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A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One-Electron Atoms at Planck’S and Nano-Scales

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A New Relativistic Bound States Solution for Modified Pseudoharmonic Potential in One-Electron Atoms at Planck’S and Nano-Scales 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 thspin-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22( l+ 1) Received: | Published: have also shown that, the usual states in ordinary 3-.dimensions are canceled and has been replaced by new degenerated sub-states under November 24, 2017 January 12, 2018 the pseudo spin symmetry and spin symmetry conditions, respectively in the new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 Submit Manuscript | http://medcraveonline.com 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 jmand 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 drthe 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 2for 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.
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