Bound states solution of Klein-Gordon Equation with type - I equal vector and Scalar Poschl-Teller potential for Arbitray l State

Transcription

1 AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH Sciece Huβ ISSN: 5-649X doi:.55/aji Boud tate olutio of Klei-Godo Equatio with type - I equal vecto ad Scala Pochl-Telle potetial fo Abitay l State * Akpa N. Ikot Akpa B. Udoimuk ad Loui E. Akpabio Theoetical Phyic Goup Depatmet of Phyic Uiveity of Uyo Nigeia. Depatmet of Phyic Uiveity of Calaba Nigeia. * Coepodig autho demikot5@yahoo.com ABSTRACT We olve the boud tate olutio of Klei-Godo equatio fo Pochl-Telle potetial via the Nikifoov-Uvaov method. We obtai the eegy eige value ad the wave fuctio i tem of hypegeometic fuctio. Keywod: Klei-Godo Equatio equal vecto INTRODUCTION tadad method [4] upe-ymmety ad the hape ivaiace potetial [5] ad the Nikifoov- I theoetical phyic oe of the iteetig poblem Uvaov method [6]. The pupoe of the peet i to obtai a exact olutio of the Schödige pape i to olve the Klei-Godo equatio fo equal Klei-Godo Duffi-Kemme-Petai ad Diac cala ad Vecto type-i Pochl-Telle potetial with equatio fo mixed Vecto ad Scala potetial [-]. abitay agula mometum quatum umbe uig Thee equatio ae fequetly ued to decibe the the Nikifoov-Uvaov method [6]. paticle dyamic i o-elativitic ad elativitic The ogaizatio of the pape i a follow. I ectio quatum mechaic. Howeve the eache fo the we eview the Nikifoov-Uvaov [NU] method. We exact olutio of thee equatio have a impotat peet factoizatio method i ectio. The exact eeach aea ice the bith of quatum mechaic olutio of the Klei-Godo equatio i give i [4-5]. ectio 4. Fially a bief cocluio i peeted i ectio 5. Coequetly the boud tate olutio of Klei- Godo equatio ad Diac equatio with mixed Vecto ad cala potetial have igificatly eiched oe kowledge of atomic ad ub-atomic ytem [6-7]. I ecet time a geat attetio ha bee pet to olve thee o-elativitic ad elativitic wave equatio fo vaiou potetial uig diffeet method [8-]. Relativitic equatio cotai two object: the fou-vecto liea mometum opeato ad the cala et ma. Thee allow oe to itoduce two type of potetial couplig which ae the fou vecto potetial V( ad the pace time cala potetial S(. May autho by uig diffeet method have tudied the boud tate of the Klei-Godo equatio ad the Diac equatio with mixed typical potetial ude the coditio that each of the cala potetial i equal to it vecto potetial V( []. Thee ivetigatio iclude the Hulthe potetial [-4] Moe potetial [5-6]. Pochl-Telle potetial [7-8] Katze potetial [9-] Roe Moe type potetial [] the Coulombic ighaped potetial [-]. The method iclude the Review of Nikifoov-Uvaov Method The NU method i baed o olvig a ecod ode liea diffeetial equatio by educig it to a geealized equatio of hypegeometic type [6]. Thi method ha bee ued to olve the Schödige Diac ad Klei-Godo equatio fo diffeet kid of potetial [9-]. I NU method the ecod ode diffeetial equatio ca be witte i the fom. τ ( σ ( ψ ( ψ ( ϕ( () σ ( σ ( whe σ ( ad σ ( ae polyomial at mot of ecod degee ad τ ( i a fit degee polyomial. We wite the tafomatio fo the wave fuctio i equatio () a ψ ( ϕ( χ ( ad thi educe equatio () to equatio of hypegeometic type σ( χ ( ) ( ) τ χ( λχ ( ad ( ϕ i defied a a logaithmic deivative [6] () ()

2 Am. J. Sci. Id. Re. (): 79-8 ϕ ( ) π ( ) (4) ϕ ( ) σ ( ) The othe wave fuctio χ ( i the hype geometic fuctio whoe olutio ae obtaied by the Rodique elatio [6] B d χ ( [ σ ( ] (5) d whee B i a omalizatio cotat ad ρ ( i the weight fuctio that mut atify the coditio d d The fuctio ( ( σ ( ) τ ( (6) π ad the paamete λ equied fo the NU-method ae defied a follow: σ τ σ σ π ( ± σ ( kσ ( λ k π ( I ode to fid the k-value the expeio ude the quae oot mut be the quae of a polyomial. Thu a ew eigevalue equatio fo the ecodode diffeetial equatio become dτ () ( ) d σ ( λ λ d τ ( τ ( π ( d ad it deivative i egative. Thu by the compaio of equatio (8) ad equatio (6) we obtai the eegy eigevalue. Factoizatio Method I pheical co-odiate the Klei-Godo equatio fo a paticle i geealized type -I Pochl-Telle potetial with vecto potetial V( ad Scala potetial S( ead [] (9) () [ h c ( E V ( ) ( mc S( ) ] ψ ( θ ϕ) () o explicitly we wite c h Siθ Siθ θ θ ( ) {( E V ( ) ( m ) } c S( Si ψ θ ϕ θ ϕ ψ ( θ ϕ) () whee E i the eegy pectum ad m i the et ma of the paticle. I ode to fid a exact olutio to equatio () we give pheical total wave fuctio a ψ ( θ ϕ) R( H ( θ ) φ( ϕ ) () Subtitutig equatio () ito Schödige equatio () the wave equatio epaated ito idepedet vaiable ad the followig equatio ae obtaied (7) (8) ( E V ( ) ( mc S( ) R( (4) d d λ d d h c d dh ( θ ) m H( θ ) Cotθ ( ) λ l H θ dθ dθ Si θ ad d φ( ϕ ) dϕ φ( ϕ) m l whee λ i the epaatio cotat. The olutio of equatio (5) ad (6) ae the pheical hamoic Y lm ( θϕ) ad ae well kow [] with λ l( l ) whee l i the obital agula mometum l L ad the magetic quatum umbe m l ± ± L± l. Now uig a aaltz u ( R( (7) fo the adial wave fuctio i equatio (4) tafom ito d l ( l ) ( E V ( ) ( mc S( ) ( ) (8) u d h c whee ( ) (5) (6) l l hc i the cetifugal potetial ad h c will be ue hece with. I the atual uit thi pape we demotate the applicatio of the NUmethod fo the type-i Pochl-Telle potetial. The equal vecto ad cala type Pochl-Telle potetial i give by [7] A B V ( S( Si Co (9) The Cetifugal tem togethe with the equal cala ad Vecto Type I Pochl-Telle potetial give ie to the effective potetial V eff ( ) ( l l A B Si Co m Thi effective potetial caot be olved aalytically fo l becaue of the cetifugal tem []. Howeve i ode to obtai the aalytical olutio of equatio (8) with cheme uggeted by C. S. Jia et al [7] to deal with the cetifugal tem. Whe << I we ue a modified ew impoved appoximatio cheme to deal with the cetifugal tem 4 C 4 Si whee the paamete () () C i a dimeiole cotat. Figue how the vaiatio of Type Pochl-Telle Potetial with fo S-State P-State ad D-State. 8

3 Am. J. Sci. Id. Re. (): 79-8 Fig.: Vaiatio of the effective potetial a a fuctio of. The cuve ae how fo S-tate P-tate ad D-tate. Boud State Solutio of Klei-Godo Equatio I the peece of a Vecto Potetial V( ad a Scala Potetial S( the type I Pochl Telle Potetial of equatio () we wite the Klei-Godo equatio a d A B l( l ) C ( E m ) d 4Si Si Co E m u( ( )] () whee A B ad ae poitive paamete. I ode to fid the boud tate olutio of equatio () we et S Si ad thi tafom equatio () ito d u du ( E m ) A ( E m ) B S( d d ( ( m E ) l ( l ) C l( l ) ( ) () u Now ettig the followig dimeiole paamete: ( m ) ( ) E E m V ε β A γ ( E m ) l( l ) B V δ 4 l ( l ) (4) 4 tu equatio () ito a ecod ode diffeetial equatio of the hypegeometic type. d u du [( ε δ ) d ( d 4 ( ( β δ ε ) ( β ) u() (5) C If we apply the NU method i the peet cae by compaig equatio (5) with equatio () we obtaied the followig expeio: τ σ ( ( σ ( a whee b c a ( ε δ ) ( β δ ) b ε (6) ( ) (7) c β Subtitutig equatio (6) i to equatio (7) ad accodig to the popety () that π ( i a polyomial we fid π ( a fo π ( ± fo whee c ( ) k ( ) k a 4 b 4 (8) The uitable k i equatio (8) i detemied by the polyomial τ τ π havig a egative deivative. Theefoe the polyomial π ( i computed fom equatio (8) a π [( ) ] () ( 8

4 Am. J. Sci. Id. Re. (): 79-8 ad the fuctio ( τ ( equiemet a: τ atifie thee τ ( [( ) ] () τ ( ( ). () with thi electio of τ ( ) π ( k ad the λ π λ we obtai k ( λ we have λ ( ) (4) Compaig eq. () ad (4) ad uig equatio (4) ad (9) we obtai the eegy pectum E a () ( ) Uig the elatio τ σ E ( 4v l( l ) 4 ( ( )) 8v l l 4 V V ( ( )) 4v ( ) 4 l l l( l ) l ( ( )) ( l l) C ± 8V l l 4 4 Hee the idex i a o-egative itege ad equatio (5) how that we ae dealig with a family of Poehl-Tello Potetial. I thi wok we tudy the boud tate olutio that i the adial equatio of the wave fuctio u( that u( mut atify the bouday coditio a u ( ad become fiite at. I ode to fid the wave fuctio u( we coide equatio () which ha thee igulaitie at ad we make a aaltz fo the wave fuctio a ( ε δ ) ( ) β f ( ) (5 ) u( ad u ( ad u ( ( ε δ) become βf ( ε δ β u ( f( f ( ( (6) ( () ( ε δ)( ε δ ) β () ( ε δ) f ( ) ( ε δ) f ( u f ( βf () β( β ) f ( δ β f () ( ) (7) ( ) ε Subtitutig equatio (5)-(7) ito equatio () we obtai. (5) ( f ( [ 4( ε δ ) 4( ( ε δ ) β ) ] f ( ( ε δ) β ε δ β ( ε δ) β γ f (8 [ ] () ) Thi equatio i the well-kow diffeetial equatio atified by the hypegeometic fuctio F ( a b; c ) thu () ( c) f Γ Γ( a) Γ( b) Γ( a k) Γ( b k) k Γ( c k) Whee a b ε δ β ab k k! ( ) ( ε δ ) β ε δ β ( ε δ ) β γ 4( ε δ ). c (4) (9) I tem of the vaiable we wite the uomalized adial wave fuctio fo the type I Pochl-Telle potetial a ( ε δ () ) ( ε δ ) i ( ( i )( xf ( ρ c) ( 4) R l Whee ( a b) ad ρ ab. CONCLUSION: We have ivetigated the boud tate olutio of the klei-goda with equal cala ad vecto type I Pochl Telle potetial fo abitay agula mometum umbel. We ue the Nikifoov-Uvaov method to fid exact the eegy pectum ad the uomalized wave fuctio i obtaied via a aaltz ad expeed i tem of the hypegeometic fuctio. Fially ou eult ae coitece with that of ef.[] uig aalytical method. ACKNOWLEDGEMENT Thi wok wa uppoted patially by the NANDY- LEABIO Foudatio ude gat No REFERENCES. S. Flugge Pactical Quatum Mechaic Vol. (Spige Beli 994).. M. Simek ad H. Egife J. Phy. A. Math Ge (4).. Y. Xu S. He ad C. S. Jia Phy. Sc (). 4. M. Kocak ad B. Goiil Cli Phy. Lett. 4 4 (7) 5. B. Goiil K. Kokal ad E. Baku Phy. Sc (6). 6. L. Hulthe Ak. Mat. Ata. Fu. 8A 5 (94) 7. V. P. Vahi Phy. Rev. A (99) 8

5 Am. J. Sci. Id. Re. (): H. Egife D. Demiha ad F. Buyukkhe Phyica. Scipta 59 9 (999) 9. H. Egife D. Demiha ad F. Buyukkhe Phyica Scipta 6 95 (999). M. Simek ad H. Egife J. Pha. A: Math. Ge (4). A. A. D Alhailai H. Bahlouli ad A. Al-Haa Phy. Lett. A (6). P. Albeto A. S. decato ad M. Malheio Phy. Rev. c (7). N. Saad phy. Sc (7) 4. A. Ada ad R. Seve It. J. Theo. Phy (9) 5 L. H. Zhag X. P. Li ad C. S. Jia Phy. Sc. 8 5 (9) 6. N. Roe ad P. M. Moe Phy. Rev. 4 (9) 7. C. S. Jia T. Che ad L. G. Cui Phy. Lett. A 7 6 (9) 8. T. Che Y. F. Diao ad C. S. Jia Phy. Sc (9) 9. J. Sadeghi ad B. Pouhaa EJTP 5 9 (8). Y. F. Cheg ad T. G. Dai Phy. Sc (7). A. N. Ikot ad L. E. Akpabioaccepted i Applied Phyic Reeach(APR).. O. Yeilta ad R. Seva axiv: quat-ph/74. B. M. Madal It. J. Mod. Phy. A5 5 () 4. W. Geie (Relativitic Quatum Mechaic Beli Spige 99) 5. F. Coope A. Khae ad V. Sukhatme Phy. Rep (995) 6. A. F. Nikifoov ad V. B. Uvaov (Special Fuctio of Mathematical Phyic Bikhaiie Bael 988) 7. G. Pochl ad E. Telle Z. Phy 8 4 (9) 8