An Algebraic Approach to the Harmonic Oscillator Plus an Inverse Square Potential in Three Dimensions

Transcription

1 Aeica Joual of Applie Scieces (): ISSN Sciece Publicatios 5 A Algebaic Appoach to the Haoic Oscillato Plus a Ivese Squae Potetial i Thee Diesios Shi-Hai Dog a M. ozaa-cassou Pogaa e Igeieía Molecula Istituto Meicao el Petóleo ázao Cáeas Méico D. F. Méico Abstact: The eigefuctios a eigevalues of the thee-iesioal Schöige equatio a haoic oscillato plus a ivese squae iteactio ae obtaie. A ealizatio of the lae opeatos fo the wave fuctios is stuie. It is fou that these opeatos satisfy the coutatio elatios of a SU() goup. The close aalytical epessios fo the ati eleets of iffeet fuctios a betwee the eigevalues a eigefuctios fo substitutig PACS ube(s): F Ge a..qs. ae evaluate. Aothe hie syety eploes the elatios i. Key wos: Ivese Squae Iteactio ae Opeatos SU ( ) Goup Mati Eleets INTODUCTION It is well kow that the algebaic etho has bee the subject of the iteest i the wie vaiety of fiels of physics. Systes isplayig a yaical syety ca be teate by algebaic techiques [ 3 4]. With the factoizatio etho [5 6 7] we have establishe the lae opeatos of a quatu syste soe ipotat potetials such as the Mose potetial the Pöschl-Telle oe the pseuohaoic oe the ifiitely squae-well oe a othe quatu systes [ ]. Fo those lae opeatos we ca fially costitute a suitable algeba a siultaeously obtai the ati eleets fo soe elate fuctios. It shoul be aesse that ou appoach is iffeet fo the taitioal oe whee a auiliay vaiable was itouce [9] aely we ca costuct the lae opeatos oly fo the physical vaiable out itoucig ay auiliay vaiable. ecetly the quatu syste fo the oe-iesioal haoic oscillato plus a ivese squae iteactio has bee stuie []. To ou kowlege howeve the hie syety of this quatu syste is a gap to be fille i which is the ai pupose of the peset wok. The Eigevalues a Eigefuctios: Now we ae stuyig the eact solutios which ae ecessay to be use to costuct the lae opeatos a costitute a suitable algeba. Cosie the Schöige equatio a potetial V() that epes oly o the istace fo the oigi: ψ ( θ V ( ) ψ ( θ Eψ ( θ () µ whee µ is the ass of the paticle. I the peset wok the potetial V() is take as the haoic potetial plus a ivese squae iteactio V ( ) µω () µ whee the ω is the fequecy a is the stegth of the eteal fiel. Fo siplicity the atual uits µ ω ae eploye thoughout this pape if ot eplicitly state othewise. Owig to the syety of the potetial we take the wave fuctios the fo E-ail aess: ogsh@yahoo.co E-ail aess: acelo@ 376

2 Aeica J. Applie Sci. () ψ ( θ ( ) Y ( θ (3) l whee the aial wave fuctio () satisfies the followig equatio Defiig the ew vaiable ( ) ( ) l( l ) ( ( )) ( ) E V. (4) equatio (4) ca be e-aage as l( l ) E ( ( (. (5) 4 4 Fo the behavios of the wave fuctios at the oigi a at ifiity we ca take the followig asatz fo the wave fuctios s ( e F( (6) s / ( l / ) (7) whee aothe solutio ( / ( l / ) ) / Eq. (6) ito Eq. (5) allows us to obtai s is ot a acceptable oe i physics. Substitutio of F( (s 3/ F( ( E / s 3/ 4) F( (8) whose solutio is othig but the cofluet hypegeoetic solutios F ( s E / 3/ 4s 3/ ;. Oe ca fially obtai the eigefuctios as s ( N e F ( s E / 3/ 4s 3/ ; (9) a oalize facto N to be eteie below: Fo cosieatio of the fiiteess of the solutios it is show fo Eq. (7) that the geeal quatu coitio is s E / 3/ 4... () fo which we have E ( s 3/ 4) 3/ s () which iplies that the eegy level is equiistat. ecall that whe E / s 3/ 4 is a o-egative itege the cofluet hypegeoetic fuctios ca be epesse by the associate aguee polyoials []: Γ( ) ( ) F ( ; )! Γ( ) fo which togethe the followig ipotat foula []: 377

3 Aeica J. Applie Sci. () e ( ) we ca fially obtai the oalize aial wave fuctios Γ( ) ( ) δ! s s / ( N e ( ()! N. (3) ( s / )! The costuctio of the lae opeatos: We aess the poble of fiig the ceatio a aihilatio opeatos fo the aial wave fuctios () the factoizatio etho. [8-8] the lae opeatos ca be costucte iectly fo the wave fuctios out itoucig ay auiliay vaiable aely we ite to fi iffeetial opeatos the followig popety Specifically we look fo opeatos of the fo ( ) l (. (4) A ( ) B ( (5) whee we stess that these opeatos oly epe o the physical vaiable. To this e we stat by establishig the actio of the iffeetial opeato s ( s ( N e o the wave fuctios () s / ( Oe possible elatio fo the fist eivative of the associate aguee fuctios is give i []. (6) ( ) ( ) ( ) ( ). (7) The substitutio of this epessio ito (6) eables us to obtai the followig elatio s s / N ( N ( Makig use of Eq. (3) a itoucig the ube opeato $\hat {}$ the popety. (8) ( ( (9) we ca efie the followig opeato s () the followig effect ove the wave fuctios ( l ( () whee l ( s / ). () 378

4 Aeica J. Applie Sci. () As we ca see this opeato aihilates the gou state ( ) as epecte fo a step-ow opeato. We ow pocee to fi the coespoig ceatio opeato. Befoe poceeig to o so we shoul ake use of aothe elatio betwee the associate aguee fuctios []. Substitutio of this epessio ito Eq. (6) aits us to obtai ( ) ( ) ( ) ( ) ( ). (3) s 3/ Usig Eq. (3) agai we ca efie the followig opeato ( N N (. (4) s (5) satisfyig the equatio ( ) l ( (6) l ( )( 3/ ). (7) s Theefoe it is show that the wave fuctios ca be iectly obtaie fo the ceatio opeato actig o the gou state ( ) aely ( N ( (8) N (s / )! s ( e!( s / )! (s / )!. (9) We ow stuy the algeba associate to the opeatos a : calculate the coutato [ ]. Base o the esults (-) a (6-7) we ca [ ] ( l ( (3) whee we have itouce the eigevalue 3 l s. (3) 4 We ca thus efie the opeato 3 l s. (3) 4 The opeatos a thus satisfy the coutatio elatios 379

5 Aeica J. Applie Sci. () [ ] [ ] [ ] (33) which coespo to a SU( ) algeba fo the aial wave fuctios. The Casii opeato ca be also epesse as C [ ( ( ) ] ( ( s s / 3/6) (. (34) The hailtoia Ĥ ca be epesse as: H ( s 3/ ). (35) It is show that thee ae fou seies of ieucible uitay epesetatios fo the SU( ) algeba ecept fo the ietity epesetatio []. They ae the epesetatio D ( j) a spectu boue below a above espectively; the suppleetay seies D s Q q ) a the piciple seies Q q ). Sice the eigevalues ( D p ( have the gou state the epesetatio of the yaical goup SU( ) belogs to D ( j) : I j v v j v [( v j)( v j ) ] / j v / [( v j)( v j ) ] j v I j v I j v (36) v j... j <. I copaiso Eqs. (-) (9-3) a (33-34) we have ( s 3/ 4) ( ) j v. j s 3/ 4 v a O the othe ha the followig epessios i tes of the ceatio a aihilatio opeatos a ca be obtaie as a (37a) ( 3/ ). (37b) The ati eleets of these two fuctios ca be aalytically obtaie i tes of Eqs. () (3) a (34) as ( ( ( ) ( s 3/) δ ( ) ( s / ) δ (-) (38a) ( )( s 3/ ) δ ( ) a 38

6 Aeica J. Applie Sci. () ( ( ( ) ( ) ( )( s 3/ ) δ ( s / ) δ (-) 3 δ 4 ( ) (38b) whee the itegal age ( ). It is show that this is a vey siple etho to calculate the ati eleets fo these lae opeatos. Befoe eig this sectio it shoul be poite out that thee eists aothe hie syety fo this quatu syste. As iscusse i ou pevious wok [3] it is fou that the eigefuctios () (the vaiable ) vaish ot oly as i. The substitutio i bus also as also eostates itiate coectios betwee the eegies that is to say the eigevalues E of quatu syste will chage to -E but keepig othe paaetes uchage. CONCUSION I this pape we have stuie the eigevalues a eigefuctios fo the oe-iesioal haoic oscillato plus a ivese squae iteactio a the establishe the ceatio a aihilatio opeatos iectly fo the eigefuctios () the factoizatio etho. We have eive a ealizatio of yaic goup oly i tes of the physical vaiable out itoucig ay auiliay vaiable. It is show that these opeatos satisfy a SU( ) yaic goup. The epesetatio of the bou states of this quatu syste is escibe by the epesetatio D ( j) a spectu bou below. The ati eleets of the iffeet fuctios a ae also aalytically obtaie fo the lae opeatos a. This etho ca be geealize to othe wavefuctios a epesets a siple a elegat appoach to obtai these ati eleets. ACKNOWEDGMENT This wok is suppote by CONACyT Meico. EFEENCES. Aia A. a F. Iachello 974. Iteactig boso oel of collective states I. The vibatioal liit. A. Phys. 99: 53.. Peeloov A.985.Geealize Coheet States a thei Applicatios. New Yok Spige. 3. Wyboue.G Classical Goups fo Physicists. New Yok Wiley. 4. Coope I A itegate appoach to lae a shift opeatos fo the Mose oscillato aial Coulob a aial oscillato potetials. J. Phys. A: Math. Ge. 6: Schöige E. 94. A etho of eteiig quatu echaics eigevalues a eigefuctios. Po..Iish Aca. A 46 No Ifel. a T.E. Hull 95. The factoizatio etho. ev. Mo. Phys. 3:. 7. Haiige G. N. Bessis a G. Bessis 974. Close-fo epessios of ati eleets a eigefuctios fo ae-opeato cosieatios. J. Math. Phys. 5: Dog S.H.. eus a A. Fak. ae opeatos fo the Mose potetial. It. J. Qua. Che. 86: Dog S. H. a. eus. ae opeatos fo the oifie Pöschl-Telle potetial. It. J. Qua. Che. 86: 65.. Dog S.H.. The SU() ealizatio fo the Mose potetial a its coheet states.ca. J. Phys. 8: 9.. Dog S.H.. Algebaic appoach to the Mose potetial a its coheet states. Z. Phys. Che. 6: 3.. Dog S.H. a Z.Q. Ma. The hie syety fo a quatu syste a ifiitely eep squae-well potetial. A. J. Phys. 7: Dog S.H. a Z.Q. Ma. Algebaic appoach to the pseuohaoic oscillato i D. It. J. Mo. Phys. E : Dog S. a S.H. Dog. A alteative appoach to stuy the yaical goup fo the oifie Poschl-Telle potetial. Czech. J. Phys. 5: Dog S. a S.H. Dog. A ealizatio of yaic goup fo a electo i a uifo agetic fiel. It. J. Mo. Phys. E : Dog S.H. 3. The ealizatio of yaic goup fo the pseuohaoic oscillato. Appl. Math. ett. 6: Dog S.H. 4. ealizatio of the yaical goup fo the geealize aguee fuctios. 38

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