaaisdacaWGUbGaey4kaSIaaGOmaiaadYeacqGHRaWk caaIZaGaeyypa0JaaGimaaaa@4F93@ …………(22)

and

F n,k ( r )= C n exp( M( M+ E n,k ) 2 r 2 ) r L+1 L n L+1/2 ( M( M+ E n,k ) r 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAeada Wgaaqcfasaaiaad6gacaGGSaGaam4Aaaqabaqcfa4aaeWaaeaacaWG YbaacaGLOaGaayzkaaGaeyypa0Jaam4qamaaBaaajuaibaGaamOBaa qabaqcfaOaciyzaiaacIhacaGGWbWaaeWaaeaacqGHsisldaWcaaqa amaakaaabaGaamytamaabmaabaGaamytaiabgUcaRiaadweadaWgaa qcfasaaiaad6gacaGGSaGaam4AaaqcfayabaaacaGLOaGaayzkaaaa beaaaeaacaaIYaaaaiaadkhadaahaaqcfasabeaacaaIYaaaaaqcfa OaayjkaiaawMcaaiaadkhadaahaaqcfasabeaacaWGmbGaey4kaSIa aGymaaaajuaGcaWGmbWaa0baaKqbGeaacaWGUbaabaGaamitaiabgU caRiaaigdacaGGVaGaaGOmaaaajuaGdaqadaqaamaakaaabaGaamyt amaabmaabaGaamytaiabgUcaRiaadweadaWgaaqcfasaaiaad6gaca GGSaGaam4AaaqabaaajuaGcaGLOaGaayzkaaaabeaacaWGYbWaaWba aeqajuaibaGaaGOmaaaaaKqbakaawIcacaGLPaaaaaa@690E@ …… (23)

where L n L+1/2 ( M( M+ E n,k ) r 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeada qhaaqcfasaaiaad6gaaeaacaWGmbGaey4kaSIaaGymaiaac+cacaaI YaaaaKqbaoaabmaabaWaaOaaaeaacaWGnbWaaeWaaeaacaWGnbGaey 4kaSIaamyramaaBaaajuaibaGaamOBaiaacYcacaWGRbaabeaaaKqb akaawIcacaGLPaaaaeqaaiaadkhadaahaaqabKqbGeaacaaIYaaaaa qcfaOaayjkaiaawMcaaaaa@496B@  stands for the associated Laguerre functions. For, the exact pseudospin symmetry which corresponds d( r ) dr =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaamizaiabggHiLpaabmaabaGaamOCaaGaayjkaiaawMcaaaqaaiaa dsgacaWGYbaaaiabg2da9iaaicdaaaa@3F36@ , the relativistic energy E n,k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaGGSaGaam4Aaaqcfayabaaaaa@3AB3@  and radial lower wave G nk ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEeada Wgaaqcfasaaiaad6gacaWGRbaajuaGbeaadaqadaqaaiaadkhaaiaa wIcacaGLPaaaaaa@3C85@  are given by [21]:

M 2 E n,k 2 M( M E n,k ) +4n+2 L ˜ +3=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaamytamaaCaaabeqcfasaaiaaikdaaaqcfaOaeyOeI0Iaamyramaa DaaajuaibaGaamOBaiaacYcacaWGRbaabaGaaGOmaaaaaKqbagaada Gcaaqaaiaad2eadaqadaqaaiaad2eacqGHsislcaWGfbWaaSbaaKqb GeaacaWGUbGaaiilaiaadUgaaeqaaaqcfaOaayjkaiaawMcaaaqaba aaaiabgUcaRiaaisdacaWGUbGaey4kaSIaaGOmaiqadYeagaacaiab gUcaRiaaiodacqGH9aqpcaaIWaaaaa@4FAD@ ……… (24)

and

G n,k ( r )= C n exp( M( M E n,k ) 2 r 2 ) r L ˜ +1 L n L ˜ +1/2 ( M( M E n,k ) r 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEeada Wgaaqcfasaaiaad6gacaGGSaGaam4Aaaqabaqcfa4aaeWaaeaacaWG YbaacaGLOaGaayzkaaGaeyypa0Jaam4qamaaBaaajuaibaGaamOBaa qabaqcfaOaciyzaiaacIhacaGGWbWaaeWaaeaacqGHsisldaWcaaqa amaakaaabaGaamytamaabmaabaGaamytaiabgkHiTiaadweadaWgaa qcfasaaiaad6gacaGGSaGaam4AaaqcfayabaaacaGLOaGaayzkaaaa beaaaeaacaaIYaaaaiaadkhadaahaaqcfasabeaacaaIYaaaaaqcfa OaayjkaiaawMcaaiaadkhadaahaaqcfasabeaaceWGmbGbaGaacqGH RaWkcaaIXaaaaKqbakaadYeadaqhaaqcfasaaiaad6gaaeaaceWGmb GbaGaacqGHRaWkcaaIXaGaai4laiaaikdaaaqcfa4aaeWaaeaadaGc aaqaaiaad2eadaqadaqaaiaad2eacqGHsislcaWGfbWaaSbaaKqbGe aacaWGUbGaaiilaiaadUgaaKqbagqaaaGaayjkaiaawMcaaaqabaGa amOCamaaCaaabeqcfasaaiaaikdaaaaajuaGcaGLOaGaayzkaaaaaa@6943@ ……… (25)

NC relativistic Hamiltonian for (m.a.o.)

Formalism of Bopp’s shift method

In this section I first highlight in brief the basics of the concepts of the quantum noncommutative quantum mechanics in the framework of relativistic Dirac equation for modified an harmonic oscillator V ao ( r ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfada WgaaqcfasaaiaadggacaWGVbaabeaajuaGdaqadaqaaiqadkhagaqc aaGaayjkaiaawMcaaaaa@3C9B@  on based to our works [25,28,29,32,-48]:

  1. Ordinary Dirac Hamiltonian operator H ^ ( p i , x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaabmaabaGaamiCamaaBaaajuaibaGaamyAaaqcfayabaGaaiil aiaadIhadaWgaaqcfasaaiaadMgaaeqaaaqcfaOaayjkaiaawMcaaa aa@3F17@  replace by NC Dirac Hamiltonian operator H ^ ncao ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaabaWaaSbaaKqbGeaacaWGUbGaam4yaiabgkHiTiaadgga caWGVbaajuaGbeaaaeqaamaabmaabaGabmiCayaajaWaaSbaaKqbGe aacaWGPbaajuaGbeaacaGGSaGabmiEayaajaWaaSbaaKqbGeaacaWG PbaabeaaaKqbakaawIcacaGLPaaaaaa@44D7@
  2. Ordinary spinor Ψ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI6azn aabmaabaWaa8HaaeaacaWGYbaacaGLxdcaaiaawIcacaGLPaaaaaa@3C3C@  replace by new spinor Ψ ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbfI6azz aataWaaeWaaeaadaWhdaqaaiqadkhagaWeaaGaayz4GaaacaGLOaGa ayzkaaaaaa@3C7B@ ,
  3. Ordinary relativistic energy E nk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaabweada Wgaaqcfasaaiaab6gacaqGRbaajuaGbeaaaaa@39FD@  replaces by new relativistic energy E ncao MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaamyyaiaad+gaaeqaaaaa @3C34@ and ordinary product replace by new star product MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4fIOcaaa@36E5@ .

Thus, the Dirac equation in ordinary quantum mechanics will change into the Dirac equation in extended quantum mechanics for the (m.a.o.) as follows:

H ^ ncao ( p ^ i , x ^ i ) Ψ ( r )= E ncao Ψ ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaqcfa4aaSbaaKqbGeaacaWGUbGaam4yaiabgkHi TiaadggacaWGVbaabeaaaKqbagqaamaabmaabaGabmiCayaajaWaaS baaKqbGeaacaWGPbaabeaajuaGcaGGSaGabmiEayaajaWaaSbaaKqb GeaacaWGPbaajuaGbeaaaiaawIcacaGLPaaacqGHxiIkcuqHOoqwga WeamaabmaabaWaa8XaaeaaceWGYbGbambaaiaawgoiaaGaayjkaiaa wMcaaiabg2da9iaadweadaWgaaqcfasaaiaad6gacaWGJbGaeyOeI0 Iaamyyaiaad+gaaKqbagqaaiqbfI6azzaataWaaeWaaeaadaWhdaqa aiqadkhagaWeaaGaayz4GaaacaGLOaGaayzkaaaaaa@59D5@ ... (26)

The Bopp’s shift method permutes to reduce the above NC equation to simplest form with usual product and translations applied to in space and phase operators:

H ncao ( p ^ i , x ^ i )ψ( r )= E ncao ψ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaamyyaiaad+gaaeqaaKqb aoaabmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGca GGSaGabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaaaKqbakaawIca caGLPaaacqaHipqEdaqadaqaaiqadkhagaWcaaGaayjkaiaawMcaai abg2da9iaadweadaWgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaamyy aiaad+gaaKqbagqaaiabeI8a5naabmaabaGabmOCayaalaaacaGLOa Gaayzkaaaaaa@54B5@ …… (27)

Where the new Hamiltonian operator H ncao ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaamyyaiaad+gaaeqaaKqb aoaabmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGca GGSaGabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaaaKqbakaawIca caGLPaaaaaa@44A6@  can be expressed in three general varieties: both NC space and NC phase (NC-3D: RSP), only NC space (NC-3D: RS) and only NC phase (NC: 3D-RP) as, respectively:

H ncao ( p ^ i , x ^ i )H( p ^ i = p i 1 2 θ ¯ ij x j ; x ^ i = x i 1 2 θ ij p j )   for ( NC-3D: RSP ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaamyyaiaad+gaaKqbagqa amaabmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGca GGSaGabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaaaKqbakaawIca caGLPaaacqGHHjIUcaWGibWaaeWaaeaaceWGWbGbaKaadaWgaaqcfa saaiaadMgaaeqaaKqbakabg2da9iaadchadaWgaaqcfasaaiaadMga aeqaaKqbakabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaWaa0aaae aacqaH4oqCaaWaaSbaaKqbGeaacaWGPbGaamOAaaqabaqcfaOaamiE amaaBaaajuaibaGaamOAaaqabaqcfaOaai4oaiqadIhagaqcamaaBa aajuaibaGaamyAaaqabaqcfaOaeyypa0JaamiEamaaBaaajuaibaGa amyAaaqabaqcfaOaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaacq aH4oqCdaWgaaqcfasaaiaadMgacaWGQbaabeaajuaGcaWGWbWaaSba aKqbGeaacaWGQbaabeaaaKqbakaawIcacaGLPaaafaqabeqacaaaba GaaeiiaiaabccacaqGMbGaae4Baiaabkhaaeaadaqadaqaaiaab6ea caqGdbGaaeylaiaabodacaqGebGaaeOoaiaabccacaqGsbGaae4uai aabcfaaiaawIcacaGLPaaaaaaaaa@7771@ … (28)

H ncao ( p ^ i , x ^ i )H( p ^ i = p i ; x ^ i = x i 1 2 θ ij p j )  for ( NC-3D: RS ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaamyyaiaad+gaaeqaaKqb aoaabmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGca GGSaGabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaaaKqbakaawIca caGLPaaacqGHHjIUcaWGibWaaeWaaeaaceWGWbGbaKaadaWgaaqcfa saaiaadMgaaeqaaKqbakabg2da9iaadchadaWgaaqcfasaaiaadMga aeqaaKqbakaacUdaceWG4bGbaKaadaWgaaqcfasaaiaadMgaaeqaaK qbakabg2da9iaadIhadaWgaaqcfasaaiaadMgaaeqaaKqbakabgkHi TmaalaaabaGaaGymaaqaaiaaikdaaaGaeqiUde3aaSbaaKqbGeaaca WGPbGaamOAaaqabaqcfaOaamiCamaaBaaajuaibaGaamOAaaqcfaya baaacaGLOaGaayzkaaGaaeiiaiaabccacaqGMbGaae4Baiaabkhaca qGGaWaaeWaaeaacaqGobGaae4qaiaab2cacaqGZaGaaeiraiaabQda caqGGaGaaeOuaiaabofaaiaawIcacaGLPaaaaaa@6D76@  ……(29)

H ncao ( p ^ i , x ^ i )H( p ^ i = p i 1 2 θ ¯ ij ; x j , x ^ i = x i ) for ( NC-3D: RP ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaamyyaiaad+gaaKqbagqa amaabmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGca GGSaGabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaaaKqbakaawIca caGLPaaacqGHHjIUcaWGibWaaeWaaeaaceWGWbGbaKaadaWgaaqcfa saaiaadMgaaeqaaKqbakabg2da9iaadchadaWgaaqcfasaaiaadMga aeqaaKqbakabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaWaa0aaae aacqaH4oqCaaWaaSbaaKqbGeaacaWGPbGaamOAaaqabaqcfaOaai4o aiaadIhadaWgaaqcfasaaiaadQgaaeqaaKqbakaacYcaceWG4bGbaK aadaWgaaqcfasaaiaadMgaaeqaaKqbakabg2da9iaadIhadaWgaaqc fasaaiaadMgaaeqaaaqcfaOaayjkaiaawMcaauaabeqabiaaaeaaca qGMbGaae4Baiaabkhaaeaadaqadaqaaiaab6eacaqGdbGaaeylaiaa bodacaqGebGaaeOoaiaabccacaqGsbGaaeiuaaGaayjkaiaawMcaaa aaaaa@6C60@  ……(30)

In recently work, we are interest with the second variety which present by eq. (29) and by the means of the auxiliary two variables x ^ i = x i 1 2 θ ij p j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIhaga qcamaaBaaajuaibaGaamyAaaqabaqcfaOaeyypa0JaamiEamaaBaaa juaibaGaamyAaaqabaqcfaOaeyOeI0YaaSaaaeaacaaIXaaabaGaaG OmaaaacqaH4oqCdaWgaaqcfasaaiaadMgacaWGQbaajuaGbeaacaWG WbWaaSbaaKqbGeaacaWGQbaabeaaaaa@4636@  and p ^ i = p i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadchaga qcamaaBaaajuaibaGaamyAaaqabaqcfaOaeyypa0JaamiCamaaBaaa juaibaGaamyAaaqcfayabaaaaa@3D0F@ , the new modified Hamiltonian H ncao ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaamyyaiaad+gaaeqaaKqb aoaabmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGca GGSaGabmiEayaajaWaaSbaaKqbGeaacaWGPbaajuaGbeaaaiaawIca caGLPaaaaaa@44A6@  may be written as follows

H ncao ( p ^ i , x ^ i )=α P ^ +β(M+S( r ^ ))+ V ao ( r ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaamyyaiaad+gaaKqbagqa amaabmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPbaajuaGbeaaca GGSaGabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaaaKqbakaawIca caGLPaaacqGH9aqpcqaHXoqyceqGqbGbaKaacqGHRaWkcqaHYoGyca GGOaGaamytaiabgUcaRiaadofacaGGOaGabmOCayaajaGaaiykaiaa cMcacqGHRaWkcaWGwbWaaSbaaKqbGeaacaWGHbGaam4Baaqabaqcfa 4aaeWaaeaaceWGYbGbaKaaaiaawIcacaGLPaaaaaa@57FA@ ……(31)

where the modified anharmonic oscillator V ao ( r ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfada WgaaqcfasaaiaadggacaWGVbaajuaGbeaadaqadaqaaiqadkhagaqc aaGaayjkaiaawMcaaaaa@3C9B@ is given by:

V ao ( r ^ )= 1 2 M ω 2 r ^ 2 + α 2M r ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfada WgaaqaaiaadggacaWGVbaabeaadaqadaqaaiqadkhagaqcaaGaayjk aiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaamytai abeM8a3naaCaaajuaibeqaaiaaikdaaaqcfaOabmOCayaajaWaaWba aeqajuaibaGaaGOmaaaajuaGcqGHRaWkdaWcaaqaaiabeg7aHbqaai aaikdacaWGnbGabmOCayaajaWaaWbaaKqbGeqabaGaaGOmaaaaaaaa aa@4B78@ …(32)

The Dirac equation in the presence of above interaction V ao ( r ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfada WgaaqcfasaaiaadggacaWGVbaabeaajuaGdaqadaqaaiqadkhagaqc aaGaayjkaiaawMcaaaaa@3C9B@  can be rewritten according Bopp shift method as follows:

( αP+β(M+S( r ^ )) )Ψ( r,θ,ϕ )=( E ncao V ao ( r ^ ) )Ψ( r,θ,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaeqySdeMaaeiuaiabgUcaRiabek7aIjaacIcacaWGnbGaey4kaSIa am4uaiaacIcaceWGYbGbaKaacaGGPaGaaiykaaGaayjkaiaawMcaai abfI6aznaabmaabaGaamOCaiaacYcacqaH4oqCcaGGSaGaeqy1dyga caGLOaGaayzkaaGaeyypa0ZaaeWaaeaacaWGfbWaaSbaaKqbGeaaca WGUbGaam4yaiabgkHiTiaadggacaWGVbaajuaGbeaacqGHsislcaWG wbWaaSbaaKqbGeaacaWGHbGaam4BaaqcfayabaWaaeWaaeaaceWGYb GbaKaaaiaawIcacaGLPaaaaiaawIcacaGLPaaacqqHOoqwdaqadaqa aiaadkhacaGGSaGaeqiUdeNaaiilaiabew9aMbGaayjkaiaawMcaaa aa@64FD@ ……(33)

The radial functions ( F nk ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAeada Wgaaqcfasaaiaad6gacaWGRbaabeaajuaGdaqadaqaaiaadkhaaiaa wIcacaGLPaaaaaa@3C84@ , G nk ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEeada Wgaaqcfasaaiaad6gacaWGRbaajuaGbeaadaqadaqaaiaadkhaaiaa wIcacaGLPaaaaaa@3C85@ ) are obtained by solving two equations:

[ d dr + k r ] F nk ( r )=[ M+ E nckb Δ( r ^ ) ] G nk ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba WaaSaaaeaacaWGKbaabaGaamizaiaadkhaaaGaey4kaSYaaSaaaeaa caWGRbaabaGaamOCaaaaaiaawUfacaGLDbaacaWGgbWaaSbaaKqbGe aacaWGUbGaam4Aaaqabaqcfa4aaeWaaeaacaWGYbaacaGLOaGaayzk aaGaeyypa0ZaamWaaeaacaWGnbGaey4kaSIaamyramaaBaaajuaiba GaamOBaiaadogacqGHsislcaWGRbGaamOyaaqabaqcfaOaeyOeI0Ia euiLdq0aaeWaaeaaceWGYbGbaKaaaiaawIcacaGLPaaaaiaawUfaca GLDbaacaWGhbWaaSbaaKqbGeaacaWGUbGaam4AaaqcfayabaWaaeWa aeaacaWGYbaacaGLOaGaayzkaaaaaa@5A09@ ..... . (34)

[ d dr + k r ] G nk ( r )=[ M E nckb +Σ( r ^ ) ] G nk ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba WaaSaaaeaacaWGKbaabaGaamizaiaadkhaaaGaey4kaSYaaSaaaeaa caWGRbaabaGaamOCaaaaaiaawUfacaGLDbaacaWGhbWaaSbaaKqbGe aacaWGUbGaam4Aaaqabaqcfa4aaeWaaeaacaWGYbaacaGLOaGaayzk aaGaeyypa0ZaamWaaeaacaWGnbGaeyOeI0IaamyramaaBaaajuaiba GaamOBaiaadogacqGHsislcaWGRbGaamOyaaqabaqcfaOaey4kaSIa eu4Odm1aaeWaaeaaceWGYbGbaKaaaiaawIcacaGLPaaaaiaawUfaca GLDbaacaWGhbWaaSbaaKqbGeaacaWGUbGaam4Aaaqabaqcfa4aaeWa aeaacaWGYbaacaGLOaGaayzkaaaaaa@5A28@ ... .. (35)

with Δ( r ^ )=V( r ^ )S( r ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfs5aen aabmaabaGabmOCayaajaaacaGLOaGaayzkaaGaeyypa0JaamOvamaa bmaabaGabmOCayaajaaacaGLOaGaayzkaaGaeyOeI0Iaam4uamaabm aabaGabmOCayaajaaacaGLOaGaayzkaaaaaa@4335@ and Σ( r ^ )=V( r ^ )+S( r ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfo6atn aabmaabaGabmOCayaajaaacaGLOaGaayzkaaGaeyypa0JaamOvamaa bmaabaGabmOCayaajaaacaGLOaGaayzkaaGaey4kaSIaam4uamaabm aabaGabmOCayaajaaacaGLOaGaayzkaaaaaa@4348@ , eliminating F nk ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAeada Wgaaqcfasaaiaad6gacaWGRbaabeaajuaGdaqadaqaaiaadkhaaiaa wIcacaGLPaaaaaa@3C84@ and G nk ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEeada Wgaaqcfasaaiaad6gacaWGRbaajuaGbeaadaqadaqaaiaadkhaaiaa wIcacaGLPaaaaaa@3C85@ from Eqs. (34) and (35), we can obtain the following two Schrödinger-like differential equations in (NC-3D: RS) symmetries as follows:

[ d 2 d r 2 k(k+1) r 2 ( M+ E ncao Δ( r ^ ) )( M E ncao +Σ( r ^ ) ) ] F nk ( r )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba WaaSaaaeaacaWGKbWaaWbaaKqbGeqabaGaaGOmaaaaaKqbagaacaWG KbGaamOCamaaCaaajuaibeqaaiaaikdaaaaaaKqbakabgkHiTmaala aabaGaam4AaiaacIcacaWGRbGaey4kaSIaaGymaiaacMcaaeaacaWG YbWaaWbaaKqbGeqabaGaaGOmaaaaaaqcfaOaeyOeI0YaaeWaaeaaca WGnbGaey4kaSIaamyramaaBaaajuaibaGaamOBaiaadogacqGHsisl caWGHbGaam4BaaqabaqcfaOaeyOeI0IaeuiLdq0aaeWaaeaaceWGYb GbaKaaaiaawIcacaGLPaaaaiaawIcacaGLPaaadaqadaqaaiaad2ea cqGHsislcaWGfbWaaSbaaKqbGeaacaWGUbGaam4yaiabgkHiTiaadg gacaWGVbaajuaGbeaacqGHRaWkcqqHJoWudaqadaqaaiqadkhagaqc aaGaayjkaiaawMcaaaGaayjkaiaawMcaaaGaay5waiaaw2faaiaadA eadaWgaaqcfasaaiaad6gacaWGRbaajuaGbeaadaqadaqaaiaadkha aiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@6C84@ ……(36)

and

[ d 2 d r 2 k(k1) r 2 ( M+ E ncao Δ( r ^ ) )( M E ncao +Σ( r ^ ) ) ] G nk ( r )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba WaaSaaaeaacaWGKbWaaWbaaeqajuaibaGaaGOmaaaaaKqbagaacaWG KbGaamOCamaaCaaajuaibeqaaiaaikdaaaaaaKqbakabgkHiTmaala aabaGaam4AaiaacIcacaWGRbGaeyOeI0IaaGymaiaacMcaaeaacaWG YbWaaWbaaKqbGeqabaGaaGOmaaaaaaqcfaOaeyOeI0YaaeWaaeaaca WGnbGaey4kaSIaamyramaaBaaajuaibaGaamOBaiaadogacqGHsisl caWGHbGaam4BaaqabaqcfaOaeyOeI0IaeuiLdq0aaeWaaeaaceWGYb GbaKaaaiaawIcacaGLPaaaaiaawIcacaGLPaaadaqadaqaaiaad2ea cqGHsislcaWGfbWaaSbaaKqbGeaacaWGUbGaam4yaiabgkHiTiaadg gacaWGVbaabeaajuaGcqGHRaWkcqqHJoWudaqadaqaaiqadkhagaqc aaGaayjkaiaawMcaaaGaayjkaiaawMcaaaGaay5waiaaw2faaiaadE eadaWgaaqcfasaaiaad6gacaWGRbaajuaGbeaadaqadaqaaiaadkha aiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@6C90@ ………………(37)

After straightforward calculations one can obtains the following two terms: 1 2 M ω 2 r ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaaGymaaqaaiaaikdaaaGaamytaiabeM8a3naaCaaabeqaaiaaikda aaGabmOCayaajaWaaWbaaeqabaGaaGOmaaaaaaa@3D62@  and α 2M r ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeqySdegabaGaaGOmaiaad2eaceWGYbGbaKaadaahaaqcfasabeaa caaIYaaaaaaaaaa@3BC9@  in (NC-3D: RS) as follows:

1 2 M ω 2 r ^ 2 = 1 2 M ω 2 r 2 1 2 M ω 2 L Θ α 2M r ^ 2 = α 2M r 2 + α L Θ 2M r 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aaS aaaeaacaaIXaaabaGaaGOmaaaacaWGnbGaeqyYdC3aaWbaaeqajuai baGaaGOmaaaajuaGceWGYbGbaKaadaahaaqabKqbGeaacaaIYaaaaK qbakabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaamytaiabeM8a 3naaCaaajuaibeqaaiaaikdaaaqcfaOaamOCamaaCaaajuaibeqaai aaikdaaaqcfaOaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaacaWG nbGaeqyYdC3aaWbaaKqbGeqabaGaaGOmaaaaieqajuaGceWFmbGbaS aacuqHyoqugaWcaaGcbaqcfa4aaSaaaeaacqaHXoqyaeaacaaIYaGa amytaiqadkhagaqcamaaCaaajuaibeqaaiaaikdaaaaaaKqbakabg2 da9maalaaabaGaeqySdegabaGaaGOmaiaad2eacaWGYbWaaWbaaKqb GeqabaGaaGOmaaaaaaqcfaOaey4kaSYaaSaaaeaacqaHXoqyceWFmb GbaSaacuqHyoqugaWcaaqaaiaaikdacaWGnbGaamOCamaaCaaabeqc fasaaiaaisdaaaaaaaaaaa@672D@ ………38)

Which allow us to writing the (m.a.o.) potential V ao ( r ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfada WgaaqcfasaaiaadggacaWGVbaabeaajuaGdaqadaqaaiqadkhagaqc aaGaayjkaiaawMcaaaaa@3C9B@ in (NC-3D: RS) as follows:

V ao ( r ^ )= 1 2 M ω 2 r 2 + α 2M r 2 +{ V ^ 1pao ( r,Θ,M,ω )=( α 2M r 4 1 2 M ω 2 ) L Θ   for the    spin symmetric    case  V ^ 2pao ( r,Θ,M,ω )=( α 2M r 4 1 2 M ω 2 ) L ˜ Θ   for the p-spin symmetric    case  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfada WgaaqaaiaadggacaWGVbaabeaadaqadaqaaiqadkhagaqcaaGaayjk aiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaamytai abeM8a3naaCaaajuaibeqaaiaaikdaaaqcfaOaamOCamaaCaaajuai beqaaiaaikdaaaqcfaOaey4kaSYaaSaaaeaacqaHXoqyaeaacaaIYa GaamytaiaadkhadaahaaqcfasabeaacaaIYaaaaaaajuaGcqGHRaWk daGabaabaeqabaGabmOvayaajaWaaSbaaKqbGeaacaaIXaGaamiCai abgkHiTiaadggacaWGVbaajuaGbeaadaqadaqaaiaadkhacaGGSaGa euiMdeLaaiilaiaad2eacaGGSaGaeqyYdChacaGLOaGaayzkaaGaey ypa0ZaaeWaaeaadaWcaaqaaiabeg7aHbqaaiaaikdacaWGnbGaamOC amaaCaaajuaibeqaaiaaisdaaaaaaKqbakabgkHiTmaalaaabaGaaG ymaaqaaiaaikdaaaGaamytaiabeM8a3naaCaaajuaibeqaaiaaikda aaaajuaGcaGLOaGaayzkaaacbeGab8htayaalaGafuiMdeLbaSaafa qabeqacaaabaGaaeiiaiaabccacaqGMbGaae4BaiaabkhaaeaacaqG 0bGaaeiAaiaabwgacaqGGaGaaeiiaiaabccacaqGGaGaae4Caiaabc hacaqGPbGaaeOBaiaabccacaqGZbGaaeyEaiaab2gacaqGTbGaaeyz aiaabshacaqGYbGaaeyAaiaabogacaqGGaGaaeiiaiaabccacaqGGa Gaae4yaiaabggacaqGZbGaaeyzaiaabccaaaaabaGabmOvayaajaWa aSbaaKqbGeaacaaIYaGaamiCaiabgkHiTiaadggacaWGVbaabeaaju aGdaqadaqaaiaadkhacaGGSaGaeuiMdeLaaiilaiaad2eacaGGSaGa eqyYdChacaGLOaGaayzkaaGaeyypa0ZaaeWaaeaadaWcaaqaaiabeg 7aHbqaaiaaikdacaWGnbGaamOCamaaCaaabeqcfasaaiaaisdaaaaa aKqbakabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaGaamytaiabeM 8a3naaCaaabeqcfasaaiaaikdaaaaajuaGcaGLOaGaayzkaaGab8ht ayaalyaaiaGafuiMdeLbaSaafaqabeqacaaabaGaaeiiaiaabccaca qGMbGaae4BaiaabkhaaeaacaqG0bGaaeiAaiaabwgacaqGGaGaaeiC aiaab2cacaqGZbGaaeiCaiaabMgacaqGUbGaaeiiaiaabohacaqG5b GaaeyBaiaab2gacaqGLbGaaeiDaiaabkhacaqGPbGaae4yaiaabcca caqGGaGaaeiiaiaabccacaqGJbGaaeyyaiaabohacaqGLbGaaeiiaa aaaaGaay5Eaaaaaa@C8D0@ . (39)

It’s clearly that, the first 2-terms represent the ordinary anharmonic oscillator while the rest two parts V ^ 1pao ( r,Θ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadAfaga qcamaaBaaajuaibaGaaGymaiaadchacqGHsislcaWGHbGaam4Baaqa baqcfa4aaeWaaeaacaWGYbGaaiilaiabfI5arjaacYcacaWGnbGaai ilaiabeM8a3bGaayjkaiaawMcaaaaa@455E@  and V ^ 2pao ( r,Θ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadAfaga qcamaaBaaajuaibaGaaGOmaiaadchacqGHsislcaWGHbGaam4Baaqa baqcfa4aaeWaaeaacaWGYbGaaiilaiabfI5arjaacYcacaWGnbGaai ilaiabeM8a3bGaayjkaiaawMcaaaaa@455F@  are produced by the deformation of space, this allows writing the (m.a.o.) in the NC case as an equation similarly to the usual Dirac equation of the commutative type with a non local potential. Furthermore, using the unit step function (also known as the Heaviside step function or simply the theta function) we can rewrite the modified anharmonic oscillator to the following form:

V ao ( r ^ )= 1 2 M ω 2 r 2 + α 2M r 2  +θ( E ncao ) V ^ 1pao ( r,Θ,M,ω )+θ( E ncao ) V ^ 2pao ( r,Θ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfada WgaaqcfasaaiaadggacaWGVbaabeaajuaGdaqadaqaaiqadkhagaqc aaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiaaikdaaa GaamytaiabeM8a3naaCaaajuaibeqaaiaaikdaaaqcfaOaamOCamaa CaaajuaibeqaaiaaikdaaaqcfaOaey4kaSYaaSaaaeaacqaHXoqyae aacaaIYaGaamytaiaadkhadaahaaqcfasabeaacaaIYaaaaaaajuaG caqGGaGaey4kaSIaeqiUde3aaeWaaeaacaWGfbWaaSbaaKqbGeaaca WGUbGaam4yaiabgkHiTiaadggacaWGVbaabeaaaKqbakaawIcacaGL PaaaceWGwbGbaKaadaWgaaqcfasaaiaaigdacaWGWbGaeyOeI0Iaam yyaiaad+gaaeqaaKqbaoaabmaabaGaamOCaiaacYcacqqHyoqucaGG SaGaamytaiaacYcacqaHjpWDaiaawIcacaGLPaaacqGHRaWkcqaH4o qCdaqadaqaaiabgkHiTiaadweadaWgaaqcfasaaiaad6gacaWGJbGa eyOeI0Iaamyyaiaad+gaaeqaaaqcfaOaayjkaiaawMcaaiqadAfaga qcamaaBaaajuaibaGaaGOmaiaadchacqGHsislcaWGHbGaam4Baaqc fayabaWaaeWaaeaacaWGYbGaaiilaiabfI5arjaacYcacaWGnbGaai ilaiabeM8a3bGaayjkaiaawMcaaaaa@80D1@  ………(40)

Where

θ( x )={ 1      for x 0      for x0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXn aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maaceaaeaqabeaa caaIXaqbaeqabeGaaaqaaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeOzaiaab+gacaqGYbaabaGaaeiEaiabgQYiXlaabcdacaqGGaaa aaqaaiaaicdafaqabeqacaaabaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGMbGaae4BaiaabkhaaeaacaqG4bGaeyykJeUaaeimaaaa aaGaay5Eaaaaaa@51EB@ ………(41)

We generalized the constraint for the pseudospin (p-spin) symmetry Δ( r )=V( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfs5aen aabmaabaGaamOCaaGaayjkaiaawMcaaiabg2da9iaadAfadaqadaqa aiaadkhaaiaawIcacaGLPaaaaaa@3EC0@  and Σ( r )= C ps =constants MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfo6atn aabmaabaGaamOCaaGaayjkaiaawMcaaiabg2da9iaadoeadaWgaaqc fasaaiaadchacaWGZbaabeaajuaGcqGH9aqpcaqGJbGaae4Baiaab6 gacaqGZbGaaeiDaiaabggacaqGUbGaaeiDaiaabohaaaa@4893@ which presented in refs. [18-21] into the new form Δ( r ^ )=V( r ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfs5aen aabmaabaGabmOCayaajaaacaGLOaGaayzkaaGaeyypa0JaamOvamaa bmaabaGabmOCayaajaaacaGLOaGaayzkaaaaaa@3EE0@  and Σ( r ^ )= C ^ ps =constants MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfo6atn aabmaabaGabmOCayaajaaacaGLOaGaayzkaaGaeyypa0Jabm4qayaa jaWaaSbaaKqbGeaacaWGWbGaam4CaaqcfayabaGaeyypa0Jaae4yai aab+gacaqGUbGaae4CaiaabshacaqGHbGaaeOBaiaabshacaqGZbaa aa@48B3@  in (NC-3D: RS) and inserting the potential V ao ( r ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfada WgaaqcfasaaiaadggacaWGVbaabeaajuaGdaqadaqaaiqadkhagaqc aaGaayjkaiaawMcaaaaa@3C9B@  in eq. (39) into the two Schrödinger-like differential equations (36) and (37), one obtains:

[ d 2 d r 2 k(k+1) r 2 ( M+ E nckb )( M E nckb + C ps )( a r 2 +br c r )( M E nckb + C ps )( ( c 2 r 3 b 2r a ) )( M E nckb + C ^ ps ) L Θ ] F nk ( r )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba WaaSaaaeaacaWGKbWaaWbaaKqbGeqabaGaaGOmaaaaaKqbagaacaWG KbGaamOCamaaCaaajuaibeqaaiaaikdaaaaaaKqbakabgkHiTmaala aabaGaam4AaiaacIcacaWGRbGaey4kaSIaaGymaiaacMcaaeaacaWG YbWaaWbaaeqajuaibaGaaGOmaaaaaaqcfaOaeyOeI0YaaeWaaeaaca WGnbGaey4kaSIaamyramaaBaaajuaibaGaamOBaiaadogacqGHsisl caWGRbGaamOyaaqabaaajuaGcaGLOaGaayzkaaWaaeWaaeaacaWGnb GaeyOeI0IaamyramaaBaaajuaibaGaamOBaiaadogacqGHsislcaWG RbGaamOyaaqcfayabaGaey4kaSIaam4qamaaBaaajuaibaGaamiCai aadohaaKqbagqaaaGaayjkaiaawMcaaiabgkHiTmaabmaabaGaamyy aiaadkhadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaadkgaca WGYbGaeyOeI0YaaSaaaeaacaWGJbaabaGaamOCaaaaaiaawIcacaGL Paaadaqadaqaaiaad2eacqGHsislcaWGfbWaaSbaaKqbGeaacaWGUb Gaam4yaiabgkHiTiaadUgacaWGIbaabeaajuaGcqGHRaWkcaWGdbWa aSbaaKqbGeaacaWGWbGaam4CaaqabaaajuaGcaGLOaGaayzkaaGaey OeI0YaaeWaaeaadaqadaqaamaalaaabaGaam4yaaqaaiaaikdacaWG YbWaaWbaaeqajuaibaGaaG4maaaaaaqcfaOaeyOeI0YaaSaaaeaaca WGIbaabaGaaGOmaiaadkhaaaGaeyOeI0IaamyyaaGaayjkaiaawMca aaGaayjkaiaawMcaamaabmaabaGaamytaiabgkHiTiaadweadaWgaa qcfasaaiaad6gacaWGJbGaeyOeI0Iaam4AaiaadkgaaeqaaKqbakab gUcaRiqadoeagaqcamaaBaaajuaibaGaamiCaiaadohaaeqaaaqcfa OaayjkaiaawMcaaGqabiqa=XeagaWcaiqbfI5arzaalaaacaGLBbGa ayzxaaGaamOramaaBaaajuaibaGaamOBaiaadUgaaKqbagqaamaabm aabaGaamOCaaGaayjkaiaawMcaaiabg2da9iaaicdaaaa@9ED7@  (42)

[ d 2 d r 2 k(k1) r 2 ( M+ E nckb )( M E nckb + C ps )( a r 2 +br c r ) ( M E nckb + C ps )( ( c 2 r 3 b 2r a ) ) L Θ ( M E nckb + C ^ ps ) ] G nk ( r )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba WaaSaaaeaacaWGKbWaaWbaaKqbGeqabaGaaGOmaaaaaKqbagaacaWG KbGaamOCamaaCaaabeqcfasaaiaaikdaaaaaaKqbakabgkHiTmaala aabaGaam4AaiaacIcacaWGRbGaeyOeI0IaaGymaiaacMcaaeaacaWG YbWaaWbaaeqajuaibaGaaGOmaaaaaaqcfaOaeyOeI0YaaeWaaeaaca WGnbGaey4kaSIaamyramaaBaaajuaibaGaamOBaiaadogacqGHsisl caWGRbGaamOyaaqcfayabaaacaGLOaGaayzkaaWaaeWaaeaacaWGnb GaeyOeI0IaamyramaaBaaajuaibaGaamOBaiaadogacqGHsislcaWG RbGaamOyaaqcfayabaGaey4kaSIaam4qamaaBaaajuaibaGaamiCai aadohaaKqbagqaaaGaayjkaiaawMcaaiabgkHiTmaabmaabaGaamyy aiaadkhadaahaaqcfasabeaacaaIYaaaaKqbakabgUcaRiaadkgaca WGYbGaeyOeI0YaaSaaaeaacaWGJbaabaGaamOCaaaaaiaawIcacaGL PaaacaqGGaWaaeWaaeaacaWGnbGaeyOeI0IaamyramaaBaaajuaiba GaamOBaiaadogacqGHsislcaWGRbGaamOyaaqabaqcfaOaey4kaSIa am4qamaaBaaajuaibaGaamiCaiaadohaaKqbagqaaaGaayjkaiaawM caaiabgkHiTmaabmaabaWaaeWaaeaadaWcaaqaaiaadogaaeaacaaI YaGaamOCamaaCaaajuaibeqaaiaaiodaaaaaaKqbakabgkHiTmaala aabaGaamOyaaqaaiaaikdacaWGYbaaaiabgkHiTiaadggaaiaawIca caGLPaaaaiaawIcacaGLPaaaieqaceWFmbGbaSaacuqHyoqugaWcam aabmaabaGaamytaiabgkHiTiaadweadaWgaaqcfasaaiaad6gacaWG JbGaeyOeI0Iaam4AaiaadkgaaKqbagqaaiabgUcaRiqadoeagaqcam aaBaaajuaibaGaamiCaiaadohaaKqbagqaaaGaayjkaiaawMcaaaGa ay5waiaaw2faaiaadEeadaWgaaqcfasaaiaad6gacaWGRbaajuaGbe aadaqadaqaaiaadkhaaiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@9F86@  (43)

It’s clearly that, the additive two parts V ^ 1pao ( r,Θ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadAfaga qcamaaBaaajuaibaGaaGymaiaadchacqGHsislcaWGHbGaam4Baaqa baqcfa4aaeWaaeaacaWGYbGaaiilaiabfI5arjaacYcacaWGnbGaai ilaiabeM8a3bGaayjkaiaawMcaaaaa@455E@  and V ^ 2pao ( r,Θ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadAfaga qcamaaBaaajuaibaGaaGOmaiaadchacqGHsislcaWGHbGaam4Baaqa baqcfa4aaeWaaeaacaWGYbGaaiilaiabfI5arjaacYcacaWGnbGaai ilaiabeM8a3bGaayjkaiaawMcaaaaa@455F@  are proportional with infinitesimal parameter Θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI5arb aa@37F0@ , thus we can considered as a perturbations terms.

The Exact Relativistic Spin-Orbital Hamiltonian and the Corresponding Spectrum for (m.a.o.) In (nc: 3d- rs) Symmetries for Excited States for One-Electron Atoms

The exact relativistic spin-orbital Hamiltonian for (m.a.o.) in (NC: 3D- RS) symmetries for one-electron atoms

Again, the two perturbative terms V ^ 1pao ( r,Θ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadAfaga qcamaaBaaajuaibaGaaGymaiaadchacqGHsislcaWGHbGaam4Baaqa baqcfa4aaeWaaeaacaWGYbGaaiilaiabfI5arjaacYcacaWGnbGaai ilaiabeM8a3bGaayjkaiaawMcaaaaa@455E@  and V ^ 2pao ( r,Θ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadAfaga qcamaaBaaajuaibaGaaGOmaiaadchacqGHsislcaWGHbGaam4Baaqa baqcfa4aaeWaaeaacaWGYbGaaiilaiabfI5arjaacYcacaWGnbGaai ilaiabeM8a3bGaayjkaiaawMcaaaaa@455F@  can be rewritten to the equivalent physical form for (m.a.o.) potential as follows:

{ V ^ 1pao ( r,Θ,M,ω )=Θ( α 2M r 4 1 2 M ω 2 ) L S   for the    spin symmetric    case  V ^ 2pao ( r,Θ,M,ω )=Θ( α 2M r 4 1 2 M ω 2 ) L ˜ S ˜   for the p-spin symmetric    case  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaea qabeaaceWGwbGbaKaadaWgaaqcfasaaiaaigdacaWGWbGaeyOeI0Ia amyyaiaad+gaaKqbagqaamaabmaabaGaamOCaiaacYcacqqHyoquca GGSaGaamytaiaacYcacqaHjpWDaiaawIcacaGLPaaacqGH9aqpcqqH yoqudaqadaqaamaalaaabaGaeqySdegabaGaaGOmaiaad2eacaWGYb WaaWbaaKqbGeqabaGaaGinaaaaaaqcfaOaeyOeI0YaaSaaaeaacaaI XaaabaGaaGOmaaaacaWGnbGaeqyYdC3aaWbaaeqajuaibaGaaGOmaa aaaKqbakaawIcacaGLPaaaieqaceWFmbGbaSaadaWhcaqaaiaadofa aiaawEniauaabeqabiaaaeaacaqGGaGaaeiiaiaabAgacaqGVbGaae OCaaqaaiaabshacaqGObGaaeyzaiaabccacaqGGaGaaeiiaiaabcca caqGZbGaaeiCaiaabMgacaqGUbGaaeiiaiaabohacaqG5bGaaeyBai aab2gacaqGLbGaaeiDaiaabkhacaqGPbGaae4yaiaabccacaqGGaGa aeiiaiaabccacaqGJbGaaeyyaiaabohacaqGLbGaaeiiaaaaaeaace WGwbGbaKaadaWgaaqcfasaaiaaikdacaWGWbGaeyOeI0Iaamyyaiaa d+gaaeqaaKqbaoaabmaabaGaamOCaiaacYcacqqHyoqucaGGSaGaam ytaiaacYcacqaHjpWDaiaawIcacaGLPaaacqGH9aqpcqqHyoqudaqa daqaamaalaaabaGaeqySdegabaGaaGOmaiaad2eacaWGYbWaaWbaae qajuaibaGaaGinaaaaaaqcfaOaeyOeI0YaaSaaaeaacaaIXaaabaGa aGOmaaaacaWGnbGaeqyYdC3aaWbaaKqbGeqabaGaaGOmaaaaaKqbak aawIcacaGLPaaaceWFmbGbaSGbaGaadaWhcaqaaiqadofagaacaaGa ay51GaqbaeqabeGaaaqaaiaabccacaqGGaGaaeOzaiaab+gacaqGYb aabaGaaeiDaiaabIgacaqGLbGaaeiiaiaabchacaqGTaGaae4Caiaa bchacaqGPbGaaeOBaiaabccacaqGZbGaaeyEaiaab2gacaqGTbGaae yzaiaabshacaqGYbGaaeyAaiaabogacaqGGaGaaeiiaiaabccacaqG GaGaae4yaiaabggacaqGZbGaaeyzaiaabccaaaaaaiaawUhaaaaa@B784@  ……(44)

Furthermore, the above perturbative terms V ^ 1pao ( r,Θ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadAfaga qcamaaBaaajuaibaGaaGymaiaadchacqGHsislcaWGHbGaam4Baaqa baqcfa4aaeWaaeaacaWGYbGaaiilaiabfI5arjaacYcacaWGnbGaai ilaiabeM8a3bGaayjkaiaawMcaaaaa@455E@  and V ^ 2pao ( r,Θ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadAfaga qcamaaBaaajuaibaGaaGOmaiaadchacqGHsislcaWGHbGaam4Baaqa baqcfa4aaeWaaeaacaWGYbGaaiilaiabfI5arjaacYcacaWGnbGaai ilaiabeM8a3bGaayjkaiaawMcaaaaa@455F@  can be rewritten to the following new equivalent form for (m.a.o.) potential:

{ V ^ 1pao ( r,Θ,M,ω )= 1 2 Θ( α 2M r 4 1 2 M ω 2 )( J 2 L 2 S 2 )   for the    spin symmetric    case  V ^ 2pao ( r,Θ,M,ω )= 1 2 Θ( α 2M r 4 1 2 M ω 2 )( J 2 L 2 S ˜ 2 )   for the p-spin symmetric    case  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaaea qabeaaceWGwbGbaKaadaWgaaqcfasaaiaaigdacaWGWbGaeyOeI0Ia amyyaiaad+gaaeqaaKqbaoaabmaabaGaamOCaiaacYcacqqHyoquca GGSaGaamytaiaacYcacqaHjpWDaiaawIcacaGLPaaacqGH9aqpdaWc aaqaaiaaigdaaeaacaaIYaaaaiabfI5arnaabmaabaWaaSaaaeaacq aHXoqyaeaacaaIYaGaamytaiaadkhadaahaaqcfasabeaacaaI0aaa aaaajuaGcqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaaiaad2eacq aHjpWDdaahaaqcfasabeaacaaIYaaaaaqcfaOaayjkaiaawMcaamaa bmaabaWaa8XaaeaacaWGkbaacaGLHdcadaahaaqabKqbGeaacaaIYa aaaKqbakabgkHiTmaaFmaabaGaamitaaGaayz4GaWaaWbaaKqbGeqa baGaaGOmaaaajuaGcqGHsisldaWhdaqaaiaadofaaiaawgoiamaaCa aajuaibeqaaiaaikdaaaaajuaGcaGLOaGaayzkaaqbaeqabeGaaaqa aiaabccacaqGGaGaaeOzaiaab+gacaqGYbaabaGaaeiDaiaabIgaca qGLbGaaeiiaiaabccacaqGGaGaaeiiaiaabohacaqGWbGaaeyAaiaa b6gacaqGGaGaae4CaiaabMhacaqGTbGaaeyBaiaabwgacaqG0bGaae OCaiaabMgacaqGJbGaaeiiaiaabccacaqGGaGaaeiiaiaabogacaqG HbGaae4CaiaabwgacaqGGaaaaaqaaiqadAfagaqcamaaBaaajuaiba GaaGOmaiaadchacqGHsislcaWGHbGaam4Baaqabaqcfa4aaeWaaeaa caWGYbGaaiilaiabfI5arjaacYcacaWGnbGaaiilaiabeM8a3bGaay jkaiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaeuiM de1aaeWaaeaadaWcaaqaaiabeg7aHbqaaiaaikdacaWGnbGaamOCam aaCaaajuaibeqaaiaaisdaaaaaaKqbakabgkHiTmaalaaabaGaaGym aaqaaiaaikdaaaGaamytaiabeM8a3naaCaaajuaibeqaaiaaikdaaa aajuaGcaGLOaGaayzkaaWaaeWaaeaadaWhdaqaaiaadQeaaiaawgoi amaaCaaajuaibeqaaiaaikdaaaqcfaOaeyOeI0Yaa8XaaeaacaWGmb aacaGLHdcadaahaaqabKqbGeaacaaIYaaaaKqbakabgkHiTmaaFmaa baGabm4uayaaiaaacaGLHdcadaahaaqabKqbGeaacaaIYaaaaaqcfa OaayjkaiaawMcaauaabeqabiaaaeaacaqGGaGaaeiiaiaabAgacaqG VbGaaeOCaaqaaiaabshacaqGObGaaeyzaiaabccacaqGWbGaaeylai aabohacaqGWbGaaeyAaiaab6gacaqGGaGaae4CaiaabMhacaqGTbGa aeyBaiaabwgacaqG0bGaaeOCaiaabMgacaqGJbGaaeiiaiaabccaca qGGaGaaeiiaiaabogacaqGHbGaae4CaiaabwgacaqGGaaaaaaacaGL 7baaaaa@D370@  … (45)

To the best of our knowledge, we just replace the two spin-orbital coupling S L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFmaaba Gaam4uaaGaayz4GaWaa8XaaeaacaWGmbaacaGLHdcaaaa@3BA0@  and L ˜ S ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqbakqa=X eagaWcgaacamaaFiaabaGabm4uayaaiaaacaGLxdcaaaa@3A0B@  by the expression 1 2 ( J 2 L 2 S 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaaGymaaqaaiaaikdaaaWaaeWaaeaadaWhdaqaaiaadQeaaiaawgoi amaaCaaajuaibeqaaiaaikdaaaqcfaOaeyOeI0Yaa8XaaeaacaWGmb aacaGLHdcadaahaaqcfasabeaacaaIYaaaaKqbakabgkHiTmaaFmaa baGaam4uaaGaayz4GaWaaWbaaKqbGeqabaGaaGOmaaaaaKqbakaawI cacaGLPaaaaaa@47E6@  and 1 2 ( J 2 L 2 S ˜ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaaGymaaqaaiaaikdaaaWaaeWaaeaadaWhdaqaaiaadQeaaiaawgoi amaaCaaajuaibeqaaiaaikdaaaqcfaOaeyOeI0Yaa8XaaeaacaWGmb aacaGLHdcadaahaaqabKqbGeaacaaIYaaaaKqbakabgkHiTmaaFmaa baGabm4uayaaiaaacaGLHdcadaahaaqabKqbGeaacaaIYaaaaaqcfa OaayjkaiaawMcaaaaa@47F5@ , in relativistic quantum mechanics.The set ( H nckb ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaam4AaiaadkgaaKqbagqa amaabmaabaGabmiCayaajaWaaSbaaKqbGeaacaWGPbaabeaajuaGca GGSaGabmiEayaajaWaaSbaaKqbGeaacaWGPbaabeaaaKqbakaawIca caGLPaaaaaa@44A3@ , J 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaabQeada ahaaqcfasabeaacaqGYaaaaaaa@384B@ , L 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaabYeada ahaaqcfasabeaacaaIYaaaaaaa@3854@ , S ˜ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqabofaga acamaaCaaabeqcfasaaiaaikdaaaaaaa@386A@  and J z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQeada WgaaqcfasaaiaadQhaaeqaaKqbakaacMcaaaa@39D1@  forms a complete of conserved physics quantities and the spin–orbit quantum number k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgaaa a@3769@  ( k ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadUgaga acaaaa@3778@ ) is related to the quantum numbers for spin symmetry l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgaaa a@376A@  and p-spin symmetry l ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadYgaga acaaaa@3779@  as follows [18-21]:

k={ k 1 ( l+1 )    if  -( j+1/2 ),( s 1/2 , p 3/2 ,etc )j=l+ 1 2 , aligned spin ( k0 ) k 2 +l    if   ( j=l+ 1 2 ),( p 1/2 , d 3/2 ,etc )j=l 1 2 , unaligned spin ( k0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacq GH9aqpdaGabaabaeqabaGaam4AamaaBaaajuaibaGaaGymaaqabaqc faOaeyyyIORaeyOeI0YaaeWaaeaacaWGSbGaey4kaSIaaGymaaGaay jkaiaawMcaaiaabccacaqGGaGaaeiiaiaabccacaqGPbGaaeOzaiaa bccacaqGGaGaaeylamaabmaabaGaaeOAaiabgUcaRiaabgdacaqGVa GaaeOmaaGaayjkaiaawMcaaiaabYcadaqadaqaaiaabohadaWgaaqc fasaaiaabgdacaqGVaGaaeOmaaqcfayabaGaaiilaiaadchadaWgaa qcfasaaiaaiodacaGGVaGaaGOmaaqabaqcfaOaaiilaiaadwgacaWG 0bGaam4yaaGaayjkaiaawMcaaiaabYcacaqGGaGaamOAaiabg2da9i aadYgacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaaiaacYcacaqG GaGaaeyyaiaabYgacaqGPbGaae4zaiaab6gacaqGLbGaaeizaiaabc cacaqGZbGaaeiCaiaabMgacaqGUbGaaeiiamaabmaabaGaae4Aaiab gMYiHlaabcdaaiaawIcacaGLPaaaaeaacaWGRbWaaSbaaKqbGeaaca aIYaaajuaGbeaacqGHHjIUcqGHRaWkcaWGSbGaaeiiaiaabccacaqG GaGaaeiiaiaabMgacaqGMbGaaeiiaiaabccacaqGGaWaaeWaaeaaca WGQbGaeyypa0JaamiBaiabgUcaRmaalaaabaGaaGymaaqaaiaaikda aaaacaGLOaGaayzkaaGaaiilamaabmaabaGaaeiCamaaBaaajuaiba Gaaeymaiaab+cacaqGYaaabeaajuaGcaGGSaGaamizamaaBaaajuai baGaaG4maiaac+cacaaIYaaajuaGbeaacaGGSaGaamyzaiaadshaca WGJbaacaGLOaGaayzkaaGaaeilaiaabccacaWGQbGaeyypa0JaamiB aiabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaGaaiilaiaabccaca qG1bGaaeOBaiaabggacaqGSbGaaeyAaiaabEgacaqGUbGaaeyzaiaa bsgacaqGGaGaae4CaiaabchacaqGPbGaaeOBaiaabccadaqadaqaai aabUgacqGHQms8caqGWaaacaGLOaGaayzkaaaaaiaawUhaaaaa@B123@  ……… .. ………(46)

and

k ˜ ={ k ˜ 1 l ˜     if  -( j+1/2 ),( s 1/2 , p 3/2 ,etc )j= l ˜ 1 2 , aligned spin ( k0 ) k ˜ 2 +( l ˜ +1 )    if   ( j= l ˜ + 1 2 ),( p 1/2 , d 3/2 ,etc )j= l ˜ + 1 2 , unaligned spin ( k0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadUgaga acaiabg2da9maaceaaeaqabeaaceWGRbGbaGaadaWgaaqcfasaaiaa igdaaeqaaKqbakabggMi6kabgkHiTiqadYgagaacaiaabccacaqGGa GaaeiiaiaabccacaqGPbGaaeOzaiaabccacaqGGaGaaeylamaabmaa baGaaeOAaiabgUcaRiaabgdacaqGVaGaaeOmaaGaayjkaiaawMcaai aabYcadaqadaqaaiaabohadaWgaaqcfasaaiaabgdacaqGVaGaaeOm aaqcfayabaGaaiilaiaadchadaWgaaqcfasaaiaaiodacaGGVaGaaG OmaaqcfayabaGaaiilaiaadwgacaWG0bGaam4yaaGaayjkaiaawMca aiaabYcacaqGGaGaamOAaiabg2da9iqadYgagaacaiabgkHiTmaala aabaGaaGymaaqaaiaaikdaaaGaaiilaiaabccacaqGHbGaaeiBaiaa bMgacaqGNbGaaeOBaiaabwgacaqGKbGaaeiiaiaabohacaqGWbGaae yAaiaab6gacaqGGaWaaeWaaeaacaqGRbGaeyykJeUaaeimaaGaayjk aiaawMcaaaqaaiqadUgagaacamaaBaaajuaibaGaaGOmaaqabaqcfa OaeyyyIORaey4kaSYaaeWaaeaaceWGSbGbaGaacqGHRaWkcaaIXaaa caGLOaGaayzkaaGaaeiiaiaabccacaqGGaGaaeiiaiaabMgacaqGMb GaaeiiaiaabccacaqGGaWaaeWaaeaacaWGQbGaeyypa0JabmiBayaa iaGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaaaiaawIcacaGLPa aacaGGSaWaaeWaaeaacaqGWbWaaSbaaKqbGeaacaqGXaGaae4laiaa bkdaaKqbagqaaiaacYcacaWGKbWaaSbaaKqbGeaacaaIZaGaai4lai aaikdaaeqaaKqbakaacYcacaWGLbGaamiDaiaadogaaiaawIcacaGL PaaacaqGSaGaaeiiaiaadQgacqGH9aqpceWGSbGbaGaacqGHRaWkda WcaaqaaiaaigdaaeaacaaIYaaaaiaacYcacaqGGaGaaeyDaiaab6ga caqGHbGaaeiBaiaabMgacaqGNbGaaeOBaiaabwgacaqGKbGaaeiiai aabohacaqGWbGaaeyAaiaab6gacaqGGaWaaeWaaeaacaqGRbGaeyOk JeVaaeimaaGaayjkaiaawMcaaaaacaGL7baaaaa@B19B@  …………… (47)

With k ˜ ( k ˜ 1 )= l ˜ ( l ˜ +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadUgaga acamaabmaabaGabm4AayaaiaGaeyOeI0IaaGymaaGaayjkaiaawMca aiabg2da9iqadYgagaacamaabmaabaGabmiBayaaiaGaey4kaSIaaG ymaaGaayjkaiaawMcaaaaa@41D4@  and k( k1 )=l( l+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgada qadaqaaiaadUgacqGHsislcaaIXaaacaGLOaGaayzkaaGaeyypa0Ja amiBamaabmaabaGaamiBaiabgUcaRiaaigdaaiaawIcacaGLPaaaaa a@4198@ , which allows us to form two diagonal ( 3×3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaaG4maiabgEna0kaaiodaaiaawIcacaGLPaaaaaa@3B93@  matrixes H ^ soao ( k 1 , k 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaam4Caiaad+gacqGHsislcaWGHbGaam4Baaqc fayabaWaaeWaaeaacaWGRbWaaSbaaKqbGeaacaaIXaaabeaajuaGca GGSaGaam4AamaaBaaajuaibaGaaGOmaaqabaaajuaGcaGLOaGaayzk aaaaaa@4430@  and H ˜ ^ soao ( k ˜ 1 , k ˜ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga acgaqcamaaBaaajuaibaGaam4Caiaad+gacqGHsislcaWGHbGaam4B aaqabaqcfa4aaeWaaeaaceWGRbGbaGaadaWgaaqcfasaaiaaigdaaK qbagqaaiaacYcaceWGRbGbaGaadaWgaaqcKvaG=haacaaIYaaabeaa aKqbakaawIcacaGLPaaaaaa@461F@ , for (m.a.o.), respectively, in (NC: 3D-RS) as:

( H ^ soao ) 11 ( k 1 )= k 1 Θ( α 2M r 4 1 2 M ω 2 )   if -( j+1/2 ),( s 1/2 , p 3/2 ,etc )j=l+ 1 2 , aligned spin ( k0 ) ( H ^ soao ) 22 ( k 2 )= k 2 Θ( α 2M r 4 1 2 M ω 2 )    if ( j=l+ 1 2 ),( p 1/2 , d 3/2 ,etc )j=l 1 2 , unaligned spin ( k0 ) ( H ^ soao ) 33 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aae WaaeaaceWGibGbaKaadaWgaaqcfasaaiaadohacaWGVbGaeyOeI0Ia amyyaiaad+gaaeqaaaqcfaOaayjkaiaawMcaamaaBaaajuaibaGaaG ymaiaaigdaaeqaaKqbaoaabmaabaGaam4AamaaBaaajuaibaGaaGym aaqabaaajuaGcaGLOaGaayzkaaGaeyypa0Jaam4AamaaBaaajuaiba GaaGymaaqabaqcfaOaeuiMde1aaeWaaeaadaWcaaqaaiabeg7aHbqa aiaaikdacaWGnbGaamOCamaaCaaabeqcfasaaiaaisdaaaaaaKqbak abgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaGaamytaiabeM8a3naa CaaajuaibeqaaiaaikdaaaaajuaGcaGLOaGaayzkaaGaaGPaVlaabc cacaqGGaGaaeiiaiaabMgacaqGMbGaaeiiaiaab2cadaqadaqaaiaa bQgacqGHRaWkcaqGXaGaae4laiaabkdaaiaawIcacaGLPaaacaqGSa WaaeWaaeaacaqGZbWaaSbaaKqbGeaacaqGXaGaae4laiaabkdaaeqa aKqbakaacYcacaWGWbWaaSbaaKqbGeaacaaIZaGaai4laiaaikdaaK qbagqaaiaacYcacaWGLbGaamiDaiaadogaaiaawIcacaGLPaaacaqG SaGaaeiiaiaadQgacqGH9aqpcaWGSbGaey4kaSYaaSaaaeaacaaIXa aabaGaaGOmaaaacaGGSaGaaeiiaiaabggacaqGSbGaaeyAaiaabEga caqGUbGaaeyzaiaabsgacaqGGaGaae4CaiaabchacaqGPbGaaeOBai aabccadaqadaqaaiaabUgacqGHPms4caqGWaaacaGLOaGaayzkaaaa baWaaeWaaeaaceWGibGbaKaadaWgaaqcfasaaiaadohacaWGVbGaey OeI0Iaamyyaiaad+gaaKqbagqaaaGaayjkaiaawMcaamaaBaaajuai baGaaGOmaiaaikdaaeqaaKqbaoaabmaabaGaam4AamaaBaaajuaiba GaaGOmaaqabaaajuaGcaGLOaGaayzkaaGaeyypa0Jaam4AamaaBaaa juaibaGaaGOmaaqabaqcfaOaeuiMde1aaeWaaeaadaWcaaqaaiabeg 7aHbqaaiaaikdacaWGnbGaamOCamaaCaaajuaibeqaaiaaisdaaaaa aKqbakabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaGaamytaiabeM 8a3naaCaaajuaibeqaaiaaikdaaaaajuaGcaGLOaGaayzkaaGaaeii aiaabccacaqGGaGaaeiiaiaabMgacaqGMbGaaeiiamaabmaabaGaam OAaiabg2da9iaadYgacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaa aaGaayjkaiaawMcaaiaacYcadaqadaqaaiaabchadaWgaaqcfasaai aabgdacaqGVaGaaeOmaaqcfayabaGaaiilaiaadsgadaWgaaqcfasa aiaaiodacaGGVaGaaGOmaaqabaqcfaOaaiilaiaadwgacaWG0bGaam 4yaaGaayjkaiaawMcaaiaabYcacaqGGaGaamOAaiabg2da9iaadYga cqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaaiaacYcacaqGGaGaae yDaiaab6gacaqGHbGaaeiBaiaabMgacaqGNbGaaeOBaiaabwgacaqG KbGaaeiiaiaabohacaqGWbGaaeyAaiaab6gacaqGGaWaaeWaaeaaca qGRbGaeyOkJeVaaeimaaGaayjkaiaawMcaaaGcbaqcfa4aaeWaaeaa ceWGibGbaKaadaWgaaqcfasaaiaadohacaWGVbGaeyOeI0Iaamyyai aad+gaaeqaaaqcfaOaayjkaiaawMcaamaaBaaajuaibaGaaG4maiaa iodaaeqaaKqbakabg2da9iaaicdaaaaa@ED0C@  … (48)

and

( H ^ soao ) 11 ( k ˜ 1 )= k ˜ 1 Θ( α 2M r 4 1 2 M ω 2 )   if  -( j+1/2 ),( s 1/2 , p 3/2 ,etc )j= l ˜ 1 2 , aligned spin ( k0 ) ( H ^ soao ) 22 ( k ˜ 2 )= k ˜ 2 Θ( α 2M r 4 1 2 M ω 2 )    if   ( j= l ˜ + 1 2 ),( p 1/2 , d 3/2 ,etc )j= l ˜ + 1 2 , unaligned spin ( k0 ) ( H ^ soao ) 33 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aae WaaeaaceWGibGbaKaadaWgaaqcfasaaiaadohacaWGVbGaeyOeI0Ia amyyaiaad+gaaeqaaaqcfaOaayjkaiaawMcaamaaBaaajuaibaGaaG ymaiaaigdaaeqaaKqbaoaabmaabaGabm4AayaaiaWaaSbaaKqbGeaa caaIXaaabeaaaKqbakaawIcacaGLPaaacqGH9aqpceWGRbGbaGaada WgaaqcfasaaiaaigdaaeqaaKqbakabfI5arnaabmaabaWaaSaaaeaa cqaHXoqyaeaacaaIYaGaamytaiaadkhadaahaaqcfasabeaacaaI0a aaaaaajuaGcqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaaiaad2ea cqaHjpWDdaahaaqabKqbGeaacaaIYaaaaaqcfaOaayjkaiaawMcaai aaykW7caqGGaGaaeiiaiaabccacaqGPbGaaeOzaiaabccacaqGGaGa aeylamaabmaabaGaaeOAaiabgUcaRiaabgdacaqGVaGaaeOmaaGaay jkaiaawMcaaiaabYcadaqadaqaaiaabohadaWgaaqcfasaaiaabgda caqGVaGaaeOmaaqcfayabaGaaiilaiaadchadaWgaaqcfasaaiaaio dacaGGVaGaaGOmaaqabaqcfaOaaiilaiaadwgacaWG0bGaam4yaaGa ayjkaiaawMcaaiaabYcacaqGGaGaamOAaiabg2da9iqadYgagaacai abgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaGaaiilaiaabccacaqG HbGaaeiBaiaabMgacaqGNbGaaeOBaiaabwgacaqGKbGaaeiiaiaabo hacaqGWbGaaeyAaiaab6gacaqGGaWaaeWaaeaacaqGRbGaeyykJeUa aeimaaGaayjkaiaawMcaaaqaamaabmaabaGabmisayaajaWaaSbaaK qbGeaacaWGZbGaam4BaiabgkHiTiaadggacaWGVbaajuaGbeaaaiaa wIcacaGLPaaadaWgaaqcfasaaiaaikdacaaIYaaabeaajuaGdaqada qaaiqadUgagaacamaaBaaajuaibaGaaGOmaaqcfayabaaacaGLOaGa ayzkaaGaeyypa0Jabm4AayaaiaWaaSbaaKqbGeaacaaIYaaajuaGbe aacqqHyoqudaqadaqaamaalaaabaGaeqySdegabaGaaGOmaiaad2ea caWGYbWaaWbaaKqbGeqabaGaaGinaaaaaaqcfaOaeyOeI0YaaSaaae aacaaIXaaabaGaaGOmaaaacaWGnbGaeqyYdC3aaWbaaKqbGeqabaGa aGOmaaaaaKqbakaawIcacaGLPaaacaqGGaGaaeiiaiaabccacaqGGa GaaeyAaiaabAgacaqGGaGaaeiiaiaabccadaqadaqaaiaadQgacqGH 9aqpceWGSbGbaGaacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaaa GaayjkaiaawMcaaiaacYcadaqadaqaaiaabchadaWgaaqcfasaaiaa bgdacaqGVaGaaeOmaaqcfayabaGaaiilaiaadsgadaWgaaqcfasaai aaiodacaGGVaGaaGOmaaqcfayabaGaaiilaiaadwgacaWG0bGaam4y aaGaayjkaiaawMcaaiaabYcacaqGGaGaamOAaiabg2da9iqadYgaga acaiabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaGaaiilaiaabcca caqG1bGaaeOBaiaabggacaqGSbGaaeyAaiaabEgacaqGUbGaaeyzai aabsgacaqGGaGaae4CaiaabchacaqGPbGaaeOBaiaabccadaqadaqa aiaabUgacqGHQms8caqGWaaacaGLOaGaayzkaaaakeaajuaGdaqada qaaiqadIeagaqcamaaBaaajuaibaGaam4Caiaad+gacqGHsislcaWG HbGaam4BaaqcfayabaaacaGLOaGaayzkaaWaaSbaaKqbGeaacaaIZa GaaG4maaqabaqcfaOaeyypa0JaaGimaaaaaa@EF5E@ ... (49)

The exact relativistic spin-orbital spectrum for (m.a.o.) potential symmetries for n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gada ahaaqcfasabeaacaWG0bGaamiAaaaaaaa@39A2@  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 E ncper:u ( Θ, k 1 , E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaigdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaaaaa@4DE7@  and E ncper:d ( Θ, k 2 , E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadsgaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaikdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaaaaa@4DD7@  for ( -( j+1/2 ),( s 1/2 , p 3/2 ,etc )j=l+ 1 2 , aligned spin ( k0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaab2cada qadaqaaiaabQgacqGHRaWkcaqGXaGaae4laiaabkdaaiaawIcacaGL PaaacaqGSaWaaeWaaeaacaqGZbWaaSbaaKqbGeaacaqGXaGaae4lai aabkdaaeqaaKqbakaacYcacaWGWbWaaSbaaKqbGeaacaaIZaGaai4l aiaaikdaaeqaaKqbakaacYcacaWGLbGaamiDaiaadogaaiaawIcaca GLPaaacaqGSaGaaeiiaiaadQgacqGH9aqpcaWGSbGaey4kaSYaaSaa aeaacaaIXaaabaGaaGOmaaaacaGGSaGaaeiiaiaabggacaqGSbGaae yAaiaabEgacaqGUbGaaeyzaiaabsgacaqGGaGaae4CaiaabchacaqG PbGaaeOBaiaabccadaqadaqaaiaabUgacqGHPms4caqGWaaacaGLOa Gaayzkaaaaaa@6337@  and spin-up) and ( ( j=l+ 1 2 ),( p 1/2 , d 3/2 ,etc )j=l 1 2 , unaligned spin ( k0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamOAaiabg2da9iaadYgacqGHRaWkdaWcaaqaaiaaigdaaeaacaaI YaaaaaGaayjkaiaawMcaaiaacYcadaqadaqaaiaabchadaWgaaqcfa saaiaabgdacaqGVaGaaeOmaaqabaqcfaOaaiilaiaadsgadaWgaaqc fasaaiaaiodacaGGVaGaaGOmaaqcfayabaGaaiilaiaadwgacaWG0b Gaam4yaaGaayjkaiaawMcaaiaabYcacaqGGaGaamOAaiabg2da9iaa dYgacqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaaiaacYcacaqGGa GaaeyDaiaab6gacaqGHbGaaeiBaiaabMgacaqGNbGaaeOBaiaabwga caqGKbGaaeiiaiaabohacaqGWbGaaeyAaiaab6gacaqGGaWaaeWaae aacaqGRbGaeyOkJeVaaeimaaGaayjkaiaawMcaaaaa@65E3@  and spin down), respectively, at first order of infinitesimal parameter Θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI5arb aa@37F0@ , for ground state, obtained by applying the standard perturbation theory, using Eqs. (22), (44)and (45) as:

Ψ + nk ( r,θ,ϕ )[ θ( E ncao ) V ^ 1pao ( r,Θ, E nk ,M,ω )+θ( E ncao ) V ^ 2pao ( r,Θ, E nk ,M,ω ) ] Ψ nk ( r,θ,ϕ ) r 2 drdΩ = =θ( E ncao ) F * nk ( r ) V ^ 1pkb ( r,Θ, E nk ,M,ω ) F nk ( r )dr θ( E ncao ) G n k ˜ * ( r ) V ^ 2pao ( r,Θ, E nk ,M,ω ) G n k ˜ ( r )dr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aa8 qaaeaadaGfGbqabeqabaGaey4kaScabaGaeuiQdKfaamaaBaaajuai baGaamOBaiaadUgaaKqbagqaamaabmaabaGaamOCaiaacYcacqaH4o qCcaGGSaGaeqy1dygacaGLOaGaayzkaaWaamWaaeaacqaH4oqCdaqa daqaaiaadweadaWgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaamyyai aad+gaaKqbagqaaaGaayjkaiaawMcaaiqadAfagaqcamaaBaaajuai baGaaGymaiaadchacqGHsislcaWGHbGaam4Baaqabaqcfa4aaeWaae aacaWGYbGaaiilaiabfI5arjaacYcacaWGfbWaaSbaaeaacaWGUbGa am4AaaqabaGaaiilaiaad2eacaGGSaGaeqyYdChacaGLOaGaayzkaa Gaey4kaSIaeqiUde3aaeWaaeaacqGHsislcaWGfbWaaSbaaKqbGeaa caWGUbGaam4yaiabgkHiTiaadggacaWGVbaabeaaaKqbakaawIcaca GLPaaaceWGwbGbaKaadaWgaaqcfasaaiaaikdacaWGWbGaeyOeI0Ia amyyaiaad+gaaeqaaKqbaoaabmaabaGaamOCaiaacYcacqqHyoquca GGSaGaamyramaaBaaajuaibaGaamOBaiaadUgaaeqaaKqbakaacYca caWGnbGaaiilaiabeM8a3bGaayjkaiaawMcaaaGaay5waiaaw2faai abfI6aznaaBaaajuaibaGaamOBaiaadUgaaKqbagqaamaabmaabaGa amOCaiaacYcacqaH4oqCcaGGSaGaeqy1dygacaGLOaGaayzkaaGaam OCamaaCaaajuaibeqaaiaaikdaaaqcfaOaamizaiaadkhacaWGKbGa euyQdCfabeqabiabgUIiYdGaeyypa0dakeaajuaGcqGH9aqpcqaH4o qCdaqadaqaaiaadweadaWgaaqcfasaaiaad6gacaWGJbGaeyOeI0Ia amyyaiaad+gaaeqaaaqcfaOaayjkaiaawMcaamaapeaabaGaamOram aaCaaajuaibeqaaiaacQcaaaqcfa4aaSbaaKqbGeaacaWGUbGaam4A aaqabaqcfa4aaeWaaeaacaWGYbaacaGLOaGaayzkaaGabmOvayaaja WaaSbaaKqbGeaacaaIXaGaamiCaiabgkHiTiaadUgacaWGIbaajuaG beaadaqadaqaaiaadkhacaGGSaGaeuiMdeLaaiilaiaadweadaWgaa qaaiaad6gacaWGRbaabeaacaGGSaGaamytaiaacYcacqaHjpWDaiaa wIcacaGLPaaacaWGgbWaaSbaaKqbGeaacaWGUbGaam4Aaaqabaqcfa 4aaeWaaeaacaWGYbaacaGLOaGaayzkaaGaamizaiaadkhaaeqabeGa ey4kIipacqGHsislcqaH4oqCdaqadaqaaiaadweadaWgaaqcfasaai aad6gacaWGJbGaeyOeI0Iaamyyaiaad+gaaeqaaaqcfaOaayjkaiaa wMcaamaapeaabaGaam4ramaaBaaajuaibaGaamOBaiqadUgagaacaa qabaqcfa4aaWbaaKqbGeqabaGaaiOkaaaajuaGdaqadaqaaiaadkha aiaawIcacaGLPaaaceWGwbGbaKaadaWgaaqcfasaaiaaikdacaWGWb GaeyOeI0Iaamyyaiaad+gaaeqaaKqbaoaabmaabaGaamOCaiaacYca cqqHyoqucaGGSaGaamyramaaBaaajuaibaGaamOBaiaadUgaaeqaaK qbakaacYcacaWGnbGaaiilaiabeM8a3bGaayjkaiaawMcaaiaadEea daWgaaqcfasaaiaad6gaceWGRbGbaGaaaeqaaKqbaoaabmaabaGaam OCaaGaayjkaiaawMcaaiaadsgacaWGYbaabeqabiabgUIiYdaaaaa@F365@ .... (50)

The first parts represent the modifications to the energy levels for the spin symmetric cases E ncper:u ( Θ, k 1 , E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaigdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaaaaa@4DE7@  and E ncper:d ( Θ, k 2 , E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadsgaaKqbagqaamaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaikdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaaaaa@4DD7@  while the second part represent the modifications to the energy levels ( E ncper:d ( Θ, k ˜ 1 , E 0k , E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadsgaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiqadUgaga acamaaBaaajuaibaGaaGymaaqabaqcfaOaaiilaiaadweadaWgaaqc fasaaiaaicdacaWGRbaabeaajuaGcaGGSaGaamyramaaBaaajuaiba GaamOBaiaadUgaaeqaaKqbakaacYcacaWGnbGaaiilaiabeM8a3bGa ayjkaiaawMcaaaaa@51E6@ , E ncper:u ( Θ, k ˜ 2 , E 0k , E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiqadUgaga acamaaBaaajuaibaGaaGOmaaqabaqcfaOaaiilaiaadweadaWgaaqc fasaaiaaicdacaWGRbaabeaajuaGcaGGSaGaamyramaaBaaajuaiba GaamOBaiaadUgaaeqaaKqbakaacYcacaWGnbGaaiilaiabeM8a3bGa ayjkaiaawMcaaaaa@51F8@ ) for the spin spin-symmetry, then we have explicitly:

E ncper:u ( Θ, k 1 , E nk ,M,ω )θ( E nckb ) k 1 Θ F * nk ( r )( α 2M r 4 1 2 M ω 2 ) F nk ( r )dr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaKqbagqaamaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaigdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaGaeyyyIORaeqiUde3aaeWaaeaacaWGfbWaaSbaaKqb GeaacaWGUbGaam4yaiabgkHiTiaadUgacaWGIbaabeaaaKqbakaawI cacaGLPaaacaWGRbWaaSbaaKqbGeaacaaIXaaabeaajuaGcqqHyoqu daWdbaqaamaaxacabaGaamOraaqabKqbGeaacaGGQaaaaKqbaoaaBa aajuaibaGaamOBaiaadUgaaeqaaKqbaoaabmaabaGaamOCaaGaayjk aiaawMcaamaabmaabaWaaSaaaeaacqaHXoqyaeaacaaIYaGaamytai aadkhadaahaaqcfasabeaacaaI0aaaaaaajuaGcqGHsisldaWcaaqa aiaaigdaaeaacaaIYaaaaiaad2eacqaHjpWDdaahaaqcfasabeaaca aIYaaaaaqcfaOaayjkaiaawMcaaiaadAeadaWgaaqcfasaaiaad6ga caWGRbaabeaajuaGdaqadaqaaiaadkhaaiaawIcacaGLPaaacaWGKb GaamOCaaqabeqacqGHRiI8aaaa@7CBB@  ………… (51)

E ncper:u ( Θ, k 2 , E nk ,M,ω )θ( E nckb ) k 2 Θ F * nk ( r )( α 2M r 4 1 2 M ω 2 ) F nk ( r )dr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaikdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaGaeyyyIORaeqiUde3aaeWaaeaacaWGfbWaaSbaaKqb GeaacaWGUbGaam4yaiabgkHiTiaadUgacaWGIbaajuaGbeaaaiaawI cacaGLPaaacaWGRbWaaSbaaKqbGeaacaaIYaaajuaGbeaacqqHyoqu daWdbaqaamaaxacabaGaamOraaqabKqbGeaacaGGQaaaaKqbaoaaBa aajuaibaGaamOBaiaadUgaaeqaaKqbaoaabmaabaGaamOCaaGaayjk aiaawMcaamaabmaabaWaaSaaaeaacqaHXoqyaeaacaaIYaGaamytai aadkhadaahaaqabKqbGeaacaaI0aaaaaaajuaGcqGHsisldaWcaaqa aiaaigdaaeaacaaIYaaaaiaad2eacqaHjpWDdaahaaqabKqbGeaaca aIYaaaaaqcfaOaayjkaiaawMcaaiaadAeadaWgaaqcfasaaiaad6ga caWGRbaajuaGbeaadaqadaqaaiaadkhaaiaawIcacaGLPaaacaWGKb GaamOCaaqabeqacqGHRiI8aaaa@7CBD@ ……… (52)

Inserting the radial function F nk ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGUbGaam4AaaqabaGcdaqadaqaaiaadkhaaiaawIcacaGL Paaaaaa@3B5A@  given by Eq. (23) into the above two Eqs. (51) and (52) to obtain:

E ncper:u ( Θ, k 2 , E nk ,M,ω )θ( E ncao )Θ k 1 | C n | 2 0 + exp( M( M+ E n,k ) r 2 ) r 2L+2 [ L n L+1/2 ( M( M+ E n,k ) r 2 ) ] 2 ( α 2M r 4 1 2 M ω 2 )dr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaikdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaGaeyyyIORaeqiUde3aaeWaaeaacaWGfbWaaSbaaKqb GeaacaWGUbGaam4yaiabgkHiTiaadggacaWGVbaajuaGbeaaaiaawI cacaGLPaaacqqHyoqucaWGRbWaaSbaaKqbGeaacaaIXaaabeaajuaG daabdaqaaiaadoeadaWgaaqcfasaaiaad6gaaKqbagqaaaGaay5bSl aawIa7amaaCaaabeqcfasaaiaaikdaaaqcfa4aa8qCaeaaciGGLbGa aiiEaiaacchadaqadaqaaiabgkHiTmaakaaabaGaamytamaabmaaba GaamytaiabgUcaRiaadweadaWgaaqcfasaaiaad6gacaGGSaGaam4A aaqabaaajuaGcaGLOaGaayzkaaaabeaacaWGYbWaaWbaaeqajuaiba GaaGOmaaaaaKqbakaawIcacaGLPaaacaWGYbWaaWbaaKqbGeqabaGa aGOmaiaadYeacqGHRaWkcaaIYaaaaKqbaoaadmaabaGaamitamaaDa aajuaibaGaamOBaaqaaiaadYeacqGHRaWkcaaIXaGaai4laiaaikda aaqcfa4aaeWaaeaadaGcaaqaaiaad2eadaqadaqaaiaad2eacqGHRa WkcaWGfbWaaSbaaKqbGeaacaWGUbGaaiilaiaadUgaaKqbagqaaaGa ayjkaiaawMcaaaqabaGaamOCamaaCaaajuaibeqaaiaaikdaaaaaju aGcaGLOaGaayzkaaaacaGLBbGaayzxaaWaaWbaaKqbGeqabaGaaGOm aaaajuaGdaqadaqaamaalaaabaGaeqySdegabaGaaGOmaiaad2eaca WGYbWaaWbaaeqajuaibaGaaGinaaaaaaqcfaOaeyOeI0YaaSaaaeaa caaIXaaabaGaaGOmaaaacaWGnbGaeqyYdC3aaWbaaeqajuaibaGaaG OmaaaaaKqbakaawIcacaGLPaaacaWGKbGaamOCaaqcfasaaiaaicda aeaacqGHRaWkcqGHEisPaKqbakabgUIiYdaaaa@A645@ .... (53)

E ncper:u ( Θ, k 2 , E nk ,M,ω )θ( E ncao )Θ k 2 | C n | 2 0 + exp( M( M+ E n,k ) r 2 ) r 2L+2 [ L n L+1/2 ( M( M+ E n,k ) r 2 ) ] 2 ( α 2M r 4 1 2 M ω 2 )dr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaikdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqcfayabaGaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaGaeyyyIORaeqiUde3aaeWaaeaacaWGfbWaaSbaaKqb GeaacaWGUbGaam4yaiabgkHiTiaadggacaWGVbaajuaGbeaaaiaawI cacaGLPaaacqqHyoqucaWGRbWaaSbaaKqbGeaacaaIYaaabeaajuaG daabdaqaaiaadoeadaWgaaqcfasaaiaad6gaaKqbagqaaaGaay5bSl aawIa7amaaCaaajuaibeqaaiaaikdaaaqcfa4aa8qCaeaaciGGLbGa aiiEaiaacchadaqadaqaaiabgkHiTmaakaaabaGaamytamaabmaaba GaamytaiabgUcaRiaadweadaWgaaqcfasaaiaad6gacaGGSaGaam4A aaqcfayabaaacaGLOaGaayzkaaaabeaacaWGYbWaaWbaaKqbGeqaba GaaGOmaaaaaKqbakaawIcacaGLPaaacaWGYbWaaWbaaeqajuaibaGa aGOmaiaadYeacqGHRaWkcaaIYaaaaKqbaoaadmaabaGaamitamaaDa aajuaibaGaamOBaaqaaiaadYeacqGHRaWkcaaIXaGaai4laiaaikda aaqcfa4aaeWaaeaadaGcaaqaaiaad2eadaqadaqaaiaad2eacqGHRa WkcaWGfbWaaSbaaKqbGeaacaWGUbGaaiilaiaadUgaaKqbagqaaaGa ayjkaiaawMcaaaqabaGaamOCamaaCaaajuaibeqaaiaaikdaaaaaju aGcaGLOaGaayzkaaaacaGLBbGaayzxaaWaaWbaaeqajuaibaGaaGOm aaaajuaGdaqadaqaamaalaaabaGaeqySdegabaGaaGOmaiaad2eaca WGYbWaaWbaaeqajuaibaGaaGinaaaaaaqcfaOaeyOeI0YaaSaaaeaa caaIXaaabaGaaGOmaaaacaWGnbGaeqyYdC3aaWbaaKqbGeqabaGaaG OmaaaaaKqbakaawIcacaGLPaaacaWGKbGaamOCaaqcfasaaiaaicda aeaacqGHRaWkcqGHEisPaKqbakabgUIiYdaaaa@A646@  (54)

To evaluate the integrations here, we rewriting the above two integrals to the useful forms:

E ncper:u ( Θ, k 2 , E nk ,M,ω )θ( E ncao )Θ k 1 | C n | 2 μ=1 2 T ao μ ( E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaikdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqcfayabaGaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaGaeyyyIORaeqiUde3aaeWaaeaacaWGfbWaaSbaaKqb GeaacaWGUbGaam4yaiabgkHiTiaadggacaWGVbaajuaGbeaaaiaawI cacaGLPaaacqqHyoqucaWGRbWaaSbaaKqbGeaacaaIXaaabeaajuaG daabdaqaaiaadoeadaWgaaqcfasaaiaad6gaaKqbagqaaaGaay5bSl aawIa7amaaCaaajuaibeqaaiaaikdaaaqcfa4aaabCaeaacaWGubWa a0baaKqbGeaacaWGHbGaam4BaaqaaiabeY7aTbaajuaGdaqadaqaai aadweadaWgaaqcfasaaiaad6gacaWGRbaajuaGbeaacaGGSaGaamyt aiaacYcacqaHjpWDaiaawIcacaGLPaaaaKqbGeaacqaH8oqBcqGH9a qpcaaIXaaabaGaaGOmaaqcfaOaeyyeIuoaaaa@79FE@ ……… (55)

E ncper:u ( Θ, k 2 , E nk ,M,ω )θ( E ncao )Θ k 2 | C n | 2 μ=1 2 T ao μ ( E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaikdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqcfayabaGaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaGaeyyyIORaeqiUde3aaeWaaeaacaWGfbWaaSbaaKqb GeaacaWGUbGaam4yaiabgkHiTiaadggacaWGVbaajuaGbeaaaiaawI cacaGLPaaacqqHyoqucaWGRbWaaSbaaKqbGeaacaaIYaaajuaGbeaa daabdaqaaiaadoeadaWgaaqcfasaaiaad6gaaKqbagqaaaGaay5bSl aawIa7amaaCaaajuaibeqaaiaaikdaaaqcfa4aaabCaeaacaWGubWa a0baaKqbGeaacaWGHbGaam4BaaqaaiabeY7aTbaajuaGdaqadaqaai aadweadaWgaaqcfasaaiaad6gacaWGRbaabeaajuaGcaGGSaGaamyt aiaacYcacqaHjpWDaiaawIcacaGLPaaaaKqbGeaacqaH8oqBcqGH9a qpcaaIXaaabaGaaGOmaaqcfaOaeyyeIuoaaaa@79FF@  ………… (56)

Where the factors T ao μ ( E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada qhaaqcfasaaiaadggacaWGVbaabaGaeqiVd0gaaKqbaoaabmaabaGa amyramaaBaaajuaibaGaamOBaiaadUgaaKqbagqaaiaacYcacaWGnb GaaiilaiabeM8a3bGaayjkaiaawMcaaaaa@44D2@   ( μ=1,2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaeqiVd0Maeyypa0JaaGymaiaacYcacaaIYaaacaGLOaGaayzkaaaa aa@3CE5@  are given by:

T ao 1 ( E nk ,M,ω )= α 2M 0 + exp( M( M+ E n,k ) r 2 ) r 2L2 [ L n L+1/2 ( M( M+ E n,k ) r 2 ) ] 2 dr T ao 2 ( E nk ,M,ω )= 1 2 M ω 2 0 + exp( M( M+ E n,k ) r 2 ) r 2L+2 [ L n L+1/2 ( M( M+ E n,k ) r 2 ) ] 2 dr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam ivamaaDaaajuaibaGaamyyaiaad+gaaeaacaaIXaaaaKqbaoaabmaa baGaamyramaaBaaajuaibaGaamOBaiaadUgaaKqbagqaaiaacYcaca WGnbGaaiilaiabeM8a3bGaayjkaiaawMcaaiabg2da9maalaaabaGa eqySdegabaGaaGOmaiaad2eaaaWaa8qCaeaaciGGLbGaaiiEaiaacc hadaqadaqaaiabgkHiTmaakaaabaGaamytamaabmaabaGaamytaiab gUcaRiaadweadaWgaaqcfasaaiaad6gacaGGSaGaam4Aaaqabaaaju aGcaGLOaGaayzkaaaabeaacaWGYbWaaWbaaKqbGeqabaGaaGOmaaaa aKqbakaawIcacaGLPaaacaWGYbWaaWbaaKqbGeqabaGaaGOmaiaadY eacqGHsislcaaIYaaaaKqbaoaadmaabaGaamitamaaDaaajuaibaGa amOBaaqaaiaadYeacqGHRaWkcaaIXaGaai4laiaaikdaaaqcfa4aae WaaeaadaGcaaqaaiaad2eadaqadaqaaiaad2eacqGHRaWkcaWGfbWa aSbaaKqbGeaacaWGUbGaaiilaiaadUgaaKqbagqaaaGaayjkaiaawM caaaqabaGaamOCamaaCaaabeqcfasaaiaaikdaaaaajuaGcaGLOaGa ayzkaaaacaGLBbGaayzxaaWaaWbaaKqbGeqabaGaaGOmaaaajuaGca WGKbGaamOCaaqcfasaaiaaicdaaeaacqGHRaWkcqGHEisPaKqbakab gUIiYdaakeaajuaGcaWGubWaa0baaKqbGeaacaWGHbGaam4Baaqaai aaikdaaaqcfa4aaeWaaeaacaWGfbWaaSbaaKqbGeaacaWGUbGaam4A aaqabaqcfaOaaiilaiaad2eacaGGSaGaeqyYdChacaGLOaGaayzkaa Gaeyypa0JaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaacaWGnbGa eqyYdC3aaWbaaKqbGeqabaGaaGOmaaaajuaGdaWdXbqaaiGacwgaca GG4bGaaiiCamaabmaabaGaeyOeI0YaaOaaaeaacaWGnbWaaeWaaeaa caWGnbGaey4kaSIaamyramaaBaaajuaibaGaamOBaiaacYcacaWGRb aabeaaaKqbakaawIcacaGLPaaaaeqaaiaadkhadaahaaqcfasabeaa caaIYaaaaaqcfaOaayjkaiaawMcaaiaadkhadaahaaqabKqbGeaaca aIYaGaamitaiabgUcaRiaaikdaaaqcfa4aamWaaeaacaWGmbWaa0ba aKqbGeaacaWGUbaabaGaamitaiabgUcaRiaaigdacaGGVaGaaGOmaa aajuaGdaqadaqaamaakaaabaGaamytamaabmaabaGaamytaiabgUca RiaadweadaWgaaqcfasaaiaad6gacaGGSaGaam4Aaaqcfayabaaaca GLOaGaayzkaaaabeaacaWGYbWaaWbaaeqajuaibaGaaGOmaaaaaKqb akaawIcacaGLPaaaaiaawUfacaGLDbaadaahaaqcfasabeaacaaIYa aaaKqbakaadsgacaWGYbaajuaibaGaaGimaaqaaiabgUcaRiabg6Hi LcqcfaOaey4kIipaaaaa@C53C@  ……(57)

The above two equations, after employing an appropriate coordinate transformation r 2 =t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaCa aaleqabaGaaGOmaaaakiabg2da9iaadshaaaa@39DF@ , transforms to the following form:

T ao 1 ( E nk ,M,ω )= α 4M 0 + exp( M( M+ E n,k ) t ) t ( L 1 2 )1 [ L n L+1/2 ( M( M+ E n,k ) t ) ] 2 dt T ao 2 ( E nk ,M,ω )= 1 4 M ω 2 0 + exp( M( M+ E n,k ) t ) t ( L+ 3 2 )1 [ L n L+1/2 ( M( M+ E n,k ) t ) ] 2 dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam ivamaaDaaajuaibaGaamyyaiaad+gaaeaacaaIXaaaaKqbaoaabmaa baGaamyramaaBaaajuaibaGaamOBaiaadUgaaKqbagqaaiaacYcaca WGnbGaaiilaiabeM8a3bGaayjkaiaawMcaaiabg2da9maalaaabaGa eqySdegabaGaaGinaiaad2eaaaWaa8qCaeaaciGGLbGaaiiEaiaacc hadaqadaqaaiabgkHiTmaakaaabaGaamytamaabmaabaGaamytaiab gUcaRiaadweadaWgaaqcfasaaiaad6gacaGGSaGaam4Aaaqcfayaba aacaGLOaGaayzkaaaabeaacaWG0baacaGLOaGaayzkaaGaamiDamaa CaaabeqaamaabmaabaGaamitaiabgkHiTmaalaaabaGaaGymaaqaai aaikdaaaaacaGLOaGaayzkaaGaeyOeI0IaaGymaaaadaWadaqaaiaa dYeadaqhaaqcfasaaiaad6gaaeaacaWGmbGaey4kaSIaaGymaiaac+ cacaaIYaaaaKqbaoaabmaabaWaaOaaaeaacaWGnbWaaeWaaeaacaWG nbGaey4kaSIaamyramaaBaaajuaibaGaamOBaiaacYcacaWGRbaaju aGbeaaaiaawIcacaGLPaaaaeqaaiaadshaaiaawIcacaGLPaaaaiaa wUfacaGLDbaadaahaaqcfasabeaacaaIYaaaaKqbakaadsgacaWG0b aajuaibaGaaGimaaqaaiabgUcaRiabg6HiLcqcfaOaey4kIipaaOqa aKqbakaadsfadaqhaaqcfasaaiaadggacaWGVbaabaGaaGOmaaaaju aGdaqadaqaaiaadweadaWgaaqcfasaaiaad6gacaWGRbaajuaGbeaa caGGSaGaamytaiaacYcacqaHjpWDaiaawIcacaGLPaaacqGH9aqpcq GHsisldaWcaaqaaiaaigdaaeaacaaI0aaaaiaad2eacqaHjpWDdaah aaqcfasabeaacaaIYaaaaKqbaoaapehabaGaciyzaiaacIhacaGGWb WaaeWaaeaacqGHsisldaGcaaqaaiaad2eadaqadaqaaiaad2eacqGH RaWkcaWGfbWaaSbaaKqbGeaacaWGUbGaaiilaiaadUgaaKqbagqaaa GaayjkaiaawMcaaaqabaGaamiDaaGaayjkaiaawMcaaiaadshadaah aaqabeaadaqadaqaaiaadYeacqGHRaWkdaWcaaqaaiaaiodaaeaaca aIYaaaaaGaayjkaiaawMcaaiabgkHiTiaaigdaaaWaamWaaeaacaWG mbWaa0baaKqbGeaacaWGUbaabaGaamitaiabgUcaRiaaigdacaGGVa GaaGOmaaaajuaGdaqadaqaamaakaaabaGaamytamaabmaabaGaamyt aiabgUcaRiaadweadaWgaaqcfasaaiaad6gacaGGSaGaam4Aaaqcfa yabaaacaGLOaGaayzkaaaabeaacaWG0baacaGLOaGaayzkaaaacaGL BbGaayzxaaWaaWbaaKqbGeqabaGaaGOmaaaajuaGcaWGKbGaamiDaa qcfasaaiaaicdaaeaacqGHRaWkcqGHEisPaKqbakabgUIiYdaaaaa@C3F2@  ……(58)

Now, to obtain the modifications to the energy levels for n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaCa aaleqabaGaamiDaiaadIgaaaaaaa@38FC@  excited states we apply the following special integration [69]:

0 + t . α1 exp( δt ) τ m β ( δt ) τ n β ( δt )dt= δ α τ( nα+β+1 )τ( m+λ+1 ) m!n!( 1α+β )τ( 1+λ ) 3 F 2 ( m,α,αβ;n+α,λ+1;1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaapehaba GaamiDamaaDaaajuaibaGaaiOlaaqaaiabeg7aHjabgkHiTiaaigda aaaabaGaaGimaaqaaiabgUcaRiabg6HiLcqcfaOaey4kIipaciGGLb GaaiiEaiaacchadaqadaqaaiabgkHiTiabes7aKjaadshaaiaawIca caGLPaaacqaHepaDdaqhaaqcfasaaiaad2gaaeaacqaHYoGyaaqcfa 4aaeWaaeaacqaH0oazcaWG0baacaGLOaGaayzkaaGaeqiXdq3aa0ba aKqbGeaacaWGUbaabaGaeqOSdigaaKqbaoaabmaabaGaeqiTdqMaam iDaaGaayjkaiaawMcaaiaadsgacaWG0bGaeyypa0ZaaSaaaeaacqaH 0oazdaahaaqabKqbGeaacqGHsislcqaHXoqyaaqcfaOaeqiXdq3aae WaaeaacaWGUbGaeyOeI0IaeqySdeMaey4kaSIaeqOSdiMaey4kaSIa aGymaaGaayjkaiaawMcaaiabes8a0naabmaabaGaamyBaiabgUcaRi abeU7aSjabgUcaRiaaigdaaiaawIcacaGLPaaaaeaacaWGTbGaaiyi aiaad6gacaGGHaWaaeWaaeaacaaIXaGaeyOeI0IaeqySdeMaey4kaS IaeqOSdigacaGLOaGaayzkaaGaeqiXdq3aaeWaaeaacaaIXaGaey4k aSIaeq4UdWgacaGLOaGaayzkaaaaaiaaiodadaahaaqabeaacaWGgb aaamaaBaaajuaibaGaaGOmaaqabaqcfa4aaeWaaeaacqGHsislcaWG TbGaaiilaiabeg7aHjaacYcacqaHXoqycqGHsislcqaHYoGycaGG7a GaeyOeI0IaamOBaiabgUcaRiabeg7aHjaacYcacqaH7oaBcqGHRaWk caaIXaGaai4oaiaaigdaaiaawIcacaGLPaaaaaa@A14B@ …...(59)

where F 3 2 ( m,α,αβ;n+α,λ+1;1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaBeaaba GaaG4maaqabaGaamOramaaBaaajuaibaGaaGOmaaqabaqcfa4aaeWa aeaacqGHsislcaWGTbGaaiilaiabeg7aHjaacYcacqaHXoqycqGHsi slcqaHYoGycaGG7aGaeyOeI0IaamOBaiabgUcaRiabeg7aHjaacYca cqaH7oaBcqGHRaWkcaaIXaGaai4oaiaaigdaaiaawIcacaGLPaaaaa a@4EEB@  obtained from the generalized the hypergeometric function F p q ( α 1 ,..., α p , β 1 ,...., β q ,z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaBeaaju aibaGaamiCaaqcfayabaGaamOramaaBaaajuaibaGaamyCaaqabaqc fa4aaeWaaeaacqaHXoqydaWgaaqcfasaaiaaigdaaeqaaKqbakaacY cacaGGUaGaaiOlaiaac6cacaGGSaGaeqySde2aaSbaaKqbGeaacaWG WbaajuaGbeaacaGGSaGaeqOSdi2aaSbaaKqbGeaacaaIXaaabeaaju aGcaGGSaGaaiOlaiaac6cacaGGUaGaaiOlaiaacYcacqaHYoGydaWg aaqcfasaaiaadghaaeqaaKqbakaacYcacaWG6baacaGLOaGaayzkaa aaaa@53C5@  for p=3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadchacq GH9aqpcaaIZaaaaa@3931@  and q=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadghacq GH9aqpcaaIYaaaaa@3931@  while Γ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfo5ahn aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@3A67@  denote to the usual Gamma function. After straightforward calculations, we can obtain the explicitly results:

T ao 1 ( E nk ,M,ω )= α 4M [ M( M+ E n,k ) ] 2L1 4 Γ( n+2 )Γ( n+L+3/2 ) ( n! ) 2 Γ( 2 )Γ( L+3/2 ) F 3 2 ( n,L1/2,1;Ln1/2,L+3/2;1 ) T ao 2 ( E nk ,M,ω )= 1 4 M ω 2 [ M( M+ E n,k ) ] 2L+3 4 Γ( n )Γ( n+L+3/2 ) ( n! ) 2 Γ( L+3/2 ) F 3 2 ( n,L+ 3 2 ,1;Ln+3/2,L+3/2;1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam ivamaaDaaajuaibaGaamyyaiaad+gaaeaacaaIXaaaaKqbaoaabmaa baGaamyramaaBaaajuaibaGaamOBaiaadUgaaeqaaKqbakaacYcaca WGnbGaaiilaiabeM8a3bGaayjkaiaawMcaaiabg2da9maalaaabaGa eqySdegabaGaaGinaiaad2eaaaWaaSaaaeaadaWadaqaaiaad2eada qadaqaaiaad2eacqGHRaWkcaWGfbWaaSbaaKqbGeaacaWGUbGaaiil aiaadUgaaeqaaaqcfaOaayjkaiaawMcaaaGaay5waiaaw2faamaaCa aabeqaaiabgkHiTmaalaaabaGaaGOmaiaadYeacqGHsislcaaIXaaa baGaaGinaaaaaaGaeu4KdC0aaeWaaeaacaWGUbGaey4kaSIaaGOmaa GaayjkaiaawMcaaiabfo5ahnaabmaabaGaamOBaiabgUcaRiaadYea cqGHRaWkcaaIZaGaai4laiaaikdaaiaawIcacaGLPaaaaeaadaqada qaaiaad6gacaqGHaaacaGLOaGaayzkaaWaaWbaaKqbGeqabaGaaGOm aaaajuaGcqqHtoWrdaqadaqaaiaaikdaaiaawIcacaGLPaaacqqHto WrdaqadaqaaiaadYeacqGHRaWkcaaIZaGaai4laiaaikdaaiaawIca caGLPaaaaaWaaSraaKqbGeaacaaIZaaabeaajuaGcaWGgbWaaSbaaK qbGeaacaaIYaaabeaajuaGdaqadaqaaiabgkHiTiaad6gacaGGSaGa amitaiabgkHiTiaaigdacaGGVaGaaGOmaiaacYcacqGHsislcaaIXa Gaai4oaiaadYeacqGHsislcaWGUbGaeyOeI0IaaGymaiaac+cacaaI YaGaaiilaiaadYeacqGHRaWkcaaIZaGaai4laiaaikdacaGG7aGaaG ymaaGaayjkaiaawMcaaaGcbaqcfaOaamivamaaDaaajuaibaGaamyy aiaad+gaaeaacaaIYaaaaKqbaoaabmaabaGaamyramaaBaaajuaiba GaamOBaiaadUgaaKqbagqaaiaacYcacaWGnbGaaiilaiabeM8a3bGa ayjkaiaawMcaaiabg2da9iabgkHiTmaalaaabaGaaGymaaqaaiaais daaaGaamytaiabeM8a3naaCaaajuaibeqaaiaaikdaaaqcfa4aaSaa aeaadaWadaqaaiaad2eadaqadaqaaiaad2eacqGHRaWkcaWGfbWaaS baaKqbGeaacaWGUbGaaiilaiaadUgaaKqbagqaaaGaayjkaiaawMca aaGaay5waiaaw2faamaaCaaabeqaaiabgkHiTmaalaaabaGaaGOmai aadYeacqGHRaWkcaaIZaaabaGaaGinaaaaaaGaeu4KdC0aaeWaaeaa caWGUbaacaGLOaGaayzkaaGaeu4KdC0aaeWaaeaacaWGUbGaey4kaS IaamitaiabgUcaRiaaiodacaGGVaGaaGOmaaGaayjkaiaawMcaaaqa amaabmaabaGaamOBaiaabgcaaiaawIcacaGLPaaadaahaaqabKqbGe aacaaIYaaaaKqbakabfo5ahnaabmaabaGaamitaiabgUcaRiaaioda caGGVaGaaGOmaaGaayjkaiaawMcaaaaadaWgbaqcfasaaiaaiodaae qaaKqbakaadAeadaWgaaqcfasaaiaaikdaaeqaaKqbaoaabmaabaGa eyOeI0IaamOBaiaacYcacaWGmbGaey4kaSYaaSaaaeaacaaIZaaaba GaaGOmaaaacaGGSaGaaGymaiaacUdacaWGmbGaeyOeI0IaamOBaiab gUcaRiaaiodacaGGVaGaaGOmaiaacYcacaWGmbGaey4kaSIaaG4mai aac+cacaaIYaGaai4oaiaaigdaaiaawIcacaGLPaaaaaaa@E61F@  ……(60)

Hence the exact modifications E ncper:u ( Θ, k 1 , E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaKqbagqaamaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaigdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqcfayabaGaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaaaaa@4DE7@  and E ncper:d ( Θ, k 2 , E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadsgaaKqbagqaamaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaikdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaaaaa@4DD7@  of E ncper:d ( Θ, k 2 , E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadsgaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaikdaaKqbagqaaiaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaaaaa@4DD7@  excited states which produced by spin-orbital effect:

E ncper:u ( Θ, k 2 , E nk ,M,ω )θ( E ncao )Θ k 1 | C n | 2 T ao ( E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaikdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaGaeyyyIORaeqiUde3aaeWaaeaacaWGfbWaaSbaaKqb GeaacaWGUbGaam4yaiabgkHiTiaadggacaWGVbaajuaGbeaaaiaawI cacaGLPaaacqqHyoqucaWGRbWaaSbaaKqbGeaacaaIXaaabeaajuaG daabdaqaaiaadoeadaWgaaqcfasaaiaad6gaaKqbagqaaaGaay5bSl aawIa7amaaCaaajuaibeqaaiaaikdaaaqcfaOaamivamaaBaaajuai baGaamyyaiaad+gaaeqaaKqbaoaabmaabaGaamyramaaBaaajuaiba GaamOBaiaadUgaaeqaaKqbakaacYcacaWGnbGaaiilaiabeM8a3bGa ayjkaiaawMcaaaaa@712E@ ………… (61)

E ncper:u ( Θ, k 2 , E nk ,M,ω )θ( E ncao )Θ k 2 | C n | 2 T ao ( E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaKqbagqaamaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaikdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqcfayabaGaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaGaeyyyIORaeqiUde3aaeWaaeaacaWGfbWaaSbaaKqb GeaacaWGUbGaam4yaiabgkHiTiaadggacaWGVbaabeaaaKqbakaawI cacaGLPaaacqqHyoqucaWGRbWaaSbaaKqbGeaacaaIYaaajuaGbeaa daabdaqaaiaadoeadaWgaaqcfasaaiaad6gaaKqbagqaaaGaay5bSl aawIa7amaaCaaajuaibeqaaiaaikdaaaqcfaOaamivamaaBaaajuai baGaamyyaiaad+gaaKqbagqaamaabmaabaGaamyramaaBaaajuaiba GaamOBaiaadUgaaKqbagqaaiaacYcacaWGnbGaaiilaiabeM8a3bGa ayjkaiaawMcaaaaa@712F@  ……………(62)

Where T ao ( E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaadggacaWGVbaabeaajuaGdaqadaqaaiaadweadaWg aaqcfasaaiaad6gacaWGRbaajuaGbeaacaGGSaGaamytaiaacYcacq aHjpWDaiaawIcacaGLPaaaaaa@431B@  is the sum of two factors T ao 1 ( E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada qhaaqcfasaaiaadggacaWGVbaabaGaaGymaaaajuaGdaqadaqaaiaa dweadaWgaaqcfasaaiaad6gacaWGRbaabeaajuaGcaGGSaGaamytai aacYcacqaHjpWDaiaawIcacaGLPaaaaaa@43D7@  and T ao 2 ( E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada qhaaqcfasaaiaadggacaWGVbaabaGaaGOmaaaajuaGdaqadaqaaiaa dweadaWgaaqcfasaaiaad6gacaWGRbaajuaObeaajuaGcaGGSaGaam ytaiaacYcacqaHjpWDaiaawIcacaGLPaaaaaa@4486@ .

The exact relativistic magnetic spectrum for (m.a.o.) for n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaCa aaleqabaGaamiDaiaadIgaaaaaaa@38FC@  excited states for one-electron atoms in (NC: 3D- RS) symmetries:

Having obtained the exact modifications to the relativistic energy levels E ncper:u ( Θ, k 1 , E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadwhaaKqbagqaamaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaigdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaaaaa@4DE7@  and E ncper:d ( Θ, k 2 , E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamiCaiaadwgacaWGYbGa aiOoaiaadsgaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiaadUgada WgaaqcfasaaiaaikdaaeqaaKqbakaacYcacaWGfbWaaSbaaKqbGeaa caWGUbGaam4AaaqcfayabaGaaiilaiaad2eacaGGSaGaeqyYdChaca GLOaGaayzkaaaaaa@4DD7@  for n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gada ahaaqcfasabeaacaWG0bGaamiAaaaaaaa@39A2@  excited states which produced with relativistic NC spin-orbital Hamiltonian operator, our objective now, we consider another interested physically meaningful phenomena, which also can be produce from the perturbative terms of anharmonic oscillator related to the influence of an external uniform magnetic field, it’s sufficient to apply the following two replacements to describing these phenomena:

( α 2M r 4 1 2 M ω 2 ){ L Θ χ( α 2M r 4 1 2 M ω 2 ) B L    for the    spin symmetric    case  L ˜ Θ χ( α 2M r 4 1 2 M ω 2 ) B L ˜ for the p-spin symmetric    case  ΘχB   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfa4aae WaaeaadaWcaaqaaiabeg7aHbqaaiaaikdacaWGnbGaamOCamaaCaaa juaibeqaaiaaisdaaaaaaKqbakabgkHiTmaalaaabaGaaGymaaqaai aaikdaaaGaamytaiabeM8a3naaCaaajuaibeqaaiaaikdaaaaajuaG caGLOaGaayzkaaWaaiqaaqaabeqaaGqabiqa=XeagaWcaiqbfI5arz aalaGaeyOKH4QaeyOKH4Qaeq4Xdm2aaeWaaeaadaWcaaqaaiabeg7a HbqaaiaaikdacaWGnbGaamOCamaaCaaajuaibeqaaiaaisdaaaaaaK qbakabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaGaamytaiabeM8a 3naaCaaajuaibeqaaiaaikdaaaaajuaGcaGLOaGaayzkaaWaa8Haae aacaWGcbaacaGLxdcadaWhcaqaaiaadYeaaiaawEniauaabeqabiaa aeaacaqGGaGaaeiiaiaabccacaqGMbGaae4BaiaabkhaaeaacaqG0b GaaeiAaiaabwgacaqGGaGaaeiiaiaabccacaqGGaGaae4Caiaabcha caqGPbGaaeOBaiaabccacaqGZbGaaeyEaiaab2gacaqGTbGaaeyzai aabshacaqGYbGaaeyAaiaabogacaqGGaGaaeiiaiaabccacaqGGaGa ae4yaiaabggacaqGZbGaaeyzaiaabccaaaaabaWaa8HaaeaaceWFmb GbaGaaaiaawEniaiqbfI5arzaalaGaeyOKH4QaeyOKH4Qaeq4Xdm2a aeWaaeaadaWcaaqaaiabeg7aHbqaaiaaikdacaWGnbGaamOCamaaCa aabeqcfasaaiaaisdaaaaaaKqbakabgkHiTmaalaaabaGaaGymaaqa aiaaikdaaaGaamytaiabeM8a3naaCaaajuaibeqaaiaaikdaaaaaju aGcaGLOaGaayzkaaWaa8HaaeaacaWGcbaacaGLxdcaceWFmbGbaGaa caWFGaGaa8hiaiaa=bcacaWFGaGaa8hiauaabeqabiaaaeaacaqGMb Gaae4BaiaabkhaaeaacaqG0bGaaeiAaiaabwgacaqGGaGaaeiCaiaa b2cacaqGZbGaaeiCaiaabMgacaqGUbGaaeiiaiaabohacaqG5bGaae yBaiaab2gacaqGLbGaaeiDaiaabkhacaqGPbGaae4yaiaabccacaqG GaGaaeiiaiaabccacaqGJbGaaeyyaiaabohacaqGLbGaaeiiaaaaaa Gaay5EaaaakeaajuaGcqqHyoqucqGHsgIRcqaHhpWyjuaicaWGcbqc faOaaeiiaiaabccaaaaa@C0D1@  ………(63)

here χ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeE8aJb aa@3830@  is infinitesimal real proportional’s constants, and we choose the magnetic field  B =B k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFmaaba Gaaeiiaiaadkeaaiaawgoiaiabg2da9iaadkeadaWhdaqaaiaadUga aiaawgoiaaaa@3E1E@  for simplify the calculations, which allow us to introduce the modified new magnetic Hamiltonian H ^ magao ( r, E nk ,M,ω,χ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamyBaiaadggacaWGNbGaeyOeI0Iaamyyaiaa d+gaaeqaaKqbaoaabmaabaGaamOCaiaacYcacaWGfbWaaSbaaKqbGe aacaWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGaeqyYdCNa aiilaiabeE8aJbGaayjkaiaawMcaaaaa@4ADE@  on the (NC: 3D-RS), as:

H ^ magao ( r, E nk ,M,ω,χ )=χ( α 2M r 4 1 2 M ω 2 ){ ( B J S ˜ B )  for the    spin symmetric    case  ( B J S ˜ B ) for the p-spin symmetric    case  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamyBaiaadggacaWGNbGaeyOeI0Iaamyyaiaa d+gaaeqaaKqbaoaabmaabaGaamOCaiaacYcacaWGfbWaaSbaaKqbGe aacaWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGaeqyYdCNa aiilaiabeE8aJbGaayjkaiaawMcaaiabg2da9iabeE8aJnaabmaaba WaaSaaaeaacqaHXoqyaeaacaaIYaGaamytaiaadkhadaahaaqcfasa beaacaaI0aaaaaaajuaGcqGHsisldaWcaaqaaiaaigdaaeaacaaIYa aaaiaad2eacqaHjpWDdaahaaqabKqbGeaacaaIYaaaaaqcfaOaayjk aiaawMcaamaaceaaeaqabeaadaqadaqaamaaFiaabaGaamOqaaGaay 51GaWaa8HaaeaacaWGkbaacaGLxdcacqGHsisldaWhdaqaaiqadofa gaacaaGaayz4GaWaa8XaaeaacaWGcbaacaGLHdcaaiaawIcacaGLPa aafaqabeqacaaabaGaaeiiaiaabAgacaqGVbGaaeOCaaqaaiaabsha caqGObGaaeyzaiaabccacaqGGaGaaeiiaiaabccacaqGZbGaaeiCai aabMgacaqGUbGaaeiiaiaabohacaqG5bGaaeyBaiaab2gacaqGLbGa aeiDaiaabkhacaqGPbGaae4yaiaabccacaqGGaGaaeiiaiaabccaca qGJbGaaeyyaiaabohacaqGLbGaaeiiaaaaaeaadaqadaqaamaaFiaa baGaamOqaaGaay51GaWaa8HaaeaacaWGkbaacaGLxdcacqGHsislda WhcaqaaiqadofagaacaaGaay51GaWaa8XaaeaacaWGcbaacaGLHdca aiaawIcacaGLPaaafaqabeqacaaabaGaaeOzaiaab+gacaqGYbaaba GaaeiDaiaabIgacaqGLbGaaeiiaiaabchacaqGTaGaae4Caiaabcha caqGPbGaaeOBaiaabccacaqGZbGaaeyEaiaab2gacaqGTbGaaeyzai aabshacaqGYbGaaeyAaiaabogacaqGGaGaaeiiaiaabccacaqGGaGa ae4yaiaabggacaqGZbGaaeyzaiaabccaaaaaaiaawUhaaaaa@AE2A@ … (64)

where ( S B , S ˜ B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaeyOeI0Yaa8XaaeaacaWGtbaacaGLHdcadaWhdaqaaiaadkeaaiaa wgoiaiaacYcadaWhcaqaaiqadofagaacaaGaay51GaWaa8Xaaeaaca WGcbaacaGLHdcaaiaawIcacaGLPaaaaaa@43DD@  denotes to the two ordinary and pseudo Hamiltonians of Zeeman effect. To obtain the exact NC magnetic modifications of energy E mag-ao ( χ,m, E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaab2gacaqGHbGaae4zaiaab2cacaqGHbGaae4Baaqa baqcfa4aaeWaaeaacqaHhpWycaGGSaGaamyBaiaacYcacaWGfbWaaS baaKqbGeaacaWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGa eqyYdChacaGLOaGaayzkaaaaaa@4A7F@  for (m.a.o.) under spin-symmetry case which produced automatically from the effect of operator H ^ magao ( r, E nk ,M,ω,χ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamyBaiaadggacaWGNbGaeyOeI0Iaamyyaiaa d+gaaeqaaKqbaoaabmaabaGaamOCaiaacYcacaWGfbWaaSbaaKqbGe aacaWGUbGaam4AaaqcfayabaGaaiilaiaad2eacaGGSaGaeqyYdCNa aiilaiabeE8aJbGaayjkaiaawMcaaaaa@4ADE@ , we make the following two simultaneously replacements:

k 1 m    and       Θχ  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgada WgaaqcfasaaiaaigdaaeqaaKqbakabgkziUkaab2gacaqGGaGaaeii aiaabccacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiabfI5arjabgkziUkabeE8aJjaa bccaaaa@4B59@ ....(65)

Then, the relativistic magnetic modification of energy E mag-ao ( χ,m, E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaab2gacaqGHbGaae4zaiaab2cacaqGHbGaae4Baaqa baqcfa4aaeWaaeaacqaHhpWycaGGSaGaamyBaiaacYcacaWGfbWaaS baaKqbGeaacaWGUbGaam4AaaqabaqcfaOaaiilaiaad2eacaGGSaGa eqyYdChacaGLOaGaayzkaaaaaa@4A7F@  corresponding ground state on the (NC-3D: RS) symmetries, can be determined from the following relation:

E mag-ao ( χ,m, E nk ,M,ω )=θ( E ncao )χmBΘ | C n | 2 T ao ( E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaab2gacaqGHbGaae4zaiaab2cacaqGHbGaae4Baaqc fayabaWaaeWaaeaacqaHhpWycaGGSaGaamyBaiaacYcacaWGfbWaaS baaKqbGeaacaWGUbGaam4AaaqcfayabaGaaiilaiaad2eacaGGSaGa eqyYdChacaGLOaGaayzkaaGaeyypa0JaeqiUde3aaeWaaeaacaWGfb WaaSbaaKqbGeaacaWGUbGaam4yaiabgkHiTiaadggacaWGVbaajuaG beaaaiaawIcacaGLPaaacqaHhpWycaWGTbGaamOqaiabfI5arnaaem aabaGaam4qamaaBaaajuaibaGaamOBaaqcfayabaaacaGLhWUaayjc SdWaaWbaaKqbGeqabaGaaGOmaaaajuaGcaWGubWaaSbaaKqbGeaaca WGHbGaam4BaaqcfayabaWaaeWaaeaacaWGfbWaaSbaaKqbGeaacaWG UbGaam4AaaqcfayabaGaaiilaiaad2eacaGGSaGaeqyYdChacaGLOa Gaayzkaaaaaa@6DEA@ ………(66)

Where m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2gaaa a@376B@  denote to the angular momentum quantum number satisfying the interval, lm+l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTi aadYgacqGHKjYOcaWGTbGaeyizImQaey4kaSIaamiBaaaa@3E86@ , which allow us to fixing ( 2l+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaikdaca WGSbGaey4kaSIaaGymaaaa@39C3@ ) values for this quantum number.

Themain results of exact modified global spectrum for (m.a.o.) for one-electron atoms under spin-symmetry and p-spin symmetry in (NC: 3D-RS):

This principal part of the paper is devoted to the presentation of the several results obtained in the previous sections, we resume the n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gada ahaaqabKqbGeaacaWG0bGaamiAaaaaaaa@39A2@  excited states eigenenergies ( E ncu ( Θ, k 1 ,χ,n,m, E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamyDaaqabaqcfa4aaeWa aeaacqqHyoqucaGGSaGaam4AamaaBaaajuaibaGaaGymaaqabaqcfa OaaiilaiabeE8aJjaacYcacaWGUbGaaiilaiaad2gacaGGSaGaamyr amaaBaaajuaibaGaamOBaiaadUgaaeqaaKqbakaacYcacaWGnbGaai ilaiabeM8a3bGaayjkaiaawMcaaaaa@4FFF@ , E ncd ( Θ, k 2 ,χ,n,m, E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaamizaaqabaqcfa4aaeWa aeaacqqHyoqucaGGSaGaam4AamaaBaaajuaibaGaaGOmaaqabaqcfa OaaiilaiabeE8aJjaacYcacaWGUbGaaiilaiaad2gacaGGSaGaamyr amaaBaaajuaibaGaamOBaiaadUgaaKqbagqaaiaacYcacaWGnbGaai ilaiabeM8a3bGaayjkaiaawMcaaaaa@4FEF@ )of modified Dirac equation corresponding for ( -( j+1/2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaab2cada qadaqaaiaabQgacqGHRaWkcaqGXaGaae4laiaabkdaaiaawIcacaGL Paaaaaa@3C9C@ , ( s 1/2 , p 3/2 ,etc ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba Gaae4CamaaBaaajuaibaGaaeymaiaab+cacaqGYaaabeaajuaGcaGG SaGaamiCamaaBaaajuaibaGaaG4maiaac+cacaaIYaaabeaajuaGca GGSaGaamyzaiaadshacaWGJbaacaGLOaGaayzkaaaaaa@4419@ , j=l+ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabg2 da9iaadYgacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaaaaa@3B45@ , aligned spin k0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaabUgacq GHPms4caqGWaaaaa@39D3@  and spin-down) and ( j=l+ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQgacq GH9aqpcaWGSbGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaaaaa@3BC8@ , ( p 1/2 , d 3/2 ,etc ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaaeiCamaaBaaajuaibaGaaeymaiaab+cacaqGYaaabeaajuaGcaGG SaGaamizamaaBaaajuaibaGaaG4maiaac+cacaaIYaaajuaGbeaaca GGSaGaamyzaiaadshacaWGJbaacaGLOaGaayzkaaaaaa@440A@ , j=l 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQgacq GH9aqpcaWGSbGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaaaaa@3BD3@ , un aligned spin k0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaabUgacq GHQms8caqGWaaaaa@39E4@  and spin up), respectively, at first order of parameter Θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfI5arb aa@37F0@ , for (m.a.o.) potential in (NC: 3D-RS), respectively, on based to the obtained new results(61), (62) and (66), in addition to the original results (22) of energies in commutative space, we obtain the following original results:

E ncu ( Θ, k 1 ,χ,n,m, E nk ,M,ω )= E n k 1 +θ( E ncao )Θ k 1 | C n | 2 T ao ( E nk ,M,ω )+θ( E ncao )χmBΘ | C n | 2 T ao ( E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamyDaaqabaqcfa4aaeWa aeaacqqHyoqucaGGSaGaam4AamaaBaaajuaibaGaaGymaaqcfayaba GaaiilaiabeE8aJjaacYcacaWGUbGaaiilaiaad2gacaGGSaGaamyr amaaBaaajuaibaGaamOBaiaadUgaaKqbagqaaiaacYcacaWGnbGaai ilaiabeM8a3bGaayjkaiaawMcaaiabg2da9iaadweadaWgaaqaaiaa d6gacaWGRbWaaSbaaKqbGeaacaaIXaaajuaGbeaaaeqaaiabgUcaRi abeI7aXnaabmaabaGaamyramaaBaaajuaibaGaamOBaiaadogacqGH sislcaWGHbGaam4BaaqabaaajuaGcaGLOaGaayzkaaGaeuiMdeLaam 4AamaaBaaajuaibaGaaGymaaqabaqcfa4aaqWaaeaacaWGdbWaaSba aKqbGeaacaWGUbaajuaGbeaaaiaawEa7caGLiWoadaahaaqcfasabe aacaaIYaaaaKqbakaadsfadaWgaaqcfasaaiaadggacaWGVbaabeaa juaGdaqadaqaaiaadweadaWgaaqcfasaaiaad6gacaWGRbaabeaaju aGcaGGSaGaamytaiaacYcacqaHjpWDaiaawIcacaGLPaaacqGHRaWk cqaH4oqCdaqadaqaaiaadweadaWgaaqcfasaaiaad6gacaWGJbGaey OeI0Iaamyyaiaad+gaaeqaaaqcfaOaayjkaiaawMcaaiabeE8aJjaa d2gacaWGcbGaeuiMde1aaqWaaeaacaWGdbWaaSbaaKqbGeaacaWGUb aajuaGbeaaaiaawEa7caGLiWoadaahaaqcfasabeaacaaIYaaaaKqb akaadsfadaWgaaqcfasaaiaadggacaWGVbaabeaajuaGdaqadaqaai aadweadaWgaaqcfasaaiaad6gacaWGRbaabeaajuaGcaGGSaGaamyt aiaacYcacqaHjpWDaiaawIcacaGLPaaaaaa@9B11@  (67)

and

E ncd ( Θ, k 2 ,χ,n,m, E nk ,M,ω )= E n k 2 +θ( E ncao )Θ k 2 | C n | 2 T ao ( E nk ,M,ω )+θ( E ncao )χmBΘ | C n | 2 T ao ( E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0Iaamizaaqabaqcfa4aaeWa aeaacqqHyoqucaGGSaGaam4AamaaBaaajuaibaGaaGOmaaqcfayaba GaaiilaiabeE8aJjaacYcacaWGUbGaaiilaiaad2gacaGGSaGaamyr amaaBaaajuaibaGaamOBaiaadUgaaeqaaKqbakaacYcacaWGnbGaai ilaiabeM8a3bGaayjkaiaawMcaaiabg2da9iaadweadaWgaaqaaiaa d6gacaWGRbWaaSbaaKqbGeaacaaIYaaabeaaaKqbagqaaiabgUcaRi abeI7aXnaabmaabaGaamyramaaBaaajuaibaGaamOBaiaadogacqGH sislcaWGHbGaam4BaaqabaaajuaGcaGLOaGaayzkaaGaeuiMdeLaam 4AamaaBaaajuaibaGaaGOmaaqabaqcfa4aaqWaaeaacaWGdbWaaSba aKqbGeaacaWGUbaajuaGbeaaaiaawEa7caGLiWoadaahaaqcfasabe aacaaIYaaaaKqbakaadsfadaWgaaqcfasaaiaadggacaWGVbaajuaG beaadaqadaqaaiaadweadaWgaaqcfasaaiaad6gacaWGRbaajuaGbe aacaGGSaGaamytaiaacYcacqaHjpWDaiaawIcacaGLPaaacqGHRaWk cqaH4oqCdaqadaqaaiaadweadaWgaaqcfasaaiaad6gacaWGJbGaey OeI0Iaamyyaiaad+gaaKqbagqaaaGaayjkaiaawMcaaiabeE8aJjaa d2gacaWGcbGaeuiMde1aaqWaaeaacaWGdbWaaSbaaKqbGeaacaWGUb aajuaGbeaaaiaawEa7caGLiWoadaahaaqabKqbGeaacaaIYaaaaKqb akaadsfadaWgaaqcfasaaiaadggacaWGVbaabeaajuaGdaqadaqaai aadweadaWgaaqcfasaaiaad6gacaWGRbaajuaGbeaacaGGSaGaamyt aiaacYcacqaHjpWDaiaawIcacaGLPaaaaaa@9B03@  (68)

As it is montionated in [18], in view of exact spin symmetry in commutative space ( E nk E nk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGUbGaam4AaaqabaGccqGHsgIRcqGHsislcaWGfbWaaSba aSqaaiaad6gacaWGRbaabeaaaaa@3E8C@ , V( r )V( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfada qadaqaaiaadkhaaiaawIcacaGLPaaacqGHsgIRcqGHsislcaWGwbWa aeWaaeaacaWGYbaacaGLOaGaayzkaaaaaa@4009@ , kk+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadUgacq GHsgIRcaWGRbGaey4kaSIaaGymaaaa@3BE3@  and F nk ( r ) G n k ˜ ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAeada Wgaaqcfasaaiaad6gacaWGRbaabeaajuaGdaqadaqaaiaadkhaaiaa wIcacaGLPaaacqGHsgIRcaWGhbWaaSbaaKqbGeaacaWGUbGabm4Aay aaiaaajuaGbeaadaqadaqaaiaadkhaaiaawIcacaGLPaaaaaa@448C@ ), we need to generalize the above translations to the case of NC three dimensional spaces, then the negative values E ncu ( Θ, k ˜ 1 ,χ,m,n, E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamyDaaqabaqcfa4aaeWa aeaacqqHyoqucaGGSaGabm4AayaaiaWaaSbaaKqbGeaacaaIXaaabe aajuaGcaGGSaGaeq4XdmMaaiilaiaad2gacaGGSaGaamOBaiaacYca caWGfbWaaSbaaKqbGeaacaWGUbGaam4AaaqabaqcfaOaaiilaiaad2 eacaGGSaGaeqyYdChacaGLOaGaayzkaaaaaa@500E@  and E ncd ( Θ, k ˜ 2 ,χ,m,n, E nk ,M,ω ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada Wgaaqcfasaaiaad6gacaWGJbGaeyOeI0IaamizaaqcfayabaWaaeWa aeaacqqHyoqucaGGSaGabm4AayaaiaWaaSbaaKqbGeaacaaIYaaabe aajuaGcaGGSaGaeq4XdmMaaiilaiaad2gacaGGSaGaamOBaiaacYca caWGfbWaaSbaaKqbGeaacaWGUbGaam4AaaqabaqcfaOaaiilaiaad2 eacaGGSaGaeqyYdChacaGLOaGaayzkaaaaaa@4FFE@  are obtained as:

E ncu ( Θ, k 1 ,χ,m,n, E nk ,M,ω ) E ncu ( Θ, k ˜ 1 ,χ,m,n, E nk ,M,ω ) E ncu ( Θ, k 1 ,χ,m,n, E nk ,M,ω ) E ncd ( Θ, k 2 ,χ,n,m, E nk ,M,ω ) E ncd ( Θ, k ˜ 2 ,χ,m,n, E nk ,M,ω ) E ncd ( Θ, k 2 ,χ,m,n, E nk ,M,ω ) V( r ^ )V( r ^ ) k ˜ 1 k 1 +1      and      k ˜ 2 k 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam yramaaBaaajuaibaGaamOBaiaadogacqGHsislcaWG1baajuaGbeaa daqadaqaaiabfI5arjaacYcacaWGRbWaaSbaaKqbGeaacaaIXaaabe aajuaGcaGGSaGaeq4XdmMaaiilaiaad2gacaGGSaGaamOBaiaacYca caWGfbWaaSbaaKqbGeaacaWGUbGaam4AaaqabaqcfaOaaiilaiaad2 eacaGGSaGaeqyYdChacaGLOaGaayzkaaGaeyOKH4QaamyramaaBaaa juaibaGaamOBaiaadogacqGHsislcaWG1baabeaajuaGdaqadaqaai abfI5arjaacYcaceWGRbGbaGaadaWgaaqcfasaaiaaigdaaeqaaKqb akaacYcacqaHhpWycaGGSaGaamyBaiaacYcacaWGUbGaaiilaiaadw eadaWgaaqcfasaaiaad6gacaWGRbaajuaGbeaacaGGSaGaamytaiaa cYcacqaHjpWDaiaawIcacaGLPaaacqGHHjIUcqGHsislcaWGfbWaaS baaKqbGeaacaWGUbGaam4yaiabgkHiTiaadwhaaeqaaKqbaoaabmaa baGaeuiMdeLaaiilaiaadUgadaWgaaqcfasaaiaaigdaaKqbagqaai aacYcacqaHhpWycaGGSaGaamyBaiaacYcacaWGUbGaaiilaiaadwea daWgaaqcfasaaiaad6gacaWGRbaajuaGbeaacaGGSaGaamytaiaacY cacqaHjpWDaiaawIcacaGLPaaaaeaacaWGfbWaaSbaaKqbGeaacaWG UbGaam4yaiabgkHiTiaadsgaaeqaaKqbaoaabmaabaGaeuiMdeLaai ilaiaadUgadaWgaaqcfasaaiaaikdaaeqaaKqbakaacYcacqaHhpWy caGGSaGaamOBaiaacYcacaWGTbGaaiilaiaadweadaWgaaqcfasaai aad6gacaWGRbaabeaajuaGcaGGSaGaamytaiaacYcacqaHjpWDaiaa wIcacaGLPaaacqGHsgIRcaWGfbWaaSbaaKqbGeaacaWGUbGaam4yai abgkHiTiaadsgaaeqaaKqbaoaabmaabaGaeuiMdeLaaiilaiqadUga gaacamaaBaaajuaibaGaaGOmaaqcfayabaGaaiilaiabeE8aJjaacY cacaWGTbGaaiilaiaad6gacaGGSaGaamyramaaBaaajuaibaGaamOB aiaadUgaaeqaaKqbakaacYcacaWGnbGaaiilaiabeM8a3bGaayjkai aawMcaaiabggMi6kabgkHiTiaadweadaWgaaqcfasaaiaad6gacaWG JbGaeyOeI0Iaamizaaqabaqcfa4aaeWaaeaacqqHyoqucaGGSaGaam 4AamaaBaaajuaibaGaaGOmaaqabaqcfaOaaiilaiabeE8aJjaacYca caWGTbGaaiilaiaad6gacaGGSaGaamyramaaBaaajuaibaGaamOBai aadUgaaKqbagqaaiaacYcacaWGnbGaaiilaiabeM8a3bGaayjkaiaa wMcaaaqaaiaadAfadaqadaqaaiqadkhagaqcaaGaayjkaiaawMcaai abgkziUkabgkHiTiaadAfadaqadaqaaiqadkhagaqcaaGaayjkaiaa wMcaaaGcbaqcfaOabm4AayaaiaWaaSbaaKqbGeaacaaIXaaabeaaju aGcqGHsgIRcaWGRbWaaSbaaKqbGeaacaaIXaaabeaajuaGcqGHRaWk caaIXaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeyyai aab6gacaqGKbGaaeiiaiaabccacaqGGaGaaeiiaiaabccaceWGRbGb aGaadaWgaaqcfasaaiaaikdaaKqbagqaaiabgkziUkaadUgadaWgaa qcfasaaiaaikdaaKqbagqaaiabgUcaRiaaigdaaaaa@FE33@ …. (69)

It’s clearly, that the obtained eigenvalues of energies are real is Hermitian; consequently, the modified quantum Hamiltonian operator H ^ ncao ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamOBaiaadogacqGHsislcaWGHbGaam4Baaqc fayabaWaaeWaaeaaceWGWbGbaKaadaWgaaqcfasaaiaadMgaaKqbag qaaiaacYcaceWG4bGbaKaadaWgaaqcfasaaiaadMgaaeqaaaqcfaOa ayjkaiaawMcaaaaa@44B6@  is Hermitian and may be expressed as follows:

H ^ ncao ( p ^ i , x ^ i )= H ^ comao ( p i , x i )+{ Θ( α 2M r 4 1 2 M ω 2 ) S L +χ( α 2M r 4 1 2 M ω 2 )( B J S ˜ B ) for the    spin symmetric    case  Θ( α 2M r 4 1 2 M ω 2 ) S ˜ L +χ( α 2M r 4 1 2 M ω 2 )( B J S ˜ B ) for the  p-spin symmetric  case  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamOBaiaadogacqGHsislcaWGHbGaam4Baaqa baqcfa4aaeWaaeaaceWGWbGbaKaadaWgaaqcfasaaiaadMgaaeqaaK qbakaacYcaceWG4bGbaKaadaWgaaqcfasaaiaadMgaaeqaaaqcfaOa ayjkaiaawMcaaiabg2da9iqadIeagaqcamaaBaaajuaibaGaam4yai aad+gacaWGTbGaeyOeI0Iaamyyaiaad+gaaeqaaKqbaoaabmaabaGa amiCamaaBaaajuaibaGaamyAaaqabaqcfaOaaiilaiaadIhadaWgaa qcfasaaiaadMgaaeqaaaqcfaOaayjkaiaawMcaaiabgUcaRmaaceaa eaqabeaacqqHyoqudaqadaqaamaalaaabaGaeqySdegabaGaaGOmai aad2eacaWGYbWaaWbaaKqbGeqabaGaaGinaaaaaaqcfaOaeyOeI0Ya aSaaaeaacaaIXaaabaGaaGOmaaaacaWGnbGaeqyYdC3aaWbaaKqbGe qabaGaaGOmaaaaaKqbakaawIcacaGLPaaadaWhdaqaaiaadofaaiaa wgoiamaaFmaabaGaamitaaGaayz4GaGaey4kaSIaeq4Xdm2aaeWaae aadaWcaaqaaiabeg7aHbqaaiaaikdacaWGnbGaamOCamaaCaaajuai beqaaiaaisdaaaaaaKqbakabgkHiTmaalaaabaGaaGymaaqaaiaaik daaaGaamytaiabeM8a3naaCaaabeqcfasaaiaaikdaaaaajuaGcaGL OaGaayzkaaWaaeWaaeaadaWhcaqaaiaadkeaaiaawEniamaaFiaaba GaamOsaaGaay51GaGaeyOeI0Yaa8XaaeaaceWGtbGbaGaaaiaawgoi amaaFmaabaGaamOqaaGaayz4GaaacaGLOaGaayzkaaqbaeqabeGaaa qaaiaabAgacaqGVbGaaeOCaaqaaiaabshacaqGObGaaeyzaiaabcca caqGGaGaaeiiaiaabccacaqGZbGaaeiCaiaabMgacaqGUbGaaeiiai aabohacaqG5bGaaeyBaiaab2gacaqGLbGaaeiDaiaabkhacaqGPbGa ae4yaiaabccacaqGGaGaaeiiaiaabccacaqGJbGaaeyyaiaabohaca qGLbGaaeiiaaaaaeaacqqHyoqudaqadaqaamaalaaabaGaeqySdega baGaaGOmaiaad2eacaWGYbWaaWbaaeqajuaibaGaaGinaaaaaaqcfa OaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaacaWGnbGaeqyYdC3a aWbaaKqbGeqabaGaaGOmaaaaaKqbakaawIcacaGLPaaadaWhcaqaai qadofagaacaaGaay51GaWaa8XaaeaacaWGmbaacaGLHdcacqGHRaWk cqaHhpWydaqadaqaamaalaaabaGaeqySdegabaGaaGOmaiaad2eaca WGYbWaaWbaaeqajuaibaGaaGinaaaaaaqcfaOaeyOeI0YaaSaaaeaa caaIXaaabaGaaGOmaaaacaWGnbGaeqyYdC3aaWbaaKqbGeqabaGaaG OmaaaaaKqbakaawIcacaGLPaaadaqadaqaamaaFiaabaGaamOqaaGa ay51GaWaa8HaaeaacaWGkbaacaGLxdcacqGHsisldaWhcaqaaiqado fagaacaaGaay51GaWaa8XaaeaacaWGcbaacaGLHdcaaiaawIcacaGL PaaafaqabeqacaaabaGaaeOzaiaab+gacaqGYbaabaGaaeiDaiaabI gacaqGLbGaaeiiaiaabccacaqGWbGaaeylaiaabohacaqGWbGaaeyA aiaab6gacaqGGaGaae4CaiaabMhacaqGTbGaaeyBaiaabwgacaqG0b GaaeOCaiaabMgacaqGJbGaaeiiaiaabccacaqGJbGaaeyyaiaaboha caqGLbGaaeiiaaaaaaGaay5Eaaaaaa@F17B@  (70)

Where H ^ comao ( p i , x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaam4yaiaad+gacaWGTbGaeyOeI0Iaamyyaiaa d+gaaeqaaKqbaoaabmaabaGaamiCamaaBaaajuaibaGaamyAaaqcfa yabaGaaiilaiaadIhadaWgaaqcfasaaiaadMgaaeqaaaqcfaOaayjk aiaawMcaaaaa@4589@  is given by:

H ^ comao ( p i , x i )=αP+β(M+S(r))+ 1 2 M ω 2 r 2 + α 2M r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaam4yaiaad+gacaWGTbGaeyOeI0Iaamyyaiaa d+gaaeqaaKqbaoaabmaabaGaamiCamaaBaaajuaibaGaamyAaaqaba qcfaOaaiilaiaadIhadaWgaaqcfasaaiaadMgaaeqaaaqcfaOaayjk aiaawMcaaiabg2da9iabeg7aHjaabcfacqGHRaWkcqaHYoGycaGGOa GaamytaiabgUcaRiaadofacaGGOaGaamOCaiaacMcacaGGPaGaey4k aSYaaSaaaeaacaaIXaaabaGaaGOmaaaacaWGnbGaeqyYdC3aaWbaaK qbGeqabaGaaGOmaaaajuaGcaWGYbWaaWbaaKqbGeqabaGaaGOmaaaa juaGcqGHRaWkdaWcaaqaaiabeg7aHbqaaiaaikdacaWGnbGaamOCam aaCaaajuaibeqaaiaaikdaaaaaaaaa@610E@ ………(71)

Denote to the ordinary Hamiltonian operator in the commutative space. In this way, one can obtain the complete energy spectra for (m.a.o.) potential in (NC: 3D-RS) symmetries. Know the following accompanying constraint relations:

a-The two quantum numbers ( m ˜ ,m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GabmyBayaaiaGaaiilaiaad2gaaiaawIcacaGLPaaaaaa@3AA5@  satisfied the two intervals: l ˜ m ˜ + l ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTi qadYgagaacaiabgsMiJkqad2gagaacaiabgsMiJkabgUcaRiqadYga gaacaaaa@3EB3@  and lm+l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTi aadYgacqGHKjYOcaWGTbGaeyizImQaey4kaSIaamiBaaaa@3E86@ , thus we have 2 l ˜ +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaikdace WGSbGbaGaacqGHRaWkcaaIXaaaaa@39D2@  and 2l+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaikdaca WGSbGaey4kaSIaaGymaaaa@39C3@  values for these quantum numbers,

b-We have also two values for p-spin symmetry j= l ˜ + 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQgacq GH9aqpceWGSbGbaGaacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaa aaaa@3BD7@  and j= l ˜ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQgacq GH9aqpceWGSbGbaGaacqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaa aaaa@3BE2@  and two values for spin symmetry j=l+ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQgacq GH9aqpcaWGSbGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaaaaa@3BC8@  and j=l 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQgacq GH9aqpcaWGSbGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaaaaa@3BD3@ .

Allow us to deduce the important original results: every state in usually three dimensional space will be replace by 4( 2 l ˜ +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaisdada qadaqaaiaaikdaceWGSbGbaGaacqGHRaWkcaaIXaaacaGLOaGaayzk aaaaaa@3C19@  and 4( 2l+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaisdada qadaqaaiaaikdacaWGSbGaey4kaSIaaGymaaGaayjkaiaawMcaaaaa @3C0A@  sub-states under p-spin symmetry and spin symmetry, which allow us to fixing the degenerated states to the 4 i=0 n1 ( 2l+1 ) 4 n 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaisdada aeWbqaamaabmaabaGaaGOmaiaadYgacqGHRaWkcaaIXaaacaGLOaGa ayzkaaaabaqcfaIaamyAaiabg2da9iaaicdaaeaacaWGUbGaeyOeI0 IaaGymaaqcfaOaeyyeIuoacqGHHjIUcaaI0aGaamOBamaaCaaabeqc fasaaiaaikdaaaaaaa@48BF@ values in (NC: 3D-RS) symmetries. It is easy to see that the obtained originally results reduce to the ordinary results described on quantum mechanics when the noncommutativity of space disappears ( Θ,χ )( 0,0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaeuiMdeLaaiilaiabeE8aJbGaayjkaiaawMcaaiabgkziUoaabmaa baGaaGimaiaacYcacaaIWaaacaGLOaGaayzkaaaaaa@417A@ , equations (67), (68) and (69) reduces to (22) and (24) and one recovers the standard textbook results. Finally one concludes; our obtained results are sufficiently accurate for practical purposes. These results are in agreement with the ones obtained previously [38, 47].

The Important Concluding Remarks

Let us summarize our results as follows:

  1. The solution procedure presented in this paper is based on the both of Bopp’s shift method and standard perturbation theory, we investigate the bound state energies of n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaCa aaleqabaGaamiDaiaadIgaaaaaaa@38FC@  excited states for (m.a.o.) described on (NC: 3D-RS).
  2.  It is found that the energy eigenvalues depend on the dimensionality of the problem and non-relativistic atomic quantum numbers ( j= l ˜ ±1/2,j=l±1/2, s ˜ =±1/2,l, l ˜ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamOAaiabg2da9iqadYgagaacaiabgglaXkaaigdacaGGVaGaaGOm aiaacYcacaWGQbGaeyypa0JaamiBaiabgglaXkaaigdacaGGVaGaaG OmaiaacYcaceWGZbGbaGaacqGH9aqpcqGHXcqScaaIXaGaai4laiaa ikdacaGGSaGaamiBaiaacYcaceWGSbGbaGaaaiaawIcacaGLPaaaaa a@50E3@ and the two angular momentum quantum numbers ( m, m ˜ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamyBaiaacYcaceWGTbGbaGaaaiaawIcacaGLPaaaaaa@3AA5@  in addition to the infinitesimal parameters ( Θ,χ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaeuiMdeLaaiilaiabeE8aJbGaayjkaiaawMcaaaaa@3BE0@ .
  3. We have also constructing the corresponding NC Hermitian Hamiltonian operator H ^ ncao ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIeaga qcamaaBaaajuaibaGaamOBaiaadogacqGHsislcaWGHbGaam4Baaqa baqcfa4aaeWaaeaaceWGWbGbaKaadaWgaaqcfasaaiaadMgaaeqaaK qbakaacYcaceWG4bGbaKaadaWgaaqcfasaaiaadMgaaeqaaaqcfaOa ayjkaiaawMcaaaaa@44B6@  which presented by eq. (70).

 The energy eigenvalues are in good agreement with the results previously. Finally, we point out that these exact results (67), (68) and (69) obtained for this new proposed form of the modified potential (39) may have some interesting applications in the study of different quantum mechanical systems, nuclear physics, atomic and molecular physics.

Acknowledgement

This work was supported with search laboratory of: Physics and Material Chemistry, in Physics department, Sciences faculty-University of M'sila, Algeria.

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