Bound state solutions of deformed generalized Deng-Fan potential plus deformed Eckart potential in D-dimensions

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1 RESEARCH Revista Mexicaa de Física MAY JUNE 03 Boud state solutios of deformed geeralized Deg-Fa potetial plus deformed Eckart potetial i D-dimesios O. A. Awoga a, A. N. Ikot a ad J. B. Emah b a Theoretical physics group, Departmet of Physics, Uiversity of Uyo, Uyo, Nigeria, ola.awoga@yahoo.com; demikotphysics@gmail.com b Departmet of Physics, Akwa Ibom State Uiversity, Ikot Akpade, joemah73@gmail.com Received 6 September 0; accepted 8 Jauary 03 I this article,we preset the approximate solutio of the D-dimesioal Schrödiger euatio for deformed geeralized Deg-Fa plus deformed Eckart potetial usig parametric Nikiforov-Uvarov method. We obtai the boud state eergy eigevalues ad the correspodig wave fuctio for arbitrary l state. Special ces of this potetial are also discussed. Keywords: Deformed geeralized Deg-Fa potetial; deformed Eckart potetial; Nikiforov-Uvarov; D-dimesios. PACS: 03.65Ge,03.65-w,03.65Ca.. Itroductio The solutios of the Schrödiger euatio SE have bee extesively studied i recet times. These solutios play a vital role i uatum mechaics sice they cotai all ecessary iformatio goverig the uatum mechaical system uder cosideratio. However, apart from the S-wave oly few potetials are exactly solvable for l 0 such harmoic potetial, Coulomb potetial ad others 3. Most of the potetials for the descriptio of physical systems are ot exactly solvable for l 0. Noetheless, may authors have applied differet approximatio to the cetrifugal term ad obtaied aalytical approximatio to the l-wave solutios of the Schrödiger euatio with some expoetial-like type potetial -0. These potetials iclude the Morse potetial Maig-Rose, Scarf Poschl-Teller ad Rose Morse potetials,3. Various methods have bee used to obtai the exact or approximate solutios of the Schrödiger euatio for some expoetial-type potetial. These methods iclude the Nikiforov-Uvarov method NU,5, factorizatio method 6,7, ymptotic iteratio method 8,9 ad others 0. Recetly, may researchers have developed iterest i D-dimesioal solutios -5,3,. I the preset work, we attempt to ivestigate the deformed geeralized Deg-Fa potetial plus deformed Eckart potetial. Followig the work of other authors -5,3, we solved the SE for this potetial i D-dimesios usig parametric Nikiforov-Uvarov method. The potetial uder ivestigatio is V r = V 0 c beαr e αr V e αr e αr + V 0e αr e αr V 0, V, V are potetial depths, is the deformatio parameter which will take values of ad -, b ad c are adjustable costat, α is the rage of the potetial. If we set = c =, V = V = 0, the potetial reduces to that of the Deg-Fa potetial 6,33 with b = e αrc r c is the euilibrium iteruclear distace ad with V 0 = 0 the potetial becomes Eckart potetial 7-9. We display the behaviour of this potetial with for = ± i Fig. -. FIGURE. Variatio of the potetial a fuctio of r for V 0 = Mev, V = 0. Mev, V = 0.5 Mev =, c = 0.5, b = 0. ad various values of α =.0, 0.9 ad 0.8 fm. FIGURE. Variatio of the potetial a fuctio of r for V 0 = Mev, V = 0. Mev, V = 0.5 Mev =, c = 0.5, b = 0. ad various values of α = 0, 9 ad 8 fm.

2 30 O. A. AWOGA, A. N.IKOT, AND J. B. EMAH Three special ces ca be obtaied from this potetial follows: i Settig c = 0, =, V = 0, b = E. reduces to V GW S = V e αr + e αr + V 0e αr + e αr which is kow geeralized Woods-Saxo potetial 30,3 which is used to describe iteractio betwee uclei 3. For V 0 = 0 we obtai the Woods-Saxo potetial V W S r = V e αr + e αr 3 ii Settig V 0 = c = V = 0 ad = E. reduces to V H r = V e αr e αr which is kow Hulthe potetial 9,3 iii If ad = E. reduces to e V MR r = b αr V 0 e αr 5 which is a kow form of Maig-Rose potetial for A=0 35. I this work, the cetrifugal term will be approximated 36 r ωeαr e αr + λe αr e αr 6 ω ad λ are adjustable dimesioless parameter. This approximatio scheme is valid for large ad small α. To show that E. 6 is a good approximatio scheme we compare /r ad the approximatio scheme with differet values of αi Figs. 3- for =, respectively. The orgaizatio of the paper is follows: I Sec., we give a brief review of the Nikiforov-Uvarov method i its parametric form. I sectio 3, we highlight the SE i D- space. I Sec., we preset the boud state solutio of the SE i D-dimesio. I Sec. 5 we give a brief discussio. Fially a brief coclusio is give i Sec. 6. FIGURE. Compariso betwee /r ad the approximatio scheme fuctios of r for ω = 5.00, λ = 0.53, = ad various values of α = 0, 9 ad 8 fm.. Review of Nikiforov-Uvarov method ad its parametric form The Nikiforov-Uvarov method NU is bed o the solutio of a geeralized secod order liear differetial euatio with special orthogoal fuctio,37. The SE ψ r + E V r ψr = 0 7 ca be solved by the NU method by trasformig this euatio ito hypergeometric-type usig the trasformatio s=sx ψ s + τs σs ψ s + σs σ ψs 8 s I order to fid the solutio to E. 8 we set the wave fuctio ψs = ϕsχ s 9 Substitutig E. 9 ito E. 8 reduces E. to σsχ s + τsχ s + λχ s = 0 0 the wave fuctio ϕs is a logarithmic fuctio ϕ s ϕs = πs σs χ s is the hypergeometric-type fuctio which satisfies the Rodrigue relatio d χ s = B ρs ds σ sρs FIGURE 3. Compariso betwee /r ad the approximatio scheme fuctios of r for ω = 5.00, λ = 0.53, = ad various values of α =.0, 0.9 ad 0.8 fm. ad B is the ormalizatio costat ad the weight fuctio ρs satisfies the coditio σsρs = τsρs 3 Rev. Mex. Fis

3 BOUND STATE SOLUTIONS OF DEFORMED GENERALIZED DENG-FAN POTENTIAL PLUS DEFORMED ECKART... 3 ad The reuired πs ad λ for the NU method are defied πs = σ τ ± σ τ σs + kσs λ = k + π s 5 respectively. It is ecessary that the term uder the suare root sig i E. 3 be the suare of a polyomial. The eigevalues i E. 5 take the form λ = λ = τ σ, = 0,,... 6 τs = τs + πs 7 The derivative of E. 6 is less tha zero for boud state. The eergies are obtaied by comparig Es. 5 ad 6. The parametric geeralizatio of the NU method that is valid for both cetral ad o-cetral expoetial potetial 38 ca be derived by comparig the geeralized hypergeometric-type euatio. ψ s + c c s s c 3 s ψs + s c 3 s ξ s + ξ s ξ 3 ψs = 0 8 Comparig E. 8 ad E. 8 we obtai the followig parametric polyomials τs = c c s 9 σ = ξ s + ξ s ξ 3 0 σs = s c 3 s Substitutig E. 9, E. 0 ad E. ito E. we obtai, πs = c c 5 s Also ± c 6 c 3 k ± s + c 7 + k ± s + c 8 c = c, c 5 = c c 3, c 6 = c 5 + ξ, 3 c 7 = c c 5 ξ, c 8 = c + ξ 3 k ± = c 7 + c 3 c 8 ± c 8 c 9 5 c 9 = c 3 c 7 + c 3c 8 + c 6 6 ad Hece, the fuctio πs becomes πs = c + c 5 s c 9 + c 3 c8 s c 8 From the relatio i E. 7 we have τ s = c + c c c 5 s 7 c 9 + c 3 c8 s c 8 8 τ s = c 5 c 9 c 3 c8 9 Solvig Es. 5 ad 6, we obtai the parametric eergy euatio c c 3 + c 3 + c c 9 + c 3 c8 + c 7 + c 3 c 8 + c 8 c 9 = 0 30 The weight fuctio is obtaied ρs = s c0 c 3 s c 3 Ad together with E. we obtai χ s = P c0,c c 3 s 3 P c0,c are the Jacobi polyomials ad the superscripts c 0 ad c are give by c 0 = c + c + c 8 33 c = c c + c 3 c9 3 The other part of the wave fuctio is obtaied from E. ϕ s = s c c 3 s c3 35 c = c + c 8 36 c 3 = c + c 3 c 9 c 5 37 Thus the total wave fuctio becomes ψ s = N S c c 3 s c3 P c0,c c 3 s 38 N is the ormalizatio costat. Rev. Mex. Fis

4 3 O. A. AWOGA, A. N.IKOT, AND J. B. EMAH 3. Schrödiger Euatio i D-dimesio The D-dimesio space SE is 39 µ D + V r Ψ lm r, Ω D = E l Ψ lm 39 D = r D r D Λ D r r r Ω D 0 is the Laplacia operator. The secod term of E. 0 is the multidimesioal space cetrifugal term. Ω D represet the agular coordiates. The operator Λ D yields hyperspherical harmoic its eigefuctio. With this we write the wave fuctio Ψ lm r, Ω D = R l ry m l Ω D R l is the radial part of the euatio ad Yl m is the agular part called hyper-spherical harmoics. The Yl m Ω D obey the eigevalue euatio Λ DYl m Ω D = l l + D Yl m Ω D U r + d U ds µe µ V 0 c DD3 + ll + D Usig the trasformatio s = e αr i E. 5 we obtai ε + + s du ds + s This ca further be simplified to hypergeometric-type euatio d U ds s + s s Substitutig Es. 0 ad ito E. 39 we obtai the radial euatio r D r r + µ D Rr r E l l + D r Rr = 0 3. Boud State solutio of Schrödiger euatio I order to obtai state solutio we have to make the trasformatio D Rr = r Ur Substitutig Es., 6 ad ito E. 3, we derive the boud state SE for l 0 for the potetial uder cosideratio beαr e V e αr αr e + V0eαr αr e αr ωe αr e αr + λeαr e αr γs s βs s φs s Ur = 0 5 U = 0 6 s s ε + γ + φs + ε + γ βs ε U = 0 7 ε = µ E c α V 0 8 γ = µ α bcv 0 + V ω D D 3 + ll + D 9 µ β = µ α V + λ D D 3 + l l + D 50 µ φ = µb V 0 α Comparig E. 7 ad E. 8, we obtai the followig 5 c =, c = c 3 =, ξ = ε + γ + φ, ξ = ε + γ β, ξ 3 = ε, c = 0, c 5 =, c 6 = ε + γ + φ +, c 7 = ε + γ β, c 8 = ε, c 9 = φ + β +, c 0 = + ε, c = + φ + β, c = ε ad c 3 = + + φ + β Rev. Mex. Fis

5 BOUND STATE SOLUTIONS OF DEFORMED GENERALIZED DENG-FAN POTENTIAL PLUS DEFORMED ECKART From E. 30, we obtai ε ld = γ + φ + σ + σ 5 Usig E. 8 we obtai the eergy eigevlaues E ld = α γ + φ µ + σ + σ + c V 0 53 σ = + + φ + β 5 From E. 35 the first part of the wave fuctio is obtaied ϕs = s ε s ad the weight fuctio from E. 3 ρs = s +ε s + φ+β 55 φ+β 56 We also obtai the secod part of the wave fuctio +ε, φ+β χ s = P s 57 Thus the total wave fuctio from E. 38 is U ld r = N lde εαr e αr P +ε, φ+β + φ+β e αr 58 The Jacobi polyomials are reported i several literatures 0-, is the ormalizatio costat ad is calculated N ld = ε + ε ε + + φ + β + +!Γ ε + + φ + β + + Γε + + Γ + φ + β φ + β We have used the ormalizatio coditio alogside E. ET II 855,9 ad E. ET II 855,9 of Ref. to obtai E. 59. The derivatio of E. 59 is give i the appedix 5. Special ces ζ = + Q µ α b V 0 V + l l + + Q 63 We ow take a look at the special ces discussed i Sec.. This is doe by makig adjustmet to the costats i the potetial. i Geeralized Woods-Saxo potetial: If we set V =c=0, ω = α, =, D = 3. Our potetial i E. reduces to the geeralized Woods-Saxo potetial i E. ad the eergy i E. 53 becomes E GW S l = α µ Q = + µ { + Q α b V 0 V + l l + + Q + ad the wave fuctio i E. 58 becomes U GW S l = B l e ζαr + e αr P } 60 µb V 0 α λ ll + 6 α +ζ, +φβ + +φβ + e αr 6 B l is the ormalizatio costat i the GWS model. If we proceed by settig ω = V 0 = 0 ad λ = α, the potetial i E. 5 reduces to Woods-Saxo potetial i E. 3 ad eergy becomes El W S = α + l + µv µ α 6 + l + ad the wave fuctio is U W S l = C l e δαr + e αr l+ δ = + l + P +δ,l+ + e αr 65 µv α + l + 66 ii Hulthe potetial: If we make the choice ω = V 0 = V = c = 0, λ = α, = ad D = 3 the potetial reduces to Hulthe potetial give i E. ad the eergy eigevalues reduces to El H = α µv µ α + l + + l + 67 Rev. Mex. Fis

6 3 O. A. AWOGA, A. N.IKOT, AND J. B. EMAH This cosistet with result obtaied i 9,36. The wave fuctio for the Hulthe potetial is obtaied from 58 U H l = D l e ναr e αr l+ P +ν,l+ e αr 68 µv ν = α + l + + l + 69 iii Maig-Rose potetial: Settig ω = V 0 = V = c = 0, λ = α, D = 3 ad = the potetial reduces to the form of Maig-Rose potetial give i E. 5 the eergy becomes El MR = α µb V + ρ µ α 70 + ρ ρ = + ad the wave fuctio is U MR l = F l e ηαr e αr P +η, l + + 8µb V 0 α l+ + 8µb V 0 α + l+ + 8µb V 0 α 7 e αr 7 µb V + ρ η = α 73 + ρ The ormalizatio costats B l, C l, D l ad F l ca be obtaied from E. 59 by makig the ecessary substitutios. 6. Coclusio I this work, we study the Deformed geeralized Deg-Fa plus Eckart potetial for o-vaishig agular mometum i D-dimesios with the aid of a approximatio scheme usig parametric Nikiforov-Uvarov method. The eigevlaues of this potetial ad that of its special ces Woods-Saxo, Hulthe ad Maig-Rose potetials, are obtaied ad the wave fuctios of each potetial are expressed i terms of the Jacobi polyomials. APPENDIX Derivatio of the Normalizatio Costat E. 59 Let y = e αr, therefore E. 58 becomes ε U y + y ld = N ld +ε, φ+β + φ+β p y A. Applyig the ormalizatio coditio to E. A. we obtai N ld ε + + φ+β y ε + + y φ+β + y +ε, φ+β p y dy = A. We have used the iterval of the Jacobi polyomials,,-, i the itegral. Writig y y ad expadig E. A. we obtai, N ld ε { +ε+ + φ+β y +ε + y + y ε + + y P φ+β +ε, φ+β P φ+β +ε, y dy We ow use the stadard orthogoal itegrals of Jacobi polyomials follows } φ+β y dy = A.3 y α + y β P α,β α+β Γα + + Γβ + + y dy =!αγα + β + + y α + y β P α,β α+β+ Γα + + Γβ + + y dy =!αγα + β + + A. A.5 Rev. Mex. Fis

7 BOUND STATE SOLUTIONS OF DEFORMED GENERALIZED DENG-FAN POTENTIAL PLUS DEFORMED ECKART Euatios A. ad A.5 are respectively ivoked from Es. ET II 856 ad ET II 85 5,9 of Ref. pages 806 ad 807 respectively. α ad β i Es. A. ad A.5 have othig to do with the α ad β i this work. Comparig the superscripts of Es. A. ad A.5 with that of E. A.3, we obtai the ormalizatio costat give E. 59 N ld = ε + ε ε + + φ + β + +!Γ ε + + φ + β φ + β + + Γε + + Γ + φ + β + +. K. J. Oyewumi, F. O. Akipelu ad A. D. Agboola, It. J. Theor. Phys B. Goul, O Ozer ad M. Kocak, Comm. Thoer. Phys L. D. Ladau ad E. M. Lifshitz, Quatum Mechaics No- Relativistic Theory, 3rd ed. Pergamo, Oxford, New York R. L. Greee ad C. Aldrich, Phys. Rev. A C. L. Pekeris, Phys. Rev W. C. Qiag ad S. H. Dog, Phy. Lett A S. M. Ikhdair, J. Maths Phys S. M. Ikhdair ad Ramaza Sever, J. Phys. A: Math Theor S. M. Ikhdair, Eur. Phy. J. A C. S. Jia, J. Y. Liu ad P. Q. Wag, Phys. Lett P. M. Morse, Phys. Rev Y. P. Varshi, Rev. Mod. Phys O. Yesilt, Phys. Scr A. F. Nikiforov ad V. B. Uvarov, Special Fuctios of Mathematical Physics Birkhauser, Bel, A. N. Ikot, Afr. Rev. Phys S. H. Dog, Factorizatio Method i Quatum Mechaics Spriger, Dordrecht, L. Ifeld, Phys. Rev O. Ozer, Chi. Phys. Lett H. Ciftci, R. L. Jall ad N Saad, J. Phys. A A. S. Davydov, Quatum Mechaics Pergamo, J. L. Cardoso ad R. Alvarez-Nodirse, J. Phys A S. H. Dog ad G. H. Su, Phys. Lett A J. L. Coelho ad R. L. G Amaral, J. Phys. A K. J. Oyewumi ad A. W. Ogusola, J. Pure Appl. Sci H. Hsaabadi M. Hamzavi, S. Zarrikamar ad A. A. Rajabi, It. J. Phys. Sci S.H. Dog, Comm. Theor. Phys C. Eckart, Phys. Rev S. H. Dog, W. C. Qiag, G. H. Su ad V. B. Bezerra, J. Phys. A: Math Theor X. Zou, Z. Yi ad C. S. Jia, Phys. Lett A C. Berkemir, A. Berkdemir ad R. Sever, arxiv: ucl.th/ A. Arai, J. Math. Aal. Appl M. Brack, Rev Mod. Phys Z. H. Deg ad Y. P. Fa, Shadog Uiv. J D. Agboola, Phys. Scr W. C. Qiag ad S. H. Dog, Phys. Lett A F. Tki ad G. Kocak, Chi. Phys. B C. Tezca ad R. Sever, It. J. Theor. Phys A. Arda, R. Sever ad C. Tezca, Chi J. Phys H. Hsaabadi, S. Zarrikamar ad A. A. Rajabi, Commu. Theor. Phys M. Abramowitz ad I. A. Stegu, Hadbook of Mathematical Fuctios with Formul, graphs ad Mathematical Table Whigto, 97.. A. D. Polyai ad A. V Mazhirov, Hadbook of Itegral euatios Whigto, I. S. Gradshetyh ad I. M. Rhyzhik, Table of Itegrals, series ad products Elsevier, Burligto, O. A. Awoga ad A. N. Ikot, Pramaa J. Phys S. H. Dog, Wave Euatios i Higher Dimesios. Spriger 0, New York. Rev. Mex. Fis