SOLUTION OF THE RADIAL N-DIMENSIONAL SCHRÖDINGER EQUATION USING HOMOTOPY PERTURBATION METHOD

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1 SOLUTIO OF THE ADIAL -DIMESIOAL SCHÖDIGE EQUATIO USIG HOMOTOPY PETUBATIO METHOD SAMI AL-JABE Depatmet of Physics, a-aah atioal Uivesity, ablus, P.O.Box 7, Palestie. eceived Jue, Homotopy Petubatio Method (HPM) is applied to fomulate aalytic solutio of the fee paticle adial depedet Schödige equatio i -dimesioal space. This method is based o the costuctio of a homotopy with a embeddig paamete δ,. The method shows its effectiveess, usefuless, ad simplicity fo obtaiig appoximate aalytic solutio. I additio, some iteestig cases ae aalyzed ad the effect of space dimesio o the solutio is poited out. Key wods: Homotopy petubatio method, highe dimesios, appoximatio methods.. ITODUCTIO The seach fo aalytic exact o appoximate solutios of diffeet kids of diffeetial ad itegal equatios has bee of geat impotace ove the yeas. ecetly, a ew aalytic techique based o basic ideas of homotopy, called Homotopy Petubatio Method (HPM), was developed by He [ 4] ad has bee of geat potetial i solvig diffeet kids of diffeetial ad itegal equatios i may aeas of applied scieces ad egieeig. The method is a couplig betwee the taditioal petubatio method ad homotopy, which is a highly iteestig ad useful cocept i topology, ad defoms cotiuously to a simple poblem which is easily solved. The HPM equies the costuctio of a homotopy with a embeddig paamete p [, ] which is cosideed as a small paamete whose age elates the iitial solutios of a poblem to its fial solutios. If p is, the poblem educes to a sufficietly simplified fom, which omally admits a athe simple solutio. As p gadually iceases to, the poblem goes though a sequece of defomatio. The fial state of defomatio is achieved whe p is ad the desied solutio is the obtaied. The method is cosideed as a summatio of a ifiite seies which usually coveges apidly to the exact solutio. The HPM om. Jou. Phys., Vol. 58, os. 3 4, P , Buchaest, 3

2 48 Sami Al-Jabe has bee vaiously developed i diffeet aeas of applied scieces: It has bee applied to diffeet types of oliea poblems [5-], to itegal equatios [3-6], ad to oliea diffeetial equatios with factioal time deivatives [7- ]. The HPM has also bee used to fid solutios to oliea Schödige equatio [-4] ad to some kid of bouday value poblems [5, 6]. Futhemoe, the HPM was employed to fid solutios to heat tasfe poblems ad heat distibutio fo systems with vaiable themal coductivity [7-3]. Ove the last two decades, poblems i the -dimesioal space ae becomig iceasigly impotat i diffeet aeas: Fo example, i quatum field theoy [3], i quatum chemisty [3], i adom walks [33], i Casimi effect [34], ad i hamoic oscillatos [35, 36]. Futhemoe, the -dimesioal adial Schödige equatio has bee examied fo diffeet kid of potetials [37-39]. I the peset pape, the HPM is poposed to povide appoximate solutios fo the fee paticle adial pat of Schödige equatio i highe dimesioal space. The outlie of this pape is as follows: Sectio gives details of He's homotopy petubatio method ad shows how it ca be applied to the dimesioal Schödige equatio. Sectio 3 pesets details of applicatio of HPM to ou poblem ad the esults obtaied. Sectio 4 is devoted fo coclusios.. HOMOTOPY PETUBATIO METHOD To illustate the basic ideas of the HPM, we coside the followig oliea diffeetial equatio: with bouday coditios: A( u) F( ), Ω () du B( u, ), Γ () d Whee A is a geeal diffeetial opeato, F( ) is a kow aalytic fuctio, B is a bouday opeato ad Γ is the bouday of the domai Ω. The opeato A ca be divided ito two pats: Liea opeato L ad oliea opeato M so that Eq. () becomes L( u) + M ( u) F( ). (3) Followig He [, ], we costuct a homotopy: V (, p) : Ω [, ] which satisfies o [ ] [ ] H ( v, p) ( p) L( v) L( u ) + p A( v) F( ), p [, ], Ω (4)

3 3 Solutio of the adial -dimesioal Schödige equatio 49 [ ] H ( v, p) L( v) L( u ) + p ( v) F( ), p [, ], Ω (5) Whee p [, ] is a embeddig paamete, u is a iitial appoximatio of Eq. () which satisfies the bouday coditios. Eq.'s (4) ad (5) give espectively; H ( v, ) L( v) L( u ) (6) H ( v, ) A( v) F( ) (7) The chagig values of p fom zeo to uity is ust that of V (, p ) fom u ( ) to u( ). I topology, this is called defomatio ad L( v) L( u ), A( v) F( ) ae called homotopic. Cosideig the embeddig paamete p, as a small paamete, ad applyig the taditioal petubatio techique, we ca assume that the solutio of Eq.(4) o (5) ca be expaded as a powe seies i p; amely 3 4 v v + pv + p v + p v + p v... (8) Settig p yields the iitial appoximatio u ( ), while p gives u( ) as u lim ( u) v + v + v + v + v +... (9) p FEE-PATICLE -DIMESIOAL ADIAL SCHÖDIGE EQUATIO The Laplacia opeato i the -dimesioal spheical coodiates (, θ, θ,..., θ, ϕ ) has the fom ( ) + Λ, whee Λ is a patial diffeetial opeato o the uit sphee S give by [4] k k + k si (si k ) si k si k k k Λ θ θ θ + θ θ θ ϕ () () The sepaatio of vaiables method sepaates the fee-paticle -dimesioal Schödige equatio ito two secod-ode diffeetial equatios [4]; amely Λ Y + β Y () d ( ) d β + + k, d d (3)

4 5 Sami Al-Jabe 4 whee k me / h ad β is a sepaatio costat whose values ae give by [4] β l( l + ). (4) The solutios to Eq.() ae the hype-spheical hamoics, S { m} Y l of degee l o the uit sphee. Fo each o-egative itege l, the umbe of hypespheical hamoics is give by [4] l (l + )( l + 3)!, l!( )! ad ae chaacteized by the iteges m, m, m3,..., m with the estictios (5) l m m m3... m. (6) The hypespheical hamoics fom a othoomal set ad thus they fom a stadad basis of the ieducible epesetatios of the otatio goup SO( ) i the space of squae itegable fuctios defied ove the suface of the -dimesioal uit sphee with the ivaiat measue [4] ( ) dω si θ dθ dϕ. (7) I ode to fid the solutio to the adial pat give i Eq.(3), we coside the diffeetial equatio Y a Y c a p c + ( bcx ) Y, + + dx x dx x d d ( ) whose solutio is give by [38] (8) a c c Y ( x) x AJ p ( bx ) + B p ( bx ), whee J p ad p ae the odiay Bessel ad euma fuctios, espectively, ad a, b, c ad p ae costats. Compaig Eq.(3) with Eq.(8) yields b k, c, a ( ) /, ad p + ( ) /. Theefoe, with the use of Eq. (9), the solutio to Eq. (3) ca be immediately witte dow: A J k B k ( ) / ( ) + ( ) / ( ) + + ( ) / ( ), whee A ad B ae costats. The fuctio ( ) must be fiite at ad thus the secod tem i Eq. () must be eected due to the sigula behavio of the euma fuctio at the oigi. Theefoe, the solutio i Eq. () educes to (9) ()

5 5 Solutio of the adial -dimesioal Schödige equatio 5 A Jl k, () ( ) / ( ) + ( ) / ( ) which ca be witte i tems of the spheical Bessel fuctio as (3 ) / ) A l ( k). () ( + ( 3) / The commo epesetatio of m (k) is give by p m p ( ) k ( m + p)! m+ p m, (3) p p! (m + p + )! ( k) A m ad thus Eq. () gives the seies fom of the adial solutio (), amely p l+ ( 3) / p ( ) k ( l + p + ( 3) / )! l p p! (l + p + )! ( ) A l+ p. (4) 4. SOLUTIO OF THE ADIAL PAT OF THE FEE-PATICLE'S SCHÖDIGE EQUATIO I DIMESIOS USIG HPM I this sectio, the HPM is implemeted, i a efficiet way, to fid appoximate solutios fo the adial pat of Schödige equatio i -dimesioal space subect to the coditio that ( ) is fiite at. Witig the Eq.(3) i the fom d d 3 + ( ) + k ( + ), d d l l (5) ad i view of Eq.(4) o (5), the homotopy fo Eq.(5) ca be costucted as d ( ) d pk l( l + ) (, ) + +, 3 H p d d (6) with p [, ]. The basic assumptio of the HPM is that the solutio ( ) ca be expessed as a powe seies i p, amely 3 3 (7) ( ) p ( ) ( ) + p ( ) + p ( ) + p ( ) +... Cosideig tems up to thid powe i the paamete p ad substitutig Eq.(7) ito Eq.(6) yields

6 5 Sami Al-Jabe p + p + p ( ) + p + p + p 3 + pk l( l + ) 3 + p + p + p 3. 3 Summig up the coefficiets of equal powe of p ad settig each sum to zeo, gives the followig equatios; 3 (8) p : + ( ) l( l + ) (9) p : + ( ) l( l + ) k (3) 3 p : + ( ) l( l + ) k (3) 3 p : + ( ) l( l + ) k (3) p : + ( ) l( l + ) k (33) 3 whee () () fo,, 3,... I ode to solve Equatios (9-3), a seies method ca be used. This gives ( ) C + C l l + 3, (34) Whee C ad C ae costats. Sice () is fiite, the the costat C must be eected ad theefoe Eq.(34) becomes ( ) C l. (35) Substitutig ( ) ito Eq.(3) ad agai applyig the seies method, we get ( ) ( l + ) k C l+. (36) Similaly, the substitutio of ( ) ito Eq. (3) ad solvig the esultig equatio gives 4 k C l+ 4 ( ) 8( l + )(l + + ). (37) A futhe substitutio of ( ) ito Eq.(3) ad solvig the esultig equatio by seies method yields

7 7 Solutio of the adial -dimesioal Schödige equatio 53 6 k C 3 ( ) 3 3!( l + )(l + + )(l + + 4) l+ 6 th I geeal the tem of the solutio is give by. (38) C ( ) k (l + )! ( l + + ( 3) / )! ( ) ( l + ( 3) / )!! (l + + )! l+. (39) The above equatio ca be poved by mathematical iductio as follows: Fo, Eq.(39) educes to ou esult give i Eq.(36). Employig Eq.(33) fo + ad usig Eq.(39), we get l( l + ) + l+ 3 k f ( ), whee f ( ) is the coefficiet of ( ) give by Eq.(39). Assumig a seies solutio of the fom which upo its substitutio ito Eq.(4) gives ad thus (4) ( ) + C, (4) 3 [ ( ) ( ) ( ) ] + ( ) + l+ + + C l l k f. (4) Equatig powes of o both sides of Eq.(4) gives l + +, (43) [ l l l l l ] [( l )( l ) l l( l )] [( l )( l ) ( l ) l l( l )] [( l )( ) ( ) l ( ) ] C [ l ] k f ( ) C ( + + )( + + ) + ( )( + + ) ( + ) which gives C C C ( + )( + + C (44) k f ( ). (45) ( + )(l + + ) The substitutio fo f ( ) by the coefficiet of ( ) of Eq. (39) yields

8 54 Sami Al-Jabe 8 C + ( + ) C ( ) k (l + )! ( l + + ( 3) / )!. (46) ( l + ( 3) / )!( + )!(l + + )(l + + )! Usig the factoial popety ( p + )! ( p + ) p!, we ca wite ( l + + ( ) / )! ( l + + ( 3) / )! ( l + + ( ) / ) (l + + )! (l + + )! (l + + ) (l + + ) The substitutio of Eq.(47) ito Eq.(46) ad pefomig simple algeba, we get C C k ( l + ( 3) / )! ( + )! (l + + )! + ( + ) ( ) (l + )! ( l + + ( ) / )! ad thus, with the help of Eq.(43), Eq.(4) immediately yields (47), (48) + ( + ) C ( ) k (l + )! ( l + + ( ) / )! l+ + + ( ). ( l + ( 3) / )! ( + )! (l + + )! (49) It is clea to ote that lettig + i Eq.(39) yields exactly the esult i Eq.(49) ad thus we poved ou assetio i Eq.(39) by mathematical iductio. I ode to compae the solutio pedicted by the HPM, give by Eq.(39), with the exact solutio give by Eq. (4), we let m l + ( 3) /, so that (l + )! (m + )! m (m + )!! (5) 3 m! ( l + )! The last step ca be poved by mathematical iductio as follows: It is tivial fo m, fo m m +, we get (m + 3)! (m + 3)(m + )(m + )!, ( m + )! ( m + ) m! which, upo usig Eq.(5), we get (m + 3)! m + ( m + )! m+ ( 3)!!, QED (5) Theefoe, upo compaig Eq.(4) with Eq.(39), we get l+ ( 3) / l+ ( 3) / C (l + )!! Al, ad thus we have A C l (l + )!! (5)

9 9 Solutio of the adial -dimesioal Schödige equatio 55 Theefoe, the HPM solutio yields the exact solutio by choosig the abitay costat C to satisfy Eq.(5). It is woth to check the covegece of the solutio ust obtaied by the HPM. Oe ca check this by calculatig the atio / +, which ca be computed fom Eq. s (39) ad (49) with the esult + k ( l ( 3) / ). ( + )(l + + )(l + + ) Usig the idetity ( p + )! ( p + ) p!, the above atio becomes + k. (53) ( + )(l + + ) It is oted that, fo a give, l,, this atio deceases as /. 5. SOME ITEESTIG CASES I this sectio, we aalyze some iteestig special cases that shed some light o the HPM solutio. I this case, Eq.(39) yields 5.. THE CASE l C ( ) k ( )!( + ( 3) / ))! (( 3) / )!!( + )!, (54) which immediately gives the fist few tems, amely C C k C k! ( + ) 4 4 (55) C k 3! ( + )( + 4) It is iteestig to wite up to the thid tem, amely

10 56 Sami Al-Jabe + + The extemes of ca be foud by settig d / d to get mi ( + ), (56) k ad oe ca easily check that d / d > so that exhibits miimum at the above value of. The value of this miimum is C ( ) mi (57) It is obvious that the miimum of iceases with the space dimesio. It stats at C / 6 i the thee-dimesioal space ad C / i the ifiite-dimesioal space. It is temptig to coside the lage limit whee oe may deduce fom Eq. (55) that so that the solutio could be witte as which exhibits o exteme values. C ( ) k, (58)! k k C C exp!, (59) 5.. THE CASE l Usig Eq.(39), we get C C k ( + ) 3 (6) 3 3 Fo the lage limit we have, C k!( + )( + 4) 4 5 C k 3!( + )( + 4)( + 6) 6 7

11 Solutio of the adial -dimesioal Schödige equatio 57 ad thus the solutio becomes C ( ) k +! k C C exp ( k / )! (6) The extemes of the above solutio ca be foud by settig d / d to get max, (6) k It is a simple matte to check that d / d is egative at exhibits a maximum value, give by C e k ad thus / m (63) It is clea that as the space dimesio iceases, the positio of the maximum ad the value of the maximum shift towads highe values ad i the ifiite dimesioal space ( ), both go to ifiity. max 6. COCLUSIO I this wok, the solutio of the adial -dimesioal Schödige equatio was obtaied ad aalyzed usig HPM. The obtaied esults show the effectiveess ad usefuless of the homotopy petubatio method. It has bee demostated that the solutio obtaied by HPM yields the exact solutio by a pope choice of the abitay costat as see i Eq.(5). Fo computatioal puposes, two special cases had bee cosideed: Fo the case l, we icluded the fist thee tems i the solutio ad it was foud that the solutio exhibits a miimum value of C ( ) / at max ( + ) / k. Fo the case l, ou esults show that, fo the lage limit, the seies solutio pedicted by HPM has a closed fom. This closed fom has a maximum value of C / ke at max / k ad both ae iceasig with the icease of the space dimesio,. Futhemoe, i the ifiitedimesioal space, fo this secod case, the maximum value becomes ifiite at ifiity. Theefoe, this wok demostates the poweful of the HPM ad illustates the effect of the space dimesio o the solutio.

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