A novel exact solution of the 2+1-dimensional radial Dirac equation for the generalized Dirac oscillator with the inverse potentials

Transcription

1 A ovel exact solutio of the +-dimesioal adial Diac equatio fo the geealized Diac oscillato with the ivese potetials ZiLog Zhao ZhegWe Log ad MegYao Zhag Depatmet of Physics Guizhou Uivesity 555 Chia Abstact. The geealized Diac oscillato as oe of the exact solvable model i quatum mechaics was itoduced i + dimesioal wold i this pape. What is moe the geeal expessios of the exact solutios fo these models with the ivese cubic quatic quitic ad sixtic powe potetials i adial Diac equatio wee futhe give by meas of the fuctioal Bethe asatz. Ad fially the coespodig exact solutios i this pape wee futhe discussed. PACS:.65.Pm.65.Ge Keywods: adial Diac equatio geealized Diac oscillato Bethe asatz method ivese potetial exact solutio. Itoductio As we all kow the Diac oscillato as a ew cocept which fist came out of Ito ad collaboatos by meas of the combiatio betwee mometum ad coodiate p p im whee is the coodiates m epesets the mass of paticle ad is egaded as the fequecy of paticle [-]. It has bee successfully applied i may fields as a effective elativistic model but it ca oly be used to descibe the ifluece of aomalous magetic momet ad the liea field. Theefoe i ode to descibe the complex iteactios Dutta ad his colleagues [] poposed a ew cocept which is called the geealized Diac oscillato. So it could geealized the liea effect betwee coodiates ad mometum to oliea effect by usig this system that is to say the Diac oscillato could be egaded as a special case i the geealized Diac oscillato. As oe of the few elativistic quatum systems that ca be solved accuately the Diac oscillato has attacted much attetio [-5]. This system could solve ot oly the quak cofiig potetial [5-7] i quatum chomodyamics but also some complex iteactios [8-9]. Cosideig the pacticability of solvig the poblem the Diac oscillato has bee extesively discussed by may eseaches fom may agles: like cofomal ivaiace popeties [8] covaiace popeties [9] shift opeatos [] symmety Lie algeba [] hidde supesymmety Coespodig autho zwlog@gzu.edu.c

2 [-] completeess of wave fuctios ad so o [5-6]. Moeove this model has also bee used to explai may pheomea i Quatum Optics [7-]. The study of the elatioship betwee Diac oscillato ad elativistic Jayes-Cummigs model [ -] bidges the two uelated fields of Relativistic Quatum Mechaics ad Quatum Optics [-]. I additio some ew pheomea i codesed matte physics [6-7 -5] such as the Quatum Hall Effect ad Factioal Statistics ca also be explaied by this model. So it is iteestig to exted the Diac oscillato to the geealized Diac oscillato fo solvig the iteactio betwee mometum ad coodiate i the oliea electic field. I ode to fid the solutio of the +-dimesioal Diac equatio with some complex iteactio potetials the geealized Diac oscillato is costucted by Dutta ad his colleagues though makig a simple tasfomatio p p i ˆ f x. But ow we wat to obtai the solutio of the +-dimesioal adial Diac equatio with the geealized Diac oscillato so that the mometum opeato was tasfomed by makig a eplacemet x i ˆ ad p p i ˆ yf p p xf x y i this pape. Theefoe the coespodig complete solutios fo +-dimesioal adial Diac equatio could be give by the Bethe asatz method [6-9] whe the iteactios is popely chose. Next this pape is stuctued as follows. The +-dimesioal geealized Diac oscillato is y itoduced i Sectio. I Sectio the exact solutios of the adial Diac equatio with the geealized Diac oscillato ae give by usig the fuctioal Bethe asatz method [6-9] whe some appopiate iteactios last sectio is ou coclusio. f ae chose. Ad the. +-dimesioal geealized Diac oscillato Fo a paticle whose mass is M the Diac equatio is witte as follow i tems of two compoet spios [-] ˆ ˆ E p M () whee ˆ ad ˆ ˆ ae matices that they ae costituted by the Pauli spi matices ˆ ˆ ˆ ˆ ˆ s ˆ ˆ ˆ z () ad the paamete s takes the values ( fo spi up ad fo spi dow). The fomalism () ca tasfom two ucoupled two-compoet Diac equatios fo s o s by usig the decouplig fo the fou-compoet Diac equatio i the absece of the thid spatial coodiate. Theefoe oe of the two-compoet Diac equatios could be used to descibe the positive eegy eigestate ad aothe to descibe the egative eegy eigestate. Ad the positive eegy eigestate is what usually called the paticle state ad the coespodig egative eegy eigestate is called the atipaticle state. Due to thee is oly oe spi polaizatio i the case of +-dimesioal space so the equatio with the Diac oscillato fo the paticle state ca be

3 witte as follow [-] ˆ i ˆ E p M () ˆ M z whee the matices ad the eigefuctio satisfy the oscillato Ove the past decades the sigula potetials have attacted much attetio because they ca be used to descibe may physics poblem. Fo example followig coditios i ˆ ˆ ˆ ˆ x y. i T () the sigula potetials have ot oly bee applied widely i the eseach of the p p ad p pocedues i high-eegy physics [9-5] but also the epulsive sigula potetials could epoduce by But ow the mometum opeatos ae eplaced p p i ˆ xf ( ) ad p p i ˆ yf ( ) x x i the fee adial Diac equatio so that the coespodig adial Diac equatio with geealized oscillato is give as follow E ˆ ˆ x px ixf ˆ i ˆ ˆ y py yf zm. Accodig to the equatio () the equatio (5) ca be witte as follow E M p xf p yf x E M p xf p yf x i i y i i y. So the above equatio ca be witte the followig fomalism f Lz f f f whee E M. Next we will use the fuctioal Bethe asatz method to futhe discuss the exact solutio of the system i the followig sectio.. Explicit implemetatio of geealized Diac y y (5) (6) (7) the iteactios of scatteig ad ucleos with K-mesos [9 6]. Moeove as a example of o-elativistic quatum mechaics the questio of high eegy scatteig caused by stog sigula potetials has also bee discussed extesively by may authos [9 7-8]. Besides the sigula potetials wee also widely used to descibe the iteeactio of two atoms i molecula physics iteatomic o itemolecula foce ad chemical physics [9]. Cosideig the wide applicatios of the sigula potetial the exact solutios of the +-dimesioal adial Diac equatio with the geealized Diac oscillato ude the ivese cubic quatic quitic ad sixtic powe potetial wee discussed i this pape. Ad it is also show that the coclusio of the geealized Diac oscillato with highe ode ivese potetial ca degeeate to the coclusio of lowe ode ivese powe potetial iteactio whe the paametes ae popely selected. Next the aalytical expessios of the coespodig exact solutios ae give i sectio ad the goud state ad the fist excited state ae discussed i

4 futhe detail... Quasi exact solutio of the ivese cubic powe potetial Now let s fist discuss the ivese cubic powe potetial b d e. (8) a f This potetial has bee ivestigated i Schodige equatio by the othe authos to obtai the accuate aalytical expessio [9-]. Now we will study this poblem i the +-dimesios adial Diac equatio so the coespodig adial Diac equatio could be give by substitutig equatio (8) ito equatio (7) a b d e ab i a a ae bd b ad b a be d d e e. (9) The asymptotic behavio of the wave fuctio is extacted though a simple eplacemet ad the exact solutios of the +-dimesioal adial Diac equatio is futhe give. Next by makig a bief examiatio fo diffeetial equatio we implemet tasfomatio g exp A B D im () equatio () ito diffeetial equatio (9) we fid that the paametes satisfy the udelyig elatios m d A a D e. Theefoe ew diffeetial equatio could be witte as follow g a B e g a a B b B B ae b be Be g. () whee the paamete satisfy m d. Obviously the equatio () could also be solved accuately by usig the fuctioal Bethe asatz method [6-9] if the potetial paametes satisfy cetai costaits. Now i ode to ealize the squae itegable of wave fuctios we assume that B o a the the degee polyomials solutios of the equatio () could be give g Case oe: Bb g fo. () I this sectio the equatio () would be tasfomed ito the followig fom a a g g a e g ae (). Substitutig equatio () ito equatio () ad usig the Bethe asatz method it is easy to pove that the paametes satisfy the followig costaits whee the patametes AB ad D ae costat ad m is magetic quatum umbe. Substitutig a ae a

5 a ad the Bethe asatz equatio ca be give as follow k k a e.... Theefoe the aalytical expessios of the wave fuctios ad eegies could be witte as E M a a e () exp i m. The above wave fuctios ae squae itegable so the coespodig omalizatio costats could also be give though the stadad itegal [] exp d whee ad G Re Re Re m G pq is the Meie G-fuctio. Thus the total wave fuctios could be witte as a e exp i m ad hee the paamete was set as E px ixf ipy yf M T. Next the coespodig solutio of goud state ad fist excited state could be give diectly by usig the expessio of the exact solutio. Obviously the polyomial solutio of the equatio () satisfies g if we take. The the aalytical solutio of the goud state could be give as follow E M a e m exp i (5) with ae. The case of coespods to the fist excited state solutio of the system ad the aalytical expessios of the wave fuctio ad the eegy spectum ae futhe give E M a (6) exp i a e m i which the potetial paametes meet the followig costaits ae a a. Case two: Aa I this sectio we assume that Aa. Obviously if the potetial paametes satisfy cetai costaits the exact solutio of the system could be give by usig the Bethe asatz method [6-9]. Ad the coditio of the wave fuctio has a acceptable asymptotic behavio whe is that the paamete B must be take to be zeo. So this diffeetial equatio eads whee g B e g B B B b eb b g (7) b. Substitutig equatio () ito equatio (7) the followig elatios could be give by usig the Bethe asatz method B b 5

6 B eb B ad the Bethe asatz equatio could be witte as follow k k B e.... As metioed above the paamete B is egative so the paamete b must also be egative. Theefoe the aalytical expessios of the wave fuctios ad eegies could be witte as E M b (8) e expb i m. The wave fuctios ae squae itegable ad its coespodig omalizatio costats could also be detemied by stadad itegal [] fo exp d Bessel K Re Re total wave fuctios wee witte Re. So the e expb im. Next the goud ad the fist excited state as examples of the exact solvable systems ae ivestigated i detail. Obviously the polyomial solutio of the equatio (7) could be witte as g fo the case the eegy spectum ad wave fuctio could be witte as follow 6 8 B e m E M b exp i whee the costait satisfy eb B. The case of coespods to the fist excited state solutio of the system ad the aalytical expessios of the wave fuctio ad the eegy spectum ae futhe give B e m E M b exp i with the udelyig costait fo the potetial paametes B e B b. Hee the oot ca be futhe give by usig the Bethe asatz equatio B e. 6Be B.. Quasi exact solutio of the ivese quatic powe potetial The iteactio of the Diac equatio with ivese quatic powe potetial is cosideed i this pat b e f a a e. (9) Fom the pheomeological poit of view sigula potetial as a vey useful fom of ahamoicity could be used i may aspects of physical applicatios [-]. So the ivese quatic potetial has also bee ivestigated i may diffeet questios by lots of authos [-6]. Now the

7 pupose of ou study is to obtai the aalytical popeties of the scatteig amplitude about sigula potetial. So the coespodig +-dimesioal adial Diac equatio with ivese quatic powe potetial is give by equatio (7) b e i a a ab a ae b eb e 6. () I ode to deal with the questio the appopiate asymptotic behavio of the wave fuctio is extacted though a simple eplacemet m b hexp a e im whee the paamete satisfy mb. So this diffeetial equatio fo h eads e ae a h h a h () whee the paamete is set m b. Hee the equatio () ca be tasfomed ito a solvable fom by makig a ew vaiable t t h t at t eh t at ae ht. () So the degee polyomial solutios fo the diffeetial equatio ead h t t t ht fo () with diffeet oots t if potetial paametes fulfil cetai costaits. Thus the aalytical expessios fo eegy states ad wave fuctios could be give E M a mb t () exp a e i m. Ad the potetial paametes satisfy the followig estictive coditios a t ae the oots t could be detemied by the Bethe asazt equatios k t tk t at t e.... The wave fuctios ae squae itegable ad its coespodig omalizatio costats could be detemied by stadad itegal [] fo exp d Bessel k Re Re Re. As special cases of geeal expessios we focus o studyig goud state ad fist excited state systems i this sectio. Fist let s give the coespodig wave fuctio ad goud state eegy spectum i tems of the equatio () 7

8 E M a (5) exp i m b a e m whee the potetial paametes comply with ae. The case of coespods to fist excited state solutio of the system ad the aalytical expessios of the wave fuctios ad eegy spectum fo the fist excited state ae futhe witte E M a mb t exp a e i m (6) i which the diffeet oots t could be computed aalytically by the Bethe asazt equatio [6-9] at t e t m b ae a ad the potetial paametes satisfy the followig estictive coditio m b ae 5 m b ae... Quasi exact solutio of the ivese quitic powe potetial Now the ivese quitic powe potetial will be discussed i this sectio b d e f g g. (7) a 5 f By givig the exact solvable fom of the adial Diac equatio i tems of equatio (7) the coespodig exact solutios of adial Diac equatio with ivese quitic powe potetial could be futhe give if a a b b b b ae af be e ag bf e eg f 6 f f bg e fg 7 g ef g 5 8 (8) whee the paamete satisfy m d. Simila to the studies i the pecedig sectios the fist coditio fo obtaiig the exact solutios is to extact appopiate asymptotic behavio fom the wave fuctio. Hee we set exp k A B (9) D F G i m whee A B D F G ae costat paametes ad m is the magetic quatum umbe. Substitutig equatio (9) ito equatio (8) we fid the followig equiemets fo paametes m d A a D e F f G g. So the diffeetial equatio (8) will degeeate ito the followig fom 5 k a B e f g 7 6 a B ae b b ag f f g k k B a b 7B 9 af 6 e e () 8

9 whee B b. It is easy to fid that equatio () is exactly solvable whe we makig Bb o a ad its coespodig exact solutios could also be give by the degee polyomials k Case oe: Bb k fo. () I this case the diffeetial equatio () would E M a exp a () e f g m i. The wave fuctios ae also squaely itegable: d. The goud ad fist excited state will be discussed hee as two special cases. The case of coespods to the goud state become ito 5 k a e f g 7 k 6a a ae f af e ag k 9 6. () solutio of the system E M exp a e f g m i. () Substitutig equatio () ito equatio () ad The fist excited state solutio ca be give by usig the Bethe asatz method the equiemets of the paametes as follow a a ae makig E M a a e exp f g m i (5) a af a 5 e ag a 5 e f k k i which the oots fulfil Bethe asatz equatio 5 a e f g 7 k k whee.... Ad its exact solutios could be witte 9 i which the oot could be computed aalytically by Bethe asatz equatio a 7 e f g 5 ad whose paametes satisfy the followig elatios a ae a 8 af a 7 6e ag a 7 e f. Case two: Aa I this case we set Aa. I ode to get acceptable asymptotic behavio fo the wave fuctio whe the paamete B

10 is equied to be egative. Futhemoe accodig to be egative. Next i ode to satisfy the wave the Bethe asatz method we set B b. fuctios squae itegable we assume that g. It Thus diffeetial equatio () could be witte as k B e f g k B B b 9 6 f f g k. 7 7 b e e (6) Substitutig equatio () ito equatio (6) ad usig the Bethe asatz method the exact solutios of equatio (6) ca be give as followig E M 6 exp 7 B b (7) e f g m i ad the paametes satisfy the detemied costaits 7 B b B e B 5 B 5 e e f f f g e k k i which the oots satisfy Bethe asatz equatio B e f g 7 k k whee.... As we metioed above the paamete B is egative so the paamete b must is t difficult to use Maple fo checkig umeically. Simila to the pevious cases the goud state solutio could be witte fo expb e E M 6 b 7 with the costaits e f f g m i f g e B b 7. (8) The fist excited state solutio also could be witte by settig expb E M 8 b 9 comply with the costais e f g m i B 8 e B 9 b B 7 6e f B 7 6 f g e. Hee the oot satisfyig B 7 e f g. (9).. Quasi exact solutio of the ivese sixtic powe potetial The effect of the ivese sixth powe potetial i the adial Diac equatio is discussed as the last case i this sectio

11 d f h f a () 6 whee makig h. This potetial has played a impotat ole i the study fo paticles iteactios z y z az z z fz h y z f ah m d af z a am ad z y z 8. () i atomic molecula ad uclea physics [7-9]. Accodig to equatio (7) the coespodig adial Diac equatio could be give d f h i a 6 a ad a af d ah df f fh 8 h dh h f 6. () Simila to the discussios i the pevious sectios fo extactig appopiate asymptotic behavio fom the wave fuctio we settig y exp im a f h () whee the paamete satisfy m d. The accodig to equatio () the diffeetial equatio fo y () is witte as follow f h y a y 5 a m d af m d y f ah. () The vaiable is eplaced by z ad multiplied by z o both sides of equatio () so that the equatio ca degeeate ito the followig fom It is easy to fid that the degee polyomial solutios of the diffeetial equatio () satisfy y z z z y z fo. (5) Substitutig equatio (5) ito equatio () ad applyig the Bethe asatz method ad the aalytical expessios of the eegy spectum ad the wave fuctios could be give E M a m d z exp a (6) f h im ad the potetial paametes satisfy the followig costaits a z m d af a z m d z f ah whee the oots z ae also could be give by Bethe asatz equatios k z zk z az z fz h whee.... The above fuctios ae also squaely itegable i.e. d. It is t difficult to use Maple fo checkig umeically.

12 Obviously the goud state solutio could be give by settig E M a m d a f h (7) exp im with the costaits md af f ah. The case of coespods to the fist excited state solutio of the system ad the aalytical expessios of the eegy spectum ad the wave fuctio could be give as followig E M a m d z a exp f h im ad the potetial paametes satisfy the costais af az az z f ah. (8) Hee the oot z ca be futhe give by usig the Bethe asatz equatio. Coclusio az z fz h. I the peset wok the +-dimesioal adial Diac equatio with the geealized oscillato is itoduced ad the coespodig eegy spectums ad the wave fuctios ca be give by Bethe asatz method. Fo sigula potetial geeal expessios of exact solutios ae give by takig cubic quatic quitic ad sixth ivese powe potetials as examples. By usig the geeal expessios of exact geealized Diac oscillatos the coespodig eegy spectums ad wave fuctios of the goud state ad fist excited state ae futhe give. By compaig the geeal expessios of exact solutios of ivese powe potetials we also easily fid that the esults i sectio. could degeeate ito the coclusios fo the ivese cubic powe potetial i sectio. whe makig paametes f g. Ad if we makig the paametes b e g the esults i sectio. could also be epoduce those solutios of the ivese quatic powe potetial i sectio.. 5. Ackowledgemets This wok was suppoted by the Natioal Natue Sciece Foudatio of Chia (Gat Nos. 656). Refeeces. M.M. Stetsko. J. Math. Phys 56 (5).. M. Moshisky ad A. Szczepaiak. J. Phys. A: Math. Ge L87 (989).. D. Ito K. Moi ad E. Caiee. Nuovo. Cimeto. A 5 9 (967).. D. Dutta O. Paella ad P. Roy. A. Phys (). 5. Boumali A. Phys. Sc (7). 6. P.A. Cook. Lett. Nuovo Cimeto 9 (97). solutios of the adial Diac equatio with the

13 7. Y. Nedadi ad R.C. Baett. J. Phys. A: Math. Ge 7 (99). 8. R.P. Matiez-y-Romeo A.L. Salas-Bito. J. Math. Phys 8 (99). 9. M. Moeo ad A. Zetella. J. Phys. A: Math. Ge L8 (989).. O.L. de Lage J. Phys. A: Math. Ge 667 (99).. C. Quese ad M. Moshisky. J. Phys. A: Math. Ge 6 (99).. J. Beckes ad N. Debegh. Phys. Rev. D 55 (99).. C. Quese. It. J. Mod. Phys. A (99).. J. Beítez R.P. Matíez y Romeo H.N. Núez-Yépez ad A.L. Salas-Bito. Phys. Rev. Lett 65 85(E) (99). 5. R. Szmytkowski ad M. Guchowski. J. Phys. A: Math. Ge 99 (). 6. C. Quese ad V.M. Tkachuk. J. Phys. A: Math. Ge 8 77 (5). 7. O. Aouadi Y. Chagui ad M.S. Fayache. J. Math. Phys 57 5 (6). 8. A. Bemudez M.A. Mati-Delgado ad E. Solao. Phys. Rev. Lett 99 6 (7). 9. L. Lamata E. Solao T. Schatz ad J. Leo. Phys. Rev. Lett (7).. A. Bemudez M.A. Mati Delgado ad E.. E. Sadui J.M. Toes ad T.H. Seligma. J. Phys. A 85 ().. W. Geie. Relativistic Quatum Mechaics: Wave Equatios. Beli: Spige.99.. J.D. Boke ad S.D. Dell. Relativistic Quatum Mechaics. New Yok: McGaw-Hill V.M. Villalba. Phys. Rev. A (99). 5. G.X. Ju ad Z. Re. It. J. Mod. Phys. A (). 6. Y.Z. Zhag. J. Phys. A (). 7. P.B. Wiegma ad A.V. Zabodi. Phys. Rev. Lett 7 89 (99). 8. R. Sasaki W.L. Yag ad Y.Z. Zhag. SIGMA 5 (9). 9. D. Agboola ad Y.Z. Zhag. A. Phys 6 ().. D. Solomo. Ca. J. Phys 88 7 ().. A. Eemko L. Bizhik ad V. Loktev. A. Phys (5).. S.P. Gavilov D.M. Gitma ad J.L. Tomazelli. Eu. Phys. J. C 9 5 (5).. Y. Sucu ad N. Üal. J. Math. Phys 8 55 (7).. G. Tiktopoulous. Phys. Rev 8 B55 (965). 5. C.B. Kouis. Nuovo. Cimeto 598 (966). Solao. Phys. Rev. A 76 8 (7).

14 6. R.M. Specto ad R. Chad. Pog. Theo. Phys 9 68 (968). 7. A. Pais T.T. Wu. Phys. Rev B (96). 8. G. Esposito. J. Phys. A 99 (998). 9. T.O. Mülle H. Fiedich. Phys. Rev. A 8 6 ().. A.D. Alhaidai. Boud states of a shot-age potetial with ivese cube sigulaity. 8.. I.S. Gadshtey ad I.M. Ryzhik. Table of Itegals Seies ad Poducts. Academic Pess Lodo M. Zoil. J. Math. Phys (989).. M. Zoil. J. Math. Phys 8 (99).. G.R. Kha. Eu. Phys. J. D 5 (9). 5. A. Luge C. Neue. Moatsh. Math 8 95 (6). 6. S.H. Dog. Phys. Scipta 6 7 (). 7. B.H. Basde C.J. Joachai. Physics of Atoms ad Molecules. Logma Lodo G.C. Maitla M. Rigby E.B. Smith W.A. Wakeham. Itemolecula Foces Oxfod Uivesity Pess Oxfod B. Laid A.D. Haymet. Mol. Phys 75 7 (99).