Solutions of the D-dimensional Schrödinger equation with the Hyperbolic Pöschl Teller potential plus modified ring shaped term

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1 Solutios of the -dimesioal Schödige equatio with the Hypebolic Pöschl Telle potetial plus modified ig shaped tem Ibsal A. Assi, Akpa N. Ikot * ad E.O. Chukwuocha Physics epatmet, Kig Fahd Uivesity of Petoleum & Mieals, haha 36, Saudi Aabia epatmet of Physics, Uivesity of Pot Hacout, P.M.B 533, Choba, Pot Hacout, Nigeia * demikotphysics@gmail.com Abstact: I this pape, we solve the -dimesioal Schödige equatio with hypebolic Poschl-Telle potetial plus a geealized ig-shaped potetial. Afte the sepaatio of vaiable i the hypespheical coodiate. We used Nikifoov-Uvaov (NU) method to solve the esultig adial equatio ad obtai explicitly the eegy level ad the coespodig wave fuctio i closed fom. The solutios to the agula pat ae solved usig the NU appoach as well. PACS: 3.65.Pm, 3.65.Ge, 3.65.Fd, 3.65.Ca Keywods: -dimesioal Schodige equatio, hypebolic Pöschl Telle potetial,hypespheical coodiates, eegy spectum, ig-shaped potetials, Jacobi polyomials.. Itoductio The o-cetal potetials i ecet times have bee a active field of eseach i physics ad quatum chemisty [-3]. Fo istace, the occuece of accidetal degeeacy ad hidde symmety i the o-cetal potetials ad its applicatio i quatum chemisty ad uclea physics ae used to descibe ig-shaped molecules like bezee ad the iteactio betwee defomed pai of uclei [4-5]. It is kow that these accidetal degeeacy occuig i the ig shaped was explai by costuctig a SU () algeba [6]. Owig to these applicatios may authos have ivestigated a umbe of eal physical poblems o o-spheical oscillato [7], ig-shaped oscillato (RSO) [8] ad ig shaped o-spheical oscillato [9]. Bedemi [] had show that eithe Coulomb o hamoic oscillato will give a bette appoximatio fo udestadig the spectoscopy ad stuctue of diatomic molecules i the goud electoic state. Othe applicatios of the ig shaped potetial ca be foud i ig shaped ogaic molecules like cyclic polyees ad bezee [-]. O the othe had, Che ad og studied the Schödige equatio with a ew ig shaped potetial [3]. Cheg ad ai ivestigated modified Katze potetial plus the ew ig shaped potetial usig Nikifoov-Uvaov method [4]. Recetly, Ikot et al [5-7] ivestigated the Schödige equatio with Hulthe plus a ew ig shape potetial [3],No-spheical hamoic ad Coulomb potetial[6] ad pseudo-coulomb potetial i the cosmic stig space-time[7].may authos have used diffeet methods to obtai

2 exact solutios of the wave equatio such as the methods of Supesymmetic Quatum Mechaics (SUSY-QM) [8-], the Tidiagoal Repesetatio Appoach (TRA) [- 4], Nikifoov-Uvaov (NU) [5-9] amog othe methods Motivated by the ecet studies of the ig shaped-like potetial [3-33], we poposed a ovel hypebolical Poschl Telle potetial plus geealized ig shaped potetial of the fom, B cot cot csc csc V (, ) A tah, (.) tah whee is the sceeig paamete, A, B,,, ad ae eal potetial paametes. As a special case whe with A m, B, ad 3m m B E E,the potetial of equatio (.) tus to No-spheical hamoic oscillato 3 plus geealized ig shaped potetial cot cot csc csc V (, ) m, (.) m. -imesioal Schodige equatio i Hypespheical Coodiates The -dimesioal Schodige equatio is give below [34-35] l l E U l, l,.., l X, (.) whee is the effective mass of two iteactig paticles, is Plack s costat, E is the eegy eigevalue, U is the potetial eegy fuctio, X (,,,..., ) T is the positio vecto i -dimesios, whee,,..., is the agula positio vecto witte i tems of Hypespheical coodiates [36-37], ad dimesioal Laplacia opeato give i Appedix B. is the - The solvable potetials which allows sepaatio of vaiable i (.) must be of the fom: V U x V, (.). The sepaable wavefuctio take the followig fom: ( )/ ll X g Y ll l, l,.., l l, l,.., l, (.3) Applyig (.3) to Eq. (.) with the use of Eq. (.), we obtai the followig adial ad agula wave equatios d l E V g d 4, (.4)

3 d cos d H, (.5) d si d si d cos d l l V H, (.6) d si d si ll im whee Yl, l,.., l e H ad l l, Eq. (.6) holds fo,, with 3,. Solutios of (.6) will ot be affected by the pesece of the poposed potetial ad thus it is commo to diffeet systems ad it was doe befoe usig diffeet appoaches [38]. Cosequetly, we will oly solve equatios (.4) ad (.6) usig the method of Nikifoov-Uvaov [5,6]. 3. Nikifoov-Uvaov Method May poblems i physics leads to the followig secod ode liea diffeetial equatio [5]: d x d x u x, dx x dx x (3.) whee x x is at most liea i x. x, ae polyomials of degee at most, ad Eq. (3.) is sometimes called of hypegeometic type. Let s coside u x x y x this will tasfom equatio (3.) to the followig diffeetial equatio fo y x : x x d d dx x dx x whee we assumed the followig coditios: y x,, (3.) x x x x x x x x x x x x x x x, (3.3.a), (3.3.b), (3.3.c) x x x x x x k x, (3.3.e) k x, (3.3.f) 3

4 whee k ad ae costats chose such that i x ad x x whee x ad x x is polyomial which at most liea. This will tasfom Eq. (3.) to the followig: d d x x y x dx dx, (3.4) ae polyomials of degees ad, espectively. I this case,, whee is give below:, (3.5) Eq. (3.5) will be used to obtai the eegy spectum fomula of the quatum mechaical system. We should poit out hee that the polyomial solutios to Eq. (3.4) fo ad o the boudaies of the fiite space (the latte case is omitted fo ifiite space), ae the classical othogoal polyomials. It is well kow that each set of polyomials is associated with a weight fuctio x. Fo the polyomial solutios to Eq. (3.4), this solutios to Eq. (3.4) ae polyomials of degee, y x y x, fuctio must be bouded o the domai of the system ad must satisfy. This weight fuctio will be used to costuct the Rodigues fomula fo these polyomials which eads: B d y x x x x dx, (3.6) whee B is ust a costat obtaied by the omalizatio coditios, ad =,,,. 4. The Solutios of the -dimesioal Radial equatio We use the NU method to solve equatio (.4), i the pesece of ou potetial, Whee, cetifugal tem E A tah g( ), (4.) d tah d g( ) B l. Equatio (4.) caot be solved aalytically due to the 4. iffeet authos used diffeet appoximatio techiques to allow a appoximate aalytical solutio of (4.) ad these methods ely o Taylo expasio of the cetifugal potetial i tems of the othe compoets of the potetial of iteest [39]. I this wok, we use the followig appoximatio obtaied by Taylo expasio [4], tah, (4.) 3 3 tah 4

5 The advatage of this appoximatio it is valid ot oly fo but also fo with high accuacy. Also, it satisfies the limits o at zeo ad ifiity, that is lim RHS ad lim RHS, whee RHS deotes the ight-had-side of (4.). Now, Usig Eq. (4.) back i (4.), we get whee d d 4 B 4 A tah 4 E g, (4.3) tah 4 E E 3, 4 A A, ad 6 4 B B. Makig chage of vaiable s tah ad by witig g( s) ( s) y( s), this tasfoms (4.3) to (3.) with the polyomials beig s s, 3s /, ad Es As B. We ow use Eq. (3.3.e) to calculate ( s) which eads s s 4 4 s As Es B k s s, (4.4) The choice of k that makes (4.4) a polyomial of fist degee must satisfy whee c 6k 8 6E, c 6A 6k ad c 3 4 6B. This gives c c c, 3 Solutios of (4.7) fo k ae 5 s c 4 4 c s c s whee we will pick the egative pat i (4.5) that makes easily calculated usig (3.3.b) c c s s c s Usig (4.5) ad (4.6) i Eq. (3.5) ad Eq. (3.3.f), we get, (4.5). The fuctio s ca be, (4.6) 4k 6A 6k, (4.7)

6 4k 6A, (4.8) The ext step is to use the value of c i Eq. (4.7) with the costait o k metioed i Eq. (4.4), which is c c c, we obtai 3 6, (4.9) 8 6E 6k 4 6B 6A 6k The coditios fo boud states ae k, ad A, B /6. Usig (4.8) i (4.9), we wite the boud states fomula as follows: E l 6A 4 4 6B 6A 4 4 6A 8, (4.) I tems of the oigial paametes A, B, ad E, the spectum fomula i dimesios eads E l A B, (4.) A 4 A 6 6 whee coditios fo boud states becomes A, ad B We will oly take the (-) sig i (4.) as explaied below. The s-wave spectum fomula i thee dimesios is the oly exact solutio which is obtaied by settig i (4.). Howeve, fo othe highe states, the above solutio is acceptable with high accuacy as fa as the coditio is satisfied. The tasfomatio s tah x makes the domai of the fuctio be,. This suggests a chage of vaiable z s, to big the domai to that of Jacobi polyomials which ae well kow classical othogoal polyomials. By usig (4.4) i c5 c5 c4 c4 Eq. (3.3.a), we obtai c5 z z z 5, whee 4c4 c, ad 8c c / c. The weight fuctio ca be easily calculated usig Eq. (4.6) i, which gives c 7 ( z) z ( z) c7 c6 c6 c7, whee

7 c6 8 c ad 4 c7 c / c. The solutio of Eq. (3.4) i ou case is witte i the followig Rodigues fomula: c7 c6 c d c 7 7 c6 c7 y z C z ( z) z ( z) dz, (4.) By compaiso to Jacobi polyomials, we coclude that c6 c7, c7 P z c6 c7, c7 y z P z, whee is the Jacobi polyomial of ode i z. Thus, the boud state solutio of the adial wave equatio ow eads c6 c7 / c7 c 6 c7, c7 g tah sech P tah, (4.3) whee is ust a omalizatio costat. We must claify hee that fo Jacobi polyomials, we have to have c6 c7, c7. Thus, the paametes c ad c ae chose to satisfy c 4 c, ad c c c. Moeove, sice those paametes deped o the eegy as we metioed peviously, this yields us to eect the (+) sig i (4.) ad (4.). Cosequetly, boud states occus fo B 35 /6 (which does ot violate the old estictio B /6 ), ad E k /. The latte coditio o E is aleady satisfied as we ca see i (4.), so we do t have to woy about it. To calculate the omalizatio costat, we fist use the followig idetity of Jacobi polyomials [4] a, b a b m m P ( y) y y m m m, (4.4) Next, we use the omalizatio costait whee y tah, this gives dy g( ) d g( y) y y, a b / m m m am bm a, b y y P y dy, (4.3) To calculate the itegal i (4.3), we will use the followig vey useful itegal fomula [4] 7

8 cd c d a, b ( y) ( y) P ( y) dy ( ) ( c d ) ( a ), (4.4) 3 ( c ) ( d ) ( a ) F, a b, c ; a, c d ; whee 3 F a, b, c; d, e; f is the geealized hypegeometic fuctio [4]. By diect compaiso betwee (4.3) ad (4.4) we get /, whee is give below a b a m b m a F, a b, a m ; a, a m b ; amb ( ) ( ) ( ) / m m m ( ) (a m b ) ( a ) 3 whee a c6 c7 ad b c7., (4.5) The oly issue that is left fo discussio i this sectio is that the solutios of the adial wave equatio N g, whee N deotes the maximum umbe i which we get boud states, ae ot othogoal! But they ae omalized as we discussed above. We kow that Hemitia opeatos with distict eigevalues must have othogoal eigevectos [43]. To solve this poblem oe must use the method of Gam-Schmidt (GS) to obtai a othoomal set N by liea combiatios [44]. The latte set will be the solutios of the adial wave equatio. The pocess is a bit legthy ad we will ot be able to do it hee. Howeve, we ecouage the iteested eade to do these calculatios by efeig to the pocess of GS. 5 Solutios of the agula equatios It is well-kow fom liteatue that solutios of (7) ae witte i tems of Jacobi polyomials as follows [4] c, d H y N y y P y, (5.) whee N is ust a costat facto, / d, ad / c. Moeove, the latte paametes ae witte i tems of the quatum umbes as c d c l /, which yields l / ad l l. The above solutio was obtai usig diffeet methods icludig the NU techique [5]. Hece, solutios of (7) ae witte below si c c cos l, H N P, (5.) To solve Eq. (8), we itoduce coodiate tasfomatio as y cos, which gives, 8

9 d d y y l l U y H y dy dy y, (5.3) whee U y V y y y / y fo eal paametes,, ae elated to,, by a facto of /. Eq. (5.3) is of Hypegeometic type with the polyomials beig ( y) y, ( y) ( ) y, ad ( y) y y, whee l l, l l. The solutios of (5.3) ae witte as H ( y) ( y) Y( y), whee ( y) satisfies (3.3.a). Next, we eed to fid the fuctio ( y) usig (3.3.e), we fid that this fuctio takes the followig fom, ad ( 3 u ) y, (5.4) u y 3 whee u k, ad the paamete k defied i (3.3.f) must satisfy 4k 4 / u. The latte costait will be used late to obtai the eigevalues of Eq. (5.3). Now, we use (5.4) i (3.3.b), we get y ( u) y, which satisfies u d y / dy. Usig (3.3.a), we ca obtai ( y) to be y y y u u u u, whee u ( 3 u), ad uu. we ca also calculate the weight fuctio by, which gives ( y) y y solvig the polyomials Y ( y ) eads y y u u u u. The Rodigues fomula of d u u u u u u u u Y y y y dy, (5.5) whee is ust a costat. By diect compaiso with the Rodigues fomula of Jacobi ( u u, u u polyomials [xx], we coclude that Y y P ) ( y). As equied by Jacobi polyomials, we must impose that u u. Now, we use (5.4) ad (3.5) i (3.3.f) to obtai the followig quadatic fomula fo k The solutios of (5.6) ae give below ( 3) 3 k k ( ) 4, (5.6) k, (5.7) 9

10 Moeove, we use 3 4k 4 / k to obtai aothe solutio fo k 8k , (5.8) (( 3) 4 4 ) 6( (( 3 ) 4 )) iect compaiso betwee (5.7) ad (5.8) gives 4 4, (5.9) (( 3) 4 4 ) ( ) 4 ( (( 3) 4 )), (5.) 6 I the ext sectio, we will coside diffeet examples ad ty to obtai the ukow paametes fo each case. 6. Results ad iscussios I this sectio, we will discuss diffeet examples that ae cosideed as special cases of the potetial i (.). As a fist example, we coside the case whe,, which is equivalet to the followig ocetal Hypebolic potetial B csc, (6.) tah V (, ) A tah I this case, we have u. Thus, solutios of Eq. (5.3) eads whee u ( 3 u) u u u (, ) H N si P (cos ), (6.), 8k 3 u l l k, ad k is give below l l l (( 3) 4 4l l ) 6( (( 3) 4 l ad the coespodig eigevalues is obtai fom Eq.(5.) as, )) (6.3) ( ) 4l l (( 3) 4 4l l ) 6 ( (( 3) 4l l )) (6.4) ()The ext special case of ou potetial model is cosideed whe we choose the ig shaped paametes,, which coespods to the followig potetial

11 V (, ) A tah cot cot csc csc B (6.5) tah Ude these coditios, we have 3 u, l l, u, u, u, (6.6) 3 u l l k The k values ad the coespodig eigevalues ae obtaied as follows: 8k l l l (( 3) 4 4 4l l ) 6( (( 3) 4 4l )) (6.7) (( 3) 4 4 4l l ) ( ) 4 4l l 6 ( (( 3) 4 4l l )) The associated uomalized wave fuctio is obtai as, (6.8) u u u u ( u u, u u ) H N cos cos P (cos ) (6.9) (3) Aothe special case of ou study is whe,,, which coespods to the followig potetial B cot V (, ) A tah (6.) tah With these assumptios, we have, l l, l l, 3 u 3 (6.) u, u, u l l k Ude this special case, we obtai the k paamete, the eigevalues ad the coespodig wave fuctio as follows: k l l 4 l l l l l 8, (( 3) 4 l 4 ) 6( (( 3) 4 )) (6.)

12 (( 3) 4 l l 4 l l ) 6 ( (( 3) 4 )) (6.3) ( ) 4 l l, u u u H N si P (cos ) (6.4) (, ) l l l l Howeve, oe eeds to be caeful hee as fo thee will be o ig shaped tem ad oe eds up with Hypebolic PT potetial plus pseudo cetifugal tem B V (, ) A tah (6.5) tah (4) We coside the last special case fo,, which coespods to the followig potetial of the fom, B cot csc csc V (, ) A tah (6.6) tah The followig paametes ae obtai ude this case,, l l, l l, 3 u 3 (6.7) u, u l l k, u u Usig Eq.(6.7), we obtai the k -paamte, the eigevalues ad the coespodig wave fuctio fo this special case as follows: 8k l l 4l l l l 4l l ) 6( l l (( 3) 4 l l ) ) (( 3) 4 (6.8) l l l l ( ) 4l l (( 3) 4 l l 4l l ) ( (( 3) 4 )) (6.9) u u u u ( u u, u u ) H N cos cos P (cos ) (6.) 7. Coclusios I this pape, we have obtaied aalytically the solutios of the -dimesioal Schödige potetial with hypebolic Poschl Telle potetial plus a geealized igshaped tem. We employed NU ad tial fuctio methods to solve the adial ad agula pat of the Schödige equatio espectively. This esult is ew ad has eve bee 6

13 epoted befoe i the available liteatue to the best of ou kowledge. Fially, this esult ca fid may applicatios i atomic ad molecula physics. Appedix A: Jacobi Polyomials, Jacobi polyomials P( ) ( y) defied o, ae solutios of the followig secod ode liea diffeetial equatio [8]: d d (, ) y y P ( y) dy, (A) dy We also metio thei Othogoality elatio:,, y y P P dy m, m!, (A) Appedix B: Hypespheical Coodiates x,,... Hypespheical Catesia coodiates below [37]: The -dimesioal positio vecto is defied i tems of whee 3,4,...,, x cos, ad x cos si...si, (B) x si si...si, (B) x cos si...si, (B3) x. Fo =, this is the case of pola coodiates, with x x cos ad x y si, wheeas = 3 epesets the spheical coodiates,, x y cos si, ad x3 z cos., whee x x cos si, 3

14 The volume elemet i -dimesio is defied to be whee,,, ad, (si ) dv d d,, fo. The Laplacia opeato i dimesios is defied below, L si si si (B4) Fially, we metio the omalizatio coditios of the wavefuctio i the dimesios g d, (B5) H (si ) d, (B6) Refeeces [] M.C.Zhag,G.H.Su ad S.H.og,Phys.Lett.A 374()74 [] A..Alhaidai,J.Phys.A:Math.Ge.38(5)349 [3] C.Y.Che ad S.H.og,Phys.Lett.A 335(5)374 [4] C.Bekdemi ad R.Seve,J.Math.Che.46(9) [5] H.Hatma ad.schuch,it.j.quat.che.8(98)5 [6] G.E.agaascu,C.Campigotto ad K.Kible,Phys.Lett.A 7(99)339 [7] C.Y.Che,Y.W.Liu,HEP 3(999)865 [8] C.Quese,J.Phys.A (988)4633 [9] S.H.og,G.H.Su ad M.Lozada-Cassou,Phys.Lett.A 38(4)99 [] C.Bekdemi,J.Math.Chem.46(9)39 [] H.Hassaabadi,A.N.Ikot ad S.Zaikama,Acta Phys.Polo.A 6(4)647 [] M.C.Zhag,B.A ad H.Guo-Qig,J.Math.Chem.48()876 [3] C.Y.Che ad S.H.og,Phys.Lett.A 335(5)374 [4] Y.F.Cheg ad T.Q.dai,Phys.Sc.75(7)74 [5] A.N.Ikot,E.Olga ad H.Hassabadi,Gu J.Sci. 9(6)937 [6] A.N.Ikot,I.O.Akpa.T.M.Abbey ad H.Hassaabadi, Commu.Theo.Phys.65(6)569 [7] A.N.Ikot,T.M.Abbey,E.O.Chukwuocha ad M.C.Oyeau,Ca.J.Phys.94(6)57 4

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