An Algebraic Operator Approach to Aharonov-Bohm Effect

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1 America Joural of Moder Physics 5; 4(): Published olie February 5 5 ( doi: 648/jajmp54 ISSN: (Prit); ISSN: (Olie) A Algebraic Operator Approach to Aharoov-Bohm Effect Farri Payadeh Departmet of Physics Payame Noor Uiversity (PNU) PO BOX Tehra Ira address: Payadehfarri9@gmailcom To cite this article: Farri Payadeh A Algebraic Operator Approach to Aharoov-Bohm Effect America Joural of Moder Physics Vol 4 No 5 pp doi: 648/jajmp54 Abstract: A ew approach based o algebraic quatum operator is pursued i order to ivestigate the Aharoov-Bohm effect Itroducig a SU() dyamical ivariace algebra the discrete spectrum ad the eergy level of the quatum Aharoov- Bohm effect is obtaied This alterative method will help udergraduate studets to broader their kowledge about this iterestig quatum pheomeo Keywords: Quatum Physics Schrödiger Equatio Spherical Coordiates Hyperbolic Coordiates Aharoov-Bohm Effect Operator A Mathematical Itroductio to Aharoov-Bohm Effect The Aharoov-Bohm effect demostrates that it is ot possible to describe all electromagetic pheomea i terms of the field stregth oly This effect is observed for example i a electro double slit experimet Let us cosider the double slit experimet sketched i figure The magetic field B at the ceter is cofied to a arrow tube such that the electros move i a field-free regio Figure The Aharoov-Bohm effect This implies that classically oe would ot expect to see ay effect sice fields iteract oly locally It turs out however that there is a quatum mechaical effect Due to the vector potetiala the electro iterferece patter o the scree o the right shifts over a distace proportioal to the magetic flux [] Free particles i a magetic field are described by the Schrödiger equatio This method of splittig the wave fuctio i two parts is a semi-classical approximatio sice we igore the effects of diffractio Aharoov ad Bohm i their origial paper [] also solved the problem without splittig the wave fuctio i two parts Moreover Lee Page [3] has solved the problem of a free charged particle movig i a magetic field ad he foud a solutio which is fiite i the origi I additio to above viewpoits the Aharoov Bohm effect ca be uderstood from the fact that we ca oly measure absolute values of the wave fuctio [3-5] However by gauge ivariace it is equally valid to declare the zero mometum eigefuctio to be i ( x) e φ at the cost of represetig the i -mometum + i φ ie with a pure operator (up to a factor) as ( ) i i i gauge vector potetial A dφ [67] Therefore the Aharoov Bohm effect maifests itself as a coectio with flat space ad topologically otrivial [8-] Effects with similar mathematical iterpretatio ca be foud i other fields For example i classical statistical physics quatizatio of a molecular motor motio i a stochastic eviromet ca be iterpreted as a Aharoov Bohm effect iduced by a gauge field actig i the space of cotrol parameters [] I this paper however we deal with a differet mathematical approach to the geeralized Aharoov-Bohm effect amely the algebraic operator method This method itroduced i [3] provides the discrete spectrum ad the eergy level of the quatum Aharoov-Bohm effect applyig a SU() dyamical ivariace algebra This method has bee geeralized for coupled Aharoov-Bohm-Coulomb effects

2 America Joural of Moder Physics 5; 4(): [4] ad also for oscillatig systems [5] Therefore it appears to be a powerful mathematical method for rederivatios ad geeralizatios of quatum pheomea Note also that such approaches have bee cosidered i treatig Schrödiger equatio (for example see [6]) The paper is orgaized as follows: I sectio we review the traditioal way of solvig three dimesioal Schrödiger equatio ad we treat the Aharoov-Bohm effect usig the wave fuctios I sectio 3 we itroduce the so-called operator algebra ad revise the foudatios of the Aharoov-Bohm effect i a Coulomb field We coclude i sectio 4 ad also make a outlook Geeralized Aharoov-Bohm (AB) Effect I this sectio we deal with the Schrödiger equatios for a charged particle which is subjected to aother Coulomb potetial exerted by aother chargee ad the followig AB potetial (our approach is based o what has bee developed i [3]): Ar Aθ Aφ πrsiθ where is the flux The magetic field i the cylidrical coordiates reads as B k δ ( ρ ) πρ with δ ( ρ ) as the Dirac delta fuctio for which the flux is give by Bds Moreover the Schrödiger equatio i the presece of a electric charge ad the AB potetial is E ( i ea) Or i spherical coordiates A πr si θ φ Accordig to axial symmetry the eergy spectrum is idepedet of the azimuth quatum umberm therefore oe ca write the m -depedet part as So imφ R y e ( r θ φ ) ( r) ( θ ) () () (3) (4) (5) (6) im A πr si θ Accordigly the Schrödiger Hamiltoia becomes: Hˆ ( θ ) (7) + VA r (8) Fially the radial part of Schrödiger equatios i spherical coordiates ad i the presece of AB potetial will be ( ) R dr l l d dr + + R + + E R r dr r Which turs out that could be rearraged to ( + ) d u l l + E u + + dr r r (9) () where u ( r) rr ( r) Accordig to this the effective potetial would be V eff l l + r r ( + ) ( ) () l l + Note that is a Coulomb potetial while is a r r repulsive oe Moreover oe ca redefie the radial solutio i the followig form: where ad ( ) R r e a l H ( ρ ) ρ ρ () ( ρ ) ρ (3) H a + l + a + λ ( + )( + l + ) (4) Therefore the leadig term could be foud at the extet poits which ear e ρ l ad at ifiity behaves like ρ I other poits the leadig term is amed ( ) H ρ equatio chages to d H d ρ l + dh λ l ρ d ρ ρ + + H ( ρ ) So our (5) which has to be solved Usig the recursio relatio a+ + l + λ oe ca fid a ( + )( + l + ) ( l ) λ + + (6)

3 46 Farri Payadeh: A Algebraic Operator Approach to Aharoov-Bohm Effect ad l + + (7) for which ad l Therefore l + or l This may lead us to coclude Also the eergy eigestates are derived as ( ) mi E (8) 4 Now let us tur to the agular part of Schrödiger equatio I spherical coordiates we have ( m + β ) y siθ + y ( θ ) l ( l + ) (9) y ( θ ) siθ θ θ si θ which ca be rewritte as d dy γ + dx dx + x () ( x ) x l ( l ) y ( x) where γ ( m β ) + ad x cosθ The above equatio is the associated Legedre differetial equatio which ca be trasformed to the Hypergeometric differetial equatio usig the chage i variables Here we just poit out the importat results of both methods I the chage i variables method we put ad The we get ( ) ( ) z ( x) γ y x x () + x t () d z dz t t t z dt + + dt ( ) + ( γ + ) ( γ + ) ( γ l )( γ l ) (3) which is the Hypergeometric differetial equatio Accordig to this we will have the followig spectrum: As it is see the effect of the AB Coulomb potetial i the Schrödiger equatio is to chage the quatum orbital umber l This term is added to a ad this would be the π shift i the orbital umber Now if we set β the flux vaishes ad thereforel l + m which is the agular mometum of Hydroge electro i usual situatio whereas i the presece of a supercoductor i the regio ofb we have e l + m I this problem we solved the π Schrödiger equatio with the Hamiltoia ˆ H + VA ( r θ ) + where β VA ( r θ ) O the other r si θ had all these results could be obtaied for ˆ H (or Heff p ) but upo this coditio that istead of the quatum orbital umberl we usel I other words istead of the cetrifugal term l ( l + ) l ( l + ) we use Ideed the r r above results show that the effect of the coulomb potetial r i the Schrödiger equatio is that to shift the quatum magetic umberm by a value of ie chages m to π e m Moreover the umber of degeeracies are derived π to be or w l + + (6) π w + (7) π 3 Solvig Schrödiger Equatio i AB Potetial Usig Operator Algebra The Schrödiger equatio for a liear oscillator could be writte as ( ) F γ m + β m π F m (4) π F E m 4 π Fially the solutio of our differetial equatio will be ( γ l ) ( γ l )! ( γ + ) l γ γ + + y ( x) ( x ) ( x) (5) + m x E m x ω Defiig the followig operators: d mω a + x dx d mω a + x dx we get (8) (9)

4 America Joural of Moder Physics 5; 4(): d mω aa + + x dx a a dx d mω + x (3) Accordigly oe ca rewrite the Schrödiger equatios for two eergy levels i eighborhood i terms of the above operators We have E + ω (34) To obtai the groud state itself we must havea ad hece a a This may lead us to mωx exp (35) + mω x + ( + ) ω E m + mω x + ω E m These could be simplified to aa a a m E + ω + + m + E ω (3) (3) Oe ca show that the wave fuctios ad + satisfy the followig recursio relatios accordig to the operator equatios (3): + a m E + ω + a m E + ω (33) Also accordig to (3) the eergy levels of the system would be where π mω 3 Solvig Schrödiger Equatio for Three Dimesioal Oscillator i Spherical Coordiates (36) I spherical coordiates the Schrödiger equatio ca be writte i the followig way: rr( r) ( + ) l l + mω r + E m m r We defie two appropriate operators as d mω l a() l + r dr r d mω l a () l + r dr r (37) (38) Accordigly oe ca calculate a a adaa Like usual oe-dimesioal harmoic oscillator we shift the potetial + ω ad write the correspodig Schrödiger by ( ) equatio for the shifted potetial We have ( + ) mω l( l + ) me + r + + r + l + l + l + l + l mω l( l ) me + r + + r l l l l l (39) Accordig to (4) ad (4) oe ca ifer al () a a l a m () l + l + + E ω l + l m () ( l) l E ω + l + l (4) For the rhs of the secod relatio i (4) to be vaished we must lower the states i l to reach l sice the zero state ad its correspodig mometuml are costructig the groud state of a harmoic oscillator i a potetial shifted by a ω We do this for a sequece of potetials util we reach the state adl l I such state we will have a() l + l l m E + l + ω l a l + l m E + l + ω Equatio (44) may result i () (4)

5 48 Farri Payadeh: A Algebraic Operator Approach to Aharoov-Bohm Effect l 3 7 m E l + ω m E l + ω m E l + + ω (4) a ( l + ) a ( l + ) a ( l + ) Therefore oe must put l l + + i the secod of (4) to get l + m ( ) ( ) ω a l + + a l + + l + E l + + l + (43) hece 3 The Parabolic Couterpart E ω l + + (44) I parabolic coordiates the Schrödiger equatio ca be writte as 4 m ( ) 4 π u i Eζ + u cu (45) ζ I this case the appropriate operators ca be defied as d l Al () + i Eζ dζ d l () + i Eζ dζ A l (46) The oe dimesioal harmoic oscillator has to be solved i a potetial shifted by a4 i E( + ) The same process as it is i subsectio 3 will result i E m (47) 4 π which is the same result as it is expected from (45) Here istead of solvig the differetial equatio the operator method has bee used 4 Coclusio Our aim i this paper was the ivestigatio of the captured magetic flux whe it is applied o the eergy spectrum of a bouded charged particle i the presece of exteral oscillator or coulomb potetials As it was show the eergy spectrum ad their correspodig quatum umbers are shifted by a ad also the degeeracies are decreased π Ideed applyig the operator method we could aticipate this pheomeo which is i agreemet with the usual mathematical methods The method applied i this paper has bee also of iterest i geeralized Aharoov-Bohm effect i Coulomb ad oscillatig systems [4 5] scatterig states [7] boud states of Schrödiger equatio [8 9] ad i a graphee rig [] Therefore Aharoov Bohm Coulomb problem which was the mai subject of ivestigatio of this paper ca be regarded i two differet ways Oe by use of the traditioal method of solvig Shrodiger eqautio for the scalar couplig ad the other by applyig the operator algebra itroduced i sectio 3 for most importat coordiate systems It has bee foud that the eergy spectrum of the system demostrates that the degeeracies are dimiished ad it is clear from the spectrum of the AB potetial It is therefore of iterest for further works to geeralize this algebraic operator method for some other quatum effects like the oes i which the zero state eergy (or the vacuum eergy) is the cause of physical pheomea like the Casimir effect I this case the operators should be geeralized to Dirac particles ad their creatio-aihilatio process should be icluded as well However the metioed graphee rig i referece [] is a good example of such geeralizatio Therefore it seems that the doors are ope for further isights ito this methodical approach Refereces [] S Gasiorowicz: Quatum physics Joh Wiley (974) [] Y Aharoov D Bohm: Physical Review 5: (959) [3] L Page: Physical Review (93) [4] A Batelaa A Toomura: Physics Today 6 (9): (9) [5] E Sjöqvist: Physical Review Letters 89 (): 4 () [6] W Ehreberg RE Siday: Proceedigs of the Physical Society Series B 6: 8 (949) [7] FD Peat: Ifiite Potetial: The Life ad Times of David Bohm Addiso-Wesley ISBN (997) [8] Y Aharoov D Bohm: Physical Review 3: 5 54 (96) [9] M Peshki A Toomura: The Aharoov Bohm effect Spriger-Verlag ISBN (989)

6 America Joural of Moder Physics 5; 4(): [] L Vaidma: Physical Review A 86 (4): 4 () [] R Feyma: The Feyma Lectures o Physics pp 5 5 (964) [] VY Cheryak NA Siitsy: Joural of Chemical Physics 3(8): 8 (9) [3] HD Doeber E Papp: Physics Letters A Pages (99) [4] Gh E Dragaascu C Campigotto M Kibler: Physics Letters A (99) [6] M R Brow B J Hiley: arxiv:quat-ph/56v5 [7] S Sakoda M Omote: Advaces i Imagig ad Electro Physics 7 (999) [8] M Aktaş: Iteratioal Joural of Theoretical Physics (9) [9] M Aktaş: Joural of Mathematical Chemistry () [] E Jug M-R Hwag C-S Park D Park: Joural of Physics A: Mathematical ad Theoretical () [5] M Kibler C Campigotto: Physics Letters A 8 6 (993) H