Journal of Nanomedicine Research
A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One-electron Atoms at Planck’s and Nano-Scales
Abstract Review Article
In this paper, we present solutions of the modified Dirac equation (MDE) with modified Volume 7 Issue 1 - 2018 pseudoharmonic (MPH) potential for th spin-1/2 particles by means Bopp’s shift n Department of Physics, University of M’sila M’sila, Algeria method, in the framework of noncommutativity 3-dimensional real space (NC: 3D-RS) symmetries. The exact corrections for excited states are found straightforwardly Θ and *Corresponding author: byχ means of the standard perturbation theory. It is found that the energy eigenvalues Abdelmadjid Maireche, Laboratory strongly depend on the potentialj=±=± ll parameters,( ) 1 / 2, s two 1infinitesimal / 2, ll( ) and parameter m m ( ), which induced by position-position noncommutativity, in addition to the discreet Email:of Physics and Material Chemistry, Physics department, University of M’sila-M’sila Algeria, Tel: +213664834317; atomic quantum numbers ( ( ) ) and we 22( l+ 1) and 22( l+ 1) Received: | Published: have also shown that, the usual states in ordinary 3-.dimensions are canceled and has been replaced by new degenerated sub-states under November 24, 2017 January 12, 2018 the pseudo spin symmetry and spin symmetry conditions, respectively in the new Keywords:quantum group (NC: 3D-RS).
Dirac equation; Pseudoharmonic potential; Noncommutative space; Star product; Bopp’s shift method
µν Abbreviations: θ MPH: Modified Pseudo Harmonicpotential; (∗) denote to the The very small parameters (compared to the energy) NC: 3D-RSP: Noncommutativity Three Dimensional Real Spaces; MDE: are elementˆ of antisymmetric real matrix ˆand CCRs: Canonical Commutations Relations; NNCCRs: New fx( ) → fx( ˆ) and gx( ) → gxˆˆ( ) f( xgxˆˆˆ) ( ) ≡∗( fg) ( x) new star product, which generalized to two arbitrary functions Noncommutative Canonical Commutations Relations; ( ) ( ( fg) x t) Modified Dirac equation, SP: Schrödinger Picture; HP: Heisenberg i µν xx i µν fˆ ( xˆˆˆ) g( x) ≡( f ∗ g) ( x) ≡exp( θ ∂∂( fg) ( x,( p) ≡ fg − θθ ∂xx f ∂ g +O 2 IntroductionPicture instead of the usual productµν [17-27]: µν µ νν ( ) 2 2 (xx= )
i µν xx −∂θ f( x) ∂ gx( ) The analytical solution of Dirac equation plays a vital role in 2 µν (3) method can relativistic quantum mechanics and solving the Dirac equation The new term ( ) is induced by (space- to obtain the bound-state energies for with different potential models using various methods for example Nikiforov-Uvarov space) noncommutativity properties. A Bopp’s shift be used, instead of solving any quantum systems by using directly method, The Laplace transform approach (LTA), factorization xxˆˆ,= iθ and ppˆ , ˆ = 0 star product procedureij [18-33]:ij i j method and so on [1-6]. The quantumc algebraic= = 1 structure based to the ordinary CCRs in both of SP, HP and Dirac picture (the (xxyxzxˆˆˆˆ= 123 ,, = ˆ = ˆ ) (4)are xp, = xt( ), p( t) = iδ and xx , = xtx( ) , ( t) = pp, = pt( ), p( t) = 0 operatorsij iare jdepended ij on time),ij as i( j ): ij i j Theθθ new three-generalized θθ coordinates θθ xxppyyppzzppˆˆ=−−12 13 , =−−21 23 and ˆ =−−31 32 given by22 [22-34]:yz 22xz 22x y (1) Very recently, non-commutative geometry plays an important Where ( xyz,,) and ( pppxyz,,) 2 (5) role in modern physics and has sustained great interest, for rˆ on NC are three-usual coordinates example [7-21] and our works in this context [22-47] in the case and momentum ,which allow us to getting the operator θ of relativistic and nonrelativistic quantum mechanics. The new 22 ij threerrˆ = dimensional −LL Θ spaceswith asΘ≡ followsLLL Θ [22-33]: + Θ + Θ and Θij = quantum structure of NC space based on the following NC CCRs xz12y 23 13 2 in ∗both of SP, ∗ HP and Dirac ∗∗ picture respectively, as ∗ follows [7-25]: ∗ xˆˆij, p =xt ˆ i( ) , p ˆ j( t) = iδθ,, x ˆˆij x =xt ˆ i( ) , x ˆ j( t) = i and pˆij , p ˆ =pt ˆ i( ) ,p ˆ j( t) = 0 ij ij (6) In recent years, the study of PH potential has attracted a lot (2) of interest of different authors, it have the general features of the
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J Nanomed Res 2018, 7(1): 00168 A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One- Copyright: 11 electron Atoms at Planck’s and Nano-Scales ©2018 Maireche
Fr( ) and Gr( ) nk nk l l true interaction energy, inter atomic and dynamical properties Y (θφ, ) and Y (θφ, ) are in solid-state physics and play an important role in the history of Where, arejm the upper jm and lower k k molecular structures and interactions; this potential is considered components of the Dirac spinor, as an intermediate between harmonic oscillator and Morse- the spin and pseudo-spin spherical harmonics while ( ) is type potentials which are more realistic anharmonic potentials, l l furthermore, the PH potential is extensively used to describe the related to the total angular momentum quantum numbers for 1 bound state of the interaction systems, and has been applied for spin symmetry−+l 1 if - j +and 1/2 p-spin , s ,p symmetry , etc , j=+ l ,as aligned [6]: spin k〈 0 ( ) ( ) ( 1/2 3/2 ) ( ) both classical and modern physics [5,6]. This work is aimed at k = 2 obtaining an analytic expression for the eigenenergies of a MPH 11 +=+l if jl ,( p1/2 , d 3/2 , etcjl) , =− , unaligned spin ( k〉 0) potential in (NC: 3D-RS) symmetries using Bopp’s shift method to 22 discover the new symmetries and a possibility to obtain another 1 (11) −+l if - j 1/2 , s ,p , etc , j =− l , aligned -spin k 〈 0 applications to this potential in different fields. This work based ( ) ( 1/2 3/2 ) ( ) = 2 essentially on our previously works [22-43], it was studied in our k 11 ++(l1) if jl =+ ,( p1/2 , d 3/2 , etcjl) , =+ , unaligned- spin ( k〉 0) works [34,35] in the case of nonrelativistic case. The organization 22 scheme of the study is given as follows: In next section, we (12) briefly review the DE with PH potential on based to [6]. Sec. 3 Fr( ) Gr( ) nk nk is devoted to studying the MDE for MPH potential by applying al th Bopp’s shift method.n In the fourth section and by applying The radial functions ( , ) are obtained by solving standard perturbation theory we find the quantum corrections of the following differenti equations [6]: dr∆( )dk 2 + spectrums of the excited states in (NC-3D: RS) for relativistic d kk(+ 1) dr dr r − −ME +nk −∆( r) ME −nk +Σ( r) + F( r) = 0 spin-orbital interaction. In the next sub-section, we derive the 22 nk dr r ME+ nk −∆( r) magnetic spectrum for MPH potential. In the fifth section, we resume the global spectrum and corresponding NC Hamiltonian operator for MPH potential. Finally, the important results and the drΣ( )dk − conclusions are discussed in last section. 2 (13) Review of the Dirac Equation for PH Potential d kk(− 1) dr dr r − −ME +nk −∆( r) ME −nk +Σ( r) + G( r) = 0 22 nk dr r ME+nk −Σ( r) M The Dirac equation for a spherically symmetric potential in E (14) 3-dimensionalSr() reads for a single-nucleonVr( ) with the mass of dr∆( ) met = 0 and relativistic energy moving in an attractive scalar potential The bound state solutions of the PH potential for dr the spin (α andp++ β a ( Mrepulsive Sr ())) potential Ψ( r ,,θφ) =−Ψ( EVr in( natural)) ( r ,,unitsθφ) [6]: and cording symmetric case obtained in the exact spin symn ryl (7) and then the energy eigenvalues depend on . Ac 0 σ I × 0 01 Fr( ) α = i β = 22 σ = to LTA and asymptotic interactionnk method, which was applied in i 1 2 σ i 0 0 I22× 10 µr [6], the upper component− of the Dirac spinor gives by: Here0 −i ( 10, ), , ν +1 3 σ = and σ = are the F( r) = Nr e2 F−+ n,,ν r2 nk 11 2 i 0 3 01− 2 N and Fn(−+,2γε 1,2 r) (15) potential V (r) pinusual Dirac matrices 11 while the PH 2 for the s symmetric and the Where, are the normalization rr 2 b constant and the confluent hyper-geometric functions, the pseudo-spin V spin-symmetry( r) = D − [6]:0 ≡ar ++ c 0 rr 2 relativistic positive energy eigenvalues with the PH potential 0 r r0 2 22 under( 2theDEMMEC0 +−spin-symmetry) +− condition −(2 k + 1 )is +obtained 4 DrMEC00( as +− [6]: ) =4( n + 1/2) (8) D D and r 0 0 0
Where, are−2 two constants+2 related to the Gr (16)( ) a= Dr , b= Dr and cD= −2 nk dissociation energy of a molecule00 and00 an equilibrium distance,0 ator Hˆ an For the exactµr2 pseudo-spin symmetric case, the lower ph − respectively while , thus, ν +1 3 22D componentG( r) = Nre of2 the F− Dirac n,ν + , r2 spinor with νν( +[6]: 1) =kk ( + 1) +( M +− ECDr) and µ =0 (M +− EC) nk 1 12 00 2 the corresponding ordinary Hamiltonian oper can b r0 ˆ be expressed as:H=++(αβp ( M Sr ( ))) + V( r) ph Here N (17) Ψ (r,,θφ) (9) denote to the normalization constant and the The spinor can be written as [6]: relativistic negative energy eigenvalues with the PH potential fr l θφ nk ( ) 1 Fnk( rY) jm ( , ) under the pseudo-spin spin-symmetry condition is obtained as Ψ==(r,,θφ) r0 2 22 nk (2D+− EM) EMC − − −(2 k + 1) + 4 Dr( MEC −+) =4( n + 1/2) gr r l [6]:D 0 00 nk ( ) iGnk ( r) Y jm (θφ, ) 0 (10)
(18)
Citation:
Maireche A (2018) A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One-electron Atoms at Planck’s and Nano-Scales. J Nanomed Res 7(1): 00168. DOI: 10.15406/jnmr.2018.07.00168 A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One- Copyright: 12 electron Atoms at Planck’s and Nano-Scales ©2018 Maireche
Fr( ) Gr( ) nk nk ( p ) Ln ( x) e e It is well known that, the generalized Laguerre polynomials The dk radial functions ( , ) are obtained, in the +F( r) = ME + −∆( rˆ) G( r) absence of tensornk interaction, ncby− phsolving twonk equations: can b( p ) xpressedΓ(np ++as a1 )function of the confluent hyper- dr r LF( x) = (−+np, 1; x) geometric functionsn as: np!1Γ+( ) 1 1 (29) dk +Gr( ) = ME −nc− ph +Σ( rGrˆ) ( ) Fr (19)( ) dr r nk nk nk Gr( ) compon ∆=(rˆ) Vr( ˆˆ) − Sr( ) and Σ=(rˆ) Vr ( ˆˆ ) +(30) Sr( ) Which allow nk us to rewritten the upper component eliminating Fr( ) and Gr( ) and the lower 22ent of the Dirac spinor for the spin with , 33µµrr11 nk nk Γ+νν−−νν++Γ+ symmetric case22 andν +1 the22 pseudo-s( 22) 22pin spin-symmetry, respectively:ν +1 ( ) F( r) = Nn! r e LLr and G( r) = Nn ! r e r nk 33nn( ) nk ( ) from Eqs. (29) and (30), we can Γnn ++ννΓ ++ 22 obtain the following two Schrödinger-like differential equations d2 kk(+ 1) as follows− in (NC-3D: −ME + RS) symmetries: −∆( rˆˆ) ME − +Σ( r) F( r) = 0 ( nc−− ph )( nc ph ) nk NC Relativistic Hamiltonian Operator for MPH (20) dr22 r Potential d2 kk (− 1) (31) − −ME + −∆( rˆˆ) ME − +Σ( r) G( r) = 0 Overview of Bopp’s shift method ( nc−− ph )( nc ph ) nk dr22 r Vr( ˆ) ˆ H( pxii, ) (32) In order to obtain the MDE for MPH potential , we Ψ (r) E n replace both ordinary Hamiltonian operator , ordinary After straightforward calculations one can obtains the two ˆ a a a 22bb b operator H( pxˆˆii, ) Ψ (r ) E terms= in +(NC-3D:LL Θ+ RS)OO spacesθθ and as follows: − = − Θ+ spinor and ordinary energy by NC Hamiltonianc− ph 2 24 ( ) rrˆ 3 ( ) ∗ rˆ rr 2r , new spinor and new energy (33) ( ˆ) and the ordinary product will be replace by new star product , Vr respectively. Allow us to writing the new MED for MPH potential ab Which allow us toˆ writingΘ=− the MPH potential Θ as follows: Hˆ ( pxˆˆ, ) ∗Ψ( r) = E Ψ( r) V1 pert− ph ( r,,, ab) L for spin symmetric case as follows [22-34]:ii nc− ph ab rr432 Vr( ˆ) = −++ c 2 r (21) r ˆ ab V2 pert− ph ( r,,,Θ=− ab) L Θfor p-spin symmetric case rr432 It is worth to motioning that the Bopp’s shift method permutes to reduce theH above( equation pxˆˆii, )ψψ( rto) =simplest E the( r) form: nc−− ph nc ph (34) ψ (r ) (22) H( pxˆˆ, ) It is clearly that the star product inducing the non- Where, is a solutionnc− ph of thei iDirac equation and the new Vˆ ( r,Θ ,, ab) and Vˆ ( r,Θ ,, ab) in the commutativity1 pert is − replaced ph by the usual2 pert product− ph plus non local operator of Hamiltonian can be expressed in three corrections Vr( ˆ) al general varieties: both NC space and NC phase (NC-3D: RSP), only NC space (NC-3D: RS)11 and only NC phase (NC: 3D-RP) as, scalar potential . This lows writing the modified Dirac H( pxˆˆ, ) ≡=−=− Hp ˆ pθθij x;x ˆ x p for NC-3D: RSP respectivelync− ph i i[35-44]:i i22 j i i ij j equation in the non-commutative case as an equation similarly to the usual Dirac equation of the commutative type with a non local 1 potential. Furthermore, using the unit step function (also known H( pxˆˆ, ) ≡==− Hp ˆ px; ˆ xθ p for NC-3D: RS nc− ph ii i ii i 2 ijj (23) as the Heavisideab step function or simply the theta function) we V( rˆ) = −++c θθ E Vˆˆ( r,Θ ,, ab) + − E V( r,Θ ,, ab) (24) can rewrite the MPH( potentialnc−− ph ) 12pert−− to ph the following( form:nc ph ) pert ph 1 r2 r H( pxˆˆ, ) ≡=−= Hp ˆ pθ ij ; x , xˆ x for NC-3D: RP nc− ph ii i i2 ji i Where (25) 1 for x〉 0 (35) θ = H( pxˆˆii, ) de ( x) In recently work, we1 are interest with thenc −above ph second variety 0 for x〈 0 xxˆ = − θ p and ppˆ = : and then the newi modified i2 ij Hamiltonian j ii fined as (36) H( pxˆˆ,) =+++αβ Pˆ ( M Sr (ˆ )) V(r ˆ) a function ncof− ph ii ∆=(r) Vr( ) and Σ=(rC) =constants ps Where the MPH potential Vr( ˆ) We generalized the constraint for the pseudo-spin (p-spin) (26) ∆= ab symmetry ( (rˆˆ) Vr( ) and Vr( ˆ) = −+ is given c by: rˆ2 rˆ whichΣ=(rCˆ) presentedˆ =constants in [6] into the new form ps Vr (27)( ˆ) Vr( ˆ) in (NC-3D: RS) and inserting the potential The Dirac equation in the presence of above interaction can (beαP rewritten++ β (M Sr ())accordingˆˆ) Ψ( r ,,θφ Bopp’s) =−Ψ( EVrshift( method)) ( r ,,θφas follows:) into the two Schrödinger-like differential Eqs. (31) and (32), one obtains: (28)
Citation:
Maireche A (2018) A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One-electron Atoms at Planck’s and Nano-Scales. J Nanomed Res 7(1): 00168. DOI: 10.15406/jnmr.2018.07.00168 A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One- Copyright: 13 electron Atoms at Planck’s and Nano-Scales ©2018 Maireche
d2 kk(+ 1) a b a b − −+ME ME −+−−+−+−−Θ−+ Cˆˆ cME C L ME C ˆ F( r) =0 ( nc−− ph)( nc ph ps ) ( nc− ph ps ) ( nc− ph ps ) nk dr2 r 2 r2r r432 r (37) d2 kk(− 1) a b a b − −+ME ME − + Cˆ −−+ c ME − + C ˆˆ −− L Θ−+ME C G( r) =0 ( nc−− ph)( nc ph ps ) ( nc− ph ps ) ( nk ps ) nk dr2 r 2 r2r r432 r (38) 1 k≡−( l +1) if -( j + 1/2) ,( s ,p , etc) , j= l + , aligned spin ( k〈 0) LL→ 1 1/2 3/2 2 Vˆ ( r,Θ ,, ab) and k = 1 pert− ph 11 and two similarly equations obtained by . It’s k2 ≡+ l if jl = + ,( p1/2 , d 3/2 , etcjl) , = − , unaligned spin ( k〉 0) Vˆ ( r,Θ ,, ab) are proportion 22 clearly2 pert− ph that, the additive new parts Θ ed al with infinitesimal parameter With kk( −=11) ll ( +) and kk( −=11) ll( +) (42) (33× ) Hˆ ( kk, ) and , thus, we can consider as a perturbations terms. Our aim is so, − whichph 12 allows to derive the energy spectrum for a moving charged particle in the Hˆ ( kk, ) us so to− ph form12 two diagonal matrixes presence of a potential given by (35) analytically in a very simple way. , for MPH potential, respectively, in (NC: 3D-RS) ab 1 ˆ =Θ− + =− 〈 th symmetries(Hso− ph ) ( k1 ) as: k if -( j 1/2) ,( s1/2 ,p 3/2 , etc) , j l , aligned spin ( k 0) The Exact Relativisticn Excited States Spin-orbital for One-electron Hamiltonian Atoms and the 11 1 rr432 2 Corresponding Spectrum for MPH Potential in (NC: 3D- RS) ab 11 ˆ =Θ− =+ =+ 〉 SymmetriesThe exact relativistic for spin-orbital Hamiltonian for MPH (Hso− ph ) ( k2) k 2 if jl ,( p1/2 , d 3/2 , etcjl) , , unaligned spin ( k 0) 22 43 22 Potential in (NC: 3D- RS) symmetries for one-electron rr2 ˆ (Hso− ph ) = 0 atoms 33
ab 1 ˆ =Θ− + =+ 〈 (Hso− ph ) ( k1 ) k if -( j 1/2) ,( s1/2 ,p 3/2 , etc) , j l , aligned spin (43) ( k 0) LΘ and L Θ SL and SL and then the 11 1 rr432 2
The results (34) can be rewritten in a more accessible physical ˆ ˆ ˆ ab 11 V( r,Θ ,, ab) and V( r,Θ ,, ab) (Hso− ph ) ( k2)=Θ− k 2 if jl =+ ,( p1/2 , d 3/2 , etcjl) , =− , unaligned spin ( k〉 0) form, we replace both 1 pert− ph by 2 pert− ph 22 rr432 22 ˆ two perturbative terms (Hso− ph ) = 0 33 potential:for the spin symmetric case and the pseudo-spin spin-symmetry, th (44) respectively can be rewritten to the equivalent new form for MPH n excited states for one-electron atoms in ab ab Vˆˆ( r,,,Θ ab) =Θ−SL and V( r ,Θ , ab , ) =Θ−SL The exact relativistic spin-orbital spectrum for MPH potential 12pert−− ph pert ph (NC: 3D- RSP) symmetry rr4322rr43 symmetries for
Vˆ ( r,Θ ,, ab) Ek(Θ, 1 ) Ek(Θ, 2 ) -( j+ 1/2) 1 pert− ph (39) In this subsection,nc −we per: dare going tonc −study per: u the modifications to and Vˆ ( r,Θ ,, ab) 1 Furthermore,2 pert− ph the above perturbative terms the(s, energyp , etc levels) jl =( + , k0〈 and ) for ( 1/2 3/2 2 can be rewritten to the following new 1 1 11ab22 22ab 2 , jl= + (p,, d , etc) , alignedjl= − spin spin-down)in k0〉 and ( Vˆˆ( r,,,Θ ab) =Θ−J −− L S and V( r ,Θ , ab , ) =Θ−JS −−L2 1/2 3/2 −− 2 2 equivalent12pert ph form22 for43 MPH potential [35-4pert ph7]: 43 rr22rr Θ , , , un aligned sp and spin th up),n respectively, at first order of infinitesimal parameter , for (40) SL and SL e excited states, for the spin symmetric and the pseudo-spin To the best of our knowledge, we just replaced the coupling 1 22 2 1 2 2 2 spin-symmetry+ obtained by applying the standard perturbation JLS−− and JS−−L ˆˆ 2 spin-orbital (pseudo spin-orbital) by th two ∫ Ψ (r,,θφ) θ E−− V( r,Θ+− ,, a b) θ E− V −( r,ΘΨ ,, a b) ( r ,,θφ) r drd Ω 2 2 theory,nk using( nc Eqs. ph) 12 pert(20) ph and (35)( as:nc ph) pert ph nk 2 2 ** H( pxˆˆ, ) J L =θθE∫∫ F r Vˆˆ r,Θ− ,, a b F r dr E G r V r,Θ ,, a b G r dr expressions: nc− ph , reiispectively, ( nc−− ph) nk( ) 12 pert ph ( ) nk ( ) ( nc−− ph) nk( ) pert ph ( ) nk ( ) 2 S and J ) in relativisticz quantum mechanics. The set ( , , , k(k ) forms a complete of conserved physics quantities and (45) l l The first part represents the modifications to the energy the spin–orbit quantum number is related to the quantum Ek(Θ, 1 ) levels for the spin symmetric while the second partnc− per represent: d numbers for spin symmetry and p-spin symmetry as follows Ek(Θ, 2 ) 1 thenc − per modifications: u to the energy levels ( , [6]: k≡− l if -( j + 1/2) ,( s ,p , etc) , j= l − , aligned spin ( k〈 0) 1 1/2 3/2 2 k = ) for the pseudo-spin* spin-symmetry,ab then we E(Θ, k1 ) ≡−θ ( Enc− ph) k∫ G nk( r)− Gnk ( r) dr 11 nc− per:1 d 43 k2 ≡+( l +1) if jl = + ,( p1/2 , d 3/2 , etcjl) , = + , unaligned spin ( k〉 0) have explicitly: rr2 22 * ab E(Θ, k) ≡−θ E k∫ G( r)− G( r) dr nc− per:2 u 2 ( nc− ph) nk nk (46) rr432 (41)
Citation:
Maireche A (2018) A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One-electron Atoms at Planck’s and Nano-Scales. J Nanomed Res 7(1): 00168. DOI: 10.15406/jnmr.2018.07.00168 A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One- Copyright: 14 electron Atoms at Planck’s and Nano-Scales ©2018 Maireche
Ek(Θ, 1 ) nc− per: d Ek(Θ, 2 ) ncNow,− per: u we use eqs. (20), (40), (41) and (42) to obtain the explicit expressions for modified energy eigenvalues ( , ) for MDE with MPH potential under the pseudo spin symmetry2 conditions obtained as: 3 2 Γ+ν 1 +∞ 2 ν + 2 22νµ+−r ( 2 ) 2 ab E(Θ,! k1 ) ≡−θ ( − Enc− ph ) k Θ Nn ∫ r e Ln r − dr nc− per:1 d 3 ( ) 43 Γn ++ν 0 rr2 2 (47) 2 3 2 Γ+ν 1 +∞ 2 ν + 2 22νµ+−r ( 2 ) 2 ab E(Θ,! k2 ) ≡−θ ( − Enc− ph ) k Θ Nn ∫ r e Ln r − dr nc− per:2 u 3 ( ) 43 Γn ++ν 0 rr2 2 (48) 2 Xr= 2 3 2 And using the transformation , we have: Γ+ν 1 1 1 +∞ ν − ν + 2 −µ X ( 2 ) ab E(Θ,! k1 ) ≡−θ ( −Enc− ph ) k Θ Nn ∫ X2 e Ln ( X ) − dX nc− per:1 d 2 3 2 3 Γn ++ν 0 X 2X 2 2 (49) 2 2 3 1 Γ+ν 1 ν + 1 +∞ ν − 2 −µ X 2 ab E(Θ,! k2 ) ≡−θ ( −Enc− ph ) k Θ Nn ∫ X2 e Ln ( X ) − dX nc− per:2 u 2 3 2 3 Γn ++ν 0 X 2 2X 2 (50)
2 A direct simplification gives: 3 Γ+ν 1 1 2 E(Θ, k1) ≡−θ( − Enc− ph ) k Θ Nn! (T1−−ph ( Dr 00,,,νν 1 n)+ T 2ph ( Dr 00 ,,, 1 n)) nc− per:1 d 2 3 Γn ++ν1 2
2 3 Γ+ν (51) 1 1 2 E(Θ, k2) ≡−θν( −Enc− ph ) k Θ Nn! (L1−−ph( D 00,, r 2 , n)+ L 2ph( D 0,,rn02ν ,)) nc− per:2 u 2 3 Γn ++ν1 2 T( Dr,,,ν n) T( Dr,,,ν n) L( Dr,,ν , n) and L( Dr,,ν , n) 1− ph 00 1 2− ph 010 1− ph 00 2 2− ph 00 2 Where, the four terms , 22, are given by, respectively: 5 11 +∞ νν11+++∞ ν1− −−µµ( ) b ( ) XX22ν1−2 T( D0,,, r0 νν 1 n) = a∫∫ X2 e LLnn( X) dX, T( D00 ,,, r 1 n) = − X e ( X) dX 12−−ph ph 2 00
2 2 5 11 +∞ νν22+++∞ ν 2 − ( ) b −µ ( ) (52) −−µνX 222 2 X L( Dr00,,,νν 1 n) = a∫ X2 e LLnn( X) dX and L( Dr00 , , 1 , n) = − ∫ X e( X) dX 12−−ph ph 2 0 0
1++αβ αβ + A2 Now we apply the special integral [48]: ++α ∞ k F,1 ;1 ; aa12+ Γ++(11αβ) Γ++( αk ) 222 −+xs()αβ+ αα d B ∫ e x LLkk(a12 x) ( a x) dx = 2 kk!!Γ+( 1α ) k 1+α 1++αβ 0 dh (1−hB) (53) h=0
2 4aah aa++1 h aa+ Where A = 12 Bs= + 12 Rs+〉120 a 〉0 a 〉0 and R (αβ+) 〉−1 (1−h)2 21−h 2 1 2 , , , , , which allow us to obtaining T( Dr,,,ν n) and T( Dr,,,ν n) a : 1− ph 00 1 2− ph 00 1 s
Citation:
Maireche A (2018) A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One-electron Atoms at Planck’s and Nano-Scales. J Nanomed Res 7(1): 00168. DOI: 10.15406/jnmr.2018.07.00168 A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One- Copyright: 15 electron Atoms at Planck’s and Nano-Scales ©2018 Maireche
33 νν−−3/2 3/2 3 A2 11++ν Γνν11 − Γ ++n n F,1 ;1 ; −2 22 d 2 222 T( D,,, rν n) = Dr B 1− ph 00 1 00 33n 2 Γ+ννdh + ν − n! 11− 1 3/2 22(1 hB) 1 (54) h=0
2 3 νν11−13A +2 Γ(νν11) Γ ++n F,1++ ;ν1 ; n 2 Dr00 2 d 2 22B T( Dr00,,,ν 1 n) = − 2− ph 2 2 33n Γ+ννdh +ν n! 11− 1 22(1 hB) (55) h=0 + 2 2 4h 1 h Here νν( 1+1) =k ( k + 1) +( M +− E C) Dr A = and B =µ −+1 1 1 1 00 (1−h)2 1−h L( Dr,,ν , n) = T( Dr,,νν→ , n) and L, ( Dr,,ν , n) = T( Dr,,νν→ , n) are determined, thus, the new factors 11−−ph 00 2 ph 00 1 2 22−−ph 00 2 ph 00 1 2 by the following results: 2 33 νν22−−3/2 3/2 3 A Γνν22 − Γ ++n F,1++ ;ν 2 ; n 2 −2 22 d 2 22B L( D00,, rν 2 , n) = Dr 1− ph 00 33n 2 Γ+ννdh + ν − n! 22− 2 3/2 22(1 hB) 1 h=0
2 3 νν22−13A +2 Γ(νν22) Γ ++n F,1++ ;ν 2 ; (56) n 2 Dr00 2 d 222 B L( Dr00,,ν 2 , n) = − 2− ph 2 2 3 n 3 Γ+ν dh ν + ν n! 2 − 2 2 2 (1 hB) 2 h=0 2 νν( +1) =k ( k + 1) +( M +− E C) Dr 2 2 2 2 00 Ek(Θ, ) Ek(Θ, ) Where, nc− per: d 1 nc− per: u 2, substituting Eqs. (54), (55) and (56) into Eq. (51), we obtain the modifications Ek(Θ, ) Ek(Θ, ) to the energy levels ( , nc− per: d 1 ) producednc− per: u by2 relativistic spin-orbital effect under the pseudo-spin symmetry conditions. Now, the energy levels ( , ) produced by relativistic spin-orbital effect under the spin symmetry conditions, can be determined by means of same procedures as before, and to avoid repetition we just make the following N→ Nk, → k , k → k and θθ(− Enc−− ph ) →− (Enc ph ) steps: 1 12 2 Ek(Θ, ) Ek(Θ, ) nc− per: d 1 nc− per: u 2 (57) 2 This implies that ( , ) can be expressed3 as, respectively: Γ+ν (k2 ) 1 2 E(Θ≡, k1) θ( Enc− ph ) k Θ Nn! T1−−ph ( Dr 00,,ν( k 2) , n)+ T 2ph ( Dr 00 ,,νν( k 2) ,, n) nc− per:1 d 2 3 ( ) Γ+nkν ( 2 ) + 2
2 3 Γ+ν (k1 ) (58) 1 2 E(Θ≡, k2) θν( Enc−− ph ) k Θ Nn! L1ph ( D 00,, r( k 1) , n)+ L 2− ( D 0,r01,,ν ( kn) ) nc− per:2 u 2 3 ( ph ) Γ+nkν ( 1 ) + 2
θ −E ( nc− ph ) Ek(Θ, ) Ek(Θ, ) Ek(Θ, ) Having nc obtained− per: d 1 the nc exact− per: u modifications2 tonc − the per: d energy1 and θ (Enc− ph ) The negative and positive signs of the coefficients Ek(Θ, 2 ) levelsnc− per : (u , ) and ( th , are necessary to ensure that the modifications to n the energy levels under the pseudo spin symmetry conditions and ) under the pseudo spin symmetry conditions and spin symmetry conditions, respectively, for exited states, spin symmetryth conditions are negative and positive, respectively. n excited states for one-electron atoms in (NC: 3D- R S) produced by NC spin-orbital Hamiltonian operator, we now symmetriesThe exact relativistic magnetic spectrum for MPH potential consider another interested physically meaningful phenomena, for which also produced from the perturbative terms of PH potential
Citation:
Maireche A (2018) A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One-electron Atoms at Planck’s and Nano-Scales. J Nanomed Res 7(1): 00168. DOI: 10.15406/jnmr.2018.07.00168 A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One- Copyright: 16 electron Atoms at Planck’s and Nano-Scales ©2018 Maireche
2 3 Γ+ν1 related to the influence of an external uniform magnetic field, it’s 1 2 EE= −θχ −E Nn !T+ T( k Θ+ B m ) sufficient to apply the following two replacements to describing nc− pd ( nc− ph ) ( 12−−ph ph ) 1 nk1 2 3 −+22 −+22 −+22 Γn ++ν1 theserr phenomena: rr rr 2 D 00−L Θ→χχD 00 − BL or D 00− BL 00 0 43 43 43 2 rr22 rr rr 2 3 Γ+ν and Θ→χ B 1 1 2 EE= +θχ( −Enc− ph ) Nn!(T12−−ph+ T ph ) ( k 2 Θ+ B m ) nc− p: u nk 2 3 Here χ 2 Γ ++ν (59) n 1 B= Bk 2 is infinitesimal real proportional’sHˆ constants,( rab,,, andχ ) wein mag− ph 2 (63) choose the magnetic field , which allow us to introduce 3 Γ+ν1 the modified new magnetic−+22 Hamiltonian rr BL for pseudo spin symmetry 1 2 ˆ 00 E= E − θχ( Enc− ph ) Nn!(L12−−ph+ L ph ) ( k 1 Θ+ B m) (NC:H 3D-RS),( Dr00,, as:χχ) = D − nc− d nk1 2 3 mag− ph 0 43 Γ ++ν rr2 BL for spin symmetry n 1 2 Here (−SB) 2 (60) 3 Γ+ν1 denote to the ordinary Hamiltonian of Zeeman 1 2 E= E + θχ( Enc− ph ) Nn!(L12−−ph+ L ph ) ( k 2 Θ+ B m) E(χ,, nmD , , r) and E(χ,, nmD , , r) odi ied PH ncr− u nk mag-ph 00 mag-ph 00 2 2 3 effect. To obtain the exact NC magnetic modifications of energy Γn ++ν1 2 for m f potential, under the pseudo spin symmetry conditions and spin (64) Hˆ ( rD, ,, r χ ) symmetry conditions,m− phrespectively,00 which produced automatically Hˆ ncNow,− ph it is important to constructing the Hamiltonian operator by the effect of , we make the following two for MPH potential on based to previously obtained results. simultaneouslykk→m replacements: , → m and Θ→χ B 11 Naturally, to consider the first termHˆ in the modified Hamiltonian ph (61) operator represents the kinetic energy and the potentialˆ energy rm H( kk12, ) or χ and χ ding in ordinary commutative space of the fermionicso− ph particle E( ,, nmD ,00 , r) E( ,, nmD ,00 , r) ˆ mag-phThen, the relativisticmag-ph magnetic modification H( kk12, ) th whichso− ph presented by eq. (9), the second te n correspondetermined represents, the induced spin-orbital parts for the Hˆ ( rab,,,χ ) : excited states, in (NC-3D: RS) symmetries, can be pseudo spin symmetry conditions and spin symmetrymag− mtand the last 3 −+22 from the following relation: Γ+ν1 1 term is the modified ˆ new magneticrr00 Hamiltonian 2 Hso− ph ( k12, k)+−χ D 0 BL for pseudo spin symmetry E(χ,, n m , D00 , r ) =−− θχ( Enc− ph ) B Nn ! (L1−−ph+ L 2 ph ) m 43 mag-ph 2 3 rr2 Γn ++ν1 HHˆˆ= + − 2 nc ph ph −+ rr22 ˆ +−χ 00 Hso− ph ( k12, k) D 0 BL for spin symmetry 3 rr432 Γ+ν (k1 ) 1 2 E(χ,, n m , D00 , r) = θχ( Enc− ph ) B Nn ! (L1−−ph+ L 2 ph ) m mag-ph 2 3 (65) Γ+nkν ( 1 ) + 2 In this way, one can obtain the complete energy spectra for m and m (62) MPH potential in (NC: 3D-RS) symmetries. Know the following −lm ≤ ≤+ l and −lm ≤ ≤+ l a. Where, denotes to the angular momentum quantum accompanying constraint relations: 21l + (21l+ ) numbers , which allow us to fixing The original spectrum contains two possible values of The( ) Exactand Modified values, respectively. Global Spectrum for MPH energies in ordinary three dimensional spaces which presented Potential in (NC-3D: RS) Symmetries for One-Electron by Eqs. (16) and (18), m and m −lm ≤ ≤+ l Atoms andAs − lm mentioned ≤ ≤+ l in the previous21 l subsection,+ ( the21l+ ) quantum th n numbers satisfied the two intervals: , thus we have ( 1 ) and 1 values, jl= + and jl= − ELet ( usΘ , know1 ,χ ,, nmr resume , 00 , D ) theE excited(Θ, k2 ,χ ,, states nmr , 00 , eigenenergiesD ) ( respectively, nc− pd nc− p: u 1 1 2 2 = + and = − (Θ χ ) E(Θ, k ,χ ,, nmr , , D ) jlWe have alsojl two values for ( ) and ( Enc −d , k1, , n, m, r0 , D, 0 ncr− u 2 00) and ( 2 2 al , ) of ) for pseudo spin symmetry conditions and spin symmetry. Allow us to deduce the important origin MDE corresponding the pseudo spin symmetry conditions and 22( l + 1) and 22( l+ 1) results: every state in usually 3-dimensionaln−1 spaces will be 2 spin symmetry(Θ, χ conditions,) respectively, at first order of two replacing by sub-states.2∑ ( 21ln+≡) 2 va i=0 parameters for MPH potential in (NC: 3D-RS) symmetries, Then the degeneratedetri state cana be take lues on based to the obtained results (51), (58), and (62), in addition to the original results (18) and (20) of energies in commutative in (NC: 3D-RS) symm es. Fin lly, we resume our original results space, we obtain the following original results: in this article, the first one is the induced pseudo-spin-orbital
Citation:
Maireche A (2018) A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One-electron Atoms at Planck’s and Nano-Scales. J Nanomed Res 7(1): 00168. DOI: 10.15406/jnmr.2018.07.00168 A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One- Copyright: 17 electron Atoms at Planck’s and Nano-Scales ©2018 Maireche
ˆ H ( kk, ) and Ek(Θ, 2 ) Ek(Θ, 1 ) Ek(Θ, 2 ) so− ph 12 nc− per: u nc− per: d nc− per: u a andHˆ spin-orbital( kk, ) Hamiltonian operators ( Ek(Θ, ) , ) and ( , ), so− ph 12 nc− per: d 1 respectively s: ) and corresponding eigenvalues ( Gr( ) Gr( ) ˆ nk llθφ = Θ nk θφ (Hso−− ph ) ( ki1) Yjm ( ,) Enc per:1 d ( , ki) Yjm ( ,) 11 rr ˆˆ Grnk( ) llGrnk ( ) Hkkr( ,) Ψ⇒( ,,θφ) H kiYE(θφ,) =Θ, kiY (θφ,) so− ph 12 nk ( so−− ph ) ( 2) jm nc per: u ( 1) jm 22 rr (66) Gr ˆ nk ( ) l Hso− ph ( ki1 ) Yjm (θφ, ) =0 ( ) r 33 ˆ Frnk ( ) llFrnk ( ) (HkYEso−− ph ) ( 1)jm (θφ,) =nc per:1 d ( Θ, kY) jm (θφ,) 11 rr ˆˆFrnk ( ) llFrnk ( ) Hkkr( 12,) Ψ⇒( ,,θφ) ( HkYEkYso−− ph ) ( 2)jm (θφ,) =Θnc per: u ( , 2) jm (θφ,) so− ph nk 22 rr (67) ˆ Frnk ( ) l (Hso− ph ( kk12,)) Yjm (θφ,0) = 33 r Hˆ ( rab,,,χ ) mag− mt E(χ,, nmD , , r) and E(χ,, nmD , , r) The secondmag-ph original results00 are the inducedmag-ph the modified00 new magnetic Hamiltonian operator and corresponding eigenvalues , respectively as: Frnk ( ) l −+22l Emag-ph (χ,, nmD ,0 , r 0 ) Yjm (θφ,) χD BL 0 Fnk( rY) jm (θφ, ) ˆ 0 rr00r H( rab,,,χ) Ψ≡( r ,, θφ) − = mag− mt nk 43 l r Grnk ( ) l rr2 0 BL iGnk ( r) Y jm (θφ, ) Emag-ph (χ,, nmD ,0 , r 0 ) i Yjm (θφ,) r (68)
Θ→0 Conflict of Interest None. It is worth to mention that (in the limit ) we obtain the commutative result of the relativistic negative energy eigenvalues References and positive energy eigenvalues under pseudo spin symmetry and spin-symmetry in addition to the relativistic Hamiltonian Concludingoperator for PH Remarks potential. 1. Maghsoodi E, Hassanabadi H, Zarrinkamar S (2013) Exact Solutions of the Dirac Equation with Poschl-Teller Double-Ring Shaped Coulomb Potential via the Nikiforov-Uvarov Method. Chinese Physics B 22(3): 030302-1-030302-5. In this paper we have performed the exact analytical bound state solutions: the energy spectra and the corresponding NC 2. Biswas B, Debnath S (2016) Bound States of the Dirac-Kratzer- Hermitian Hamiltonian operator for three dimensional MDE in Fues Problem with Spin and Pseudo-Spin Symmetry via Laplace 3. spherical coordinates for MPH potential by using Bopp’s Shift Transform Approach. Bulg J Phys 43: 89-99. method and standard perturbation theory.jl=±=± It is1/2, found jl that 1/2 the Mohammad Reza Shojaei, Ali Akbar Rajabi, Maryam Farrokh, energys = ±1 /eigenvalues 2, ll , m and depend m on the dimensionality of the problem Niloufar Zoghi-Foumani (2014) Energy levels of spin-1/2 particles and new atomicΘ and quantum χ numbers ( , with Yukawa interaction. Journal of Modern Physics 5: 773-780. , ) in addition to the two infinitesimal 4. Sameer M Ikhdair, Majid Hamzavi (2012) Effects of external parameters(22( l+ 1) and 22( l),+ 1 and) we also showed that the obtained fields on a two-dimensional Klein Gordon particle under pseudo- th energy spectra degenerate and every old state will be replacedn harmonic oscillator interaction. Chin Phys B 21(11): 110302. by sub-states under the pseudo spin symmetry and spin symmetry conditions, respectively, for 5. Biswas B, Debnath S (2016) Bound States for Pseudoharmonic Potential of the Dirac Equation with Spin and Pseudo-Spin exited states. Acknowledgement Symmetry via Laplace Transform Approach. Acta Physica Polonica A 130(3): 692. 6. Connes A, Douglas MR, Schwarz A (1998) Noncommutative This work was supported with search laboratory of: Physics geometry and matrix theory: compactification on tori. and Material Chemistry, in Physics department, Sciences faculty, University of M’sila, Algeria, The author would like to thank the 7. Djemei AEF, Smail H (2004) On Quantum Mechanics on kind’s referees for the positive comments and suggestions which Noncommutative Quantum Phase Space, Commun. Theor Phys (Beijinig, China) 41: 837-844. led to the improved standard of this article.
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Maireche A (2018) A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One-electron Atoms at Planck’s and Nano-Scales. J Nanomed Res 7(1): 00168. DOI: 10.15406/jnmr.2018.07.00168 A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One- Copyright: 18 electron Atoms at Planck’s and Nano-Scales ©2018 Maireche
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Maireche A (2018) A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One-electron Atoms at Planck’s and Nano-Scales. J Nanomed Res 7(1): 00168. DOI: 10.15406/jnmr.2018.07.00168 A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One- Copyright: 19 electron Atoms at Planck’s and Nano-Scales ©2018 Maireche
39. Pihoiq (2016) Potential in Noncommutative Spaces and Phases 44. Abdelmadjid Maireche (2017) The Exact Nonrelativistic Energy Symmetries at Plank’s and Nanoscales. J Nano Electron Phys 8(1): Eigenvalues for Modified Inversely Quadratic Yukawa Potential Plus L 02027-1 -02027-10. Mie-type Potential. J Nano- Electron Phys 9(2): 02017-1- 02017-7. 40. Abdelmadjid Maireche (2016) A New Nonrelativistic Atomic 45. Abdelmadjid Maireche (2017) New Nonrelativistic Quarkonium Energy Spectrum of Energy Dependent Potential for Heavy Masses in the Two-Dimensional Space-Phase using Bopp’s shift Quarkouniom in Noncommutative Spaces and Phases Symmetries. Method and Standard Perturbation Theory. J Nano Electron Phys J Nano- Electron Phys 8(2): 02046-1 - 02046-6. 9(6): 06006-1-06006-9. ödinger Diatomic 41. Abdelmadjid Maireche (2016) A new Nonrelativistic Investigation 46. Abdelmadjid Maireche (2017) Effects of Two-Dimensional for Interactions in One-electron Atoms with Modified Vibrational- Noncommutative Theories on Bound States Schr Rotational Analysis of Supersingular plus Quadratic Potential: Molecules under New Modified Kratzer-Type Interactions, Extended Quantum Mechanics. J Nano Electron Phys 8(4): 04076- International Letters of Chemistry. Physics and Astronomy 76: 1-04076-9. 1-11. ö 42. Abdelmadjid Maireche (2015) Spectrum of Hydrogen Atom Ground 47. Gradshteyn IS, Ryzhik IM (2007) Table of Integrals, Series and State Counting Quadratic Term in Schr dinger Equation. Afr Rev Products, pp. 1200. Phys 10: 177-183. 43. Abdelmadjid Maireche (2016) New Exact Energy Eigen-values for (MIQYH) and (MIQHM) Central Potentials: Non-relativistic Solutions. Afr Rev Phys 11: 175-184.
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Maireche A (2018) A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One-electron Atoms at Planck’s and Nano-Scales. J Nanomed Res 7(1): 00168. DOI: 10.15406/jnmr.2018.07.00168