l -State Solutions of a New Four-Parameter 1/r^2 Singular Radial Non-Conventional Potential via Asymptotic Iteration Method

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1 America Joural of Computatioal ad Applied Mathematics 8, 8(): 7-3 DOI:.593/j.ajcam.88. l -State Solutios of a New Four-Parameter /r^ Sigular Radial No-Covetioal Potetial via Asymptotic Iteratio Method A. J. Sous Faculty of Techology ad Applied Scieces, Al-Quds Ope Uiversity, Nablus, Palestie Abstract I the preset work, we give a umerical solutio of the radial Schrödiger equatio for ew four-parameter radial o-covetioal potetial, which was itroduced by Alhaidari. I our calculatios, we applied the asymptotic iteratio method (AIM) to calculate the eigevalues of the potetial for arbitrary parameters ad ay l state. It is foud that this method gives highly accurate results that compares favorably with other. Moreover, some ew results were preseted i this paper. Keywords Asymptotic Iteratio Method, /r sigular potetial, igevalues, Radial Schrödiger equatio, Tridiagoal represetatios. Itroductio There has bee a growig iterest i ivestigatig the umerical solutios of the Schrödiger equatio for some physical potetial models. This is because exact solutios of the Schrödiger equatio are very limited, ad oly obtaiable for a few umber of physical potetials [-3]. The hyperbolic potetials are commoly used to model iter-atomic ad itermolecular pheomea. Amog such potetials are the Pöschl-Teller, Rose-Morse, Scarf, ad hyperbolic sigle wave potetial, which have bee studied extesively i the literatures [4-]. The hyperbolic potetial uder ivestigatio here is, i fact, a geeralizatio of the hyperbolic Pöschl-Teller potetial. This four-parameter radial potetial was itroduced by A. D. Alhaidari [, 3] ad reads as follows 4 V( r) = [ V + Vtah ( λr) + Vtah ( λr), sih ( λr) () { λ, V i } are real parameters such that λ >, V >, V, ad λ is a legth scale that determies the rage of the potetial. Near the origi, it is /r sigular, but as r it decays expoetially to zero sigifyig that it is short-rage. There are three distict physical cofiguratios of the potetial (). The first oe is whe the potetial has two local extrema (oe local miimum ad oe local maximum). I * Correspodig author: asous@qou.edu (A. J. Sous) Published olie at Copyright 8 Scietific & Academic Publishig. All Rights Reserved this cofiguratio, the potetial could have resoaces but o boud states, or it could have both. The secod cofiguratio occurs whe the two extrema coicide at a iflectio poit or whe the potetial has o local extrema. I these two cases, the potetial ca support either boud states or resoaces. I the third cofiguratio, the potetial has oe local miimum ad could support oly boud states but o resoaces. Due to the shortess of the potetial rage, we expect that the size of the boud states eergy spectrum to be fiite. These speculatios will be verified below [, 3]. If V =, potetial () becomes the well-kow hyperbolic Pöschl-Teller potetial, which belogs to the covetioal class of exactly solvable problems [, 3] ad has the followig eergy spectrum formula: ( 4 λ ) 4 λ λ = V V () where =,,..., N ad N is the largest iteger less tha or equal to. 4 V λ 4 + V λ For o-zero agular mometum, A. D. Alhaidari wrote V the four-parameter radial potetial as V() r = + V () r, r which makes V () r a o-sigular short-rage potetial ad gives the total Hamiltoia as h d ( + ) + V H = + + V () r m dr r Thus, the time-idepedet radial Schrödiger s equatio ca be writte as ( ) () () () () r ψ m r V r + + ψ r = ψ r + (3) (4)

2 8 A. J. Sous: l -State Solutios of a New Four-Parameter /r^ Sigular Radial No-Covetioal Potetial via Asymptotic Iteratio Method I this paper, we preset the Asymptotic Iteratio Method (AIM) to calculate the eigevalues ot oly for oe-dimesioal Schrödiger equatio, but also for the three-dimesioal ad spherically symmetric radial case.. Basic lemets of the Asymptotic Iteratio Method The AIM method is based o solvig a secod order differetial equatio of the form: λ f = f + s f (5) λ ad λ, s are fuctios i C ( ab, ), the prime deotes the derivative with respect to x. The variables, s ( x ) ad λ are sufficietly differetiable. To fid a geeral solutio to this equatio, we differetiate (5) with respect to x ad fid λ f = f + s f (6) = + + λ λ s λ, = + λ s s s Similarly, the secod derivative of (5) yields (4) λ (7) f = f + s f (8) = + + λ λ s λ λ, = + λ s s s quatio (5) ca be easily iterated up ( k + ) th ad ( k + ) th derivatives, k =,,3,... Therefore, we have the recurrece relatios ( k + ) f = f + sk f () ( k + ) f = λ f + s f k k k = k + k + k λ λ s λ λ, k = k + s s s (9) () From the ratio of the ( k + ) th ad ( k + ) th derivatives, we have ( k + ) ( k + ) d ( k + ) f l[ f ] = f = λ s λ [ f f ] k k + s [ f f ] k k + () We ow itroduce the asymptotic aspect of the method []. If we have, for sufficietly large k the, () reduces to sk s λ ( ) k k = = α (3) x ( k + ) ( k + ) x ( k + ) f d f l[ f ( )] = = (4) which yields the geeral solutio of (5) ca be obtaied as: x x x α λ (5) f = exp( )[ C + C exp( [ ( x ) + α( x )] ) ] The termiatio coditio of the method i (3) ca be arraged as = λ s λ s (6) k k k k k where k shows the iteratio umber. For the exactly solvable potetials, the eergy eigevalues are obtaied from the roots of (6), ad the eigevalues will ot depeds to chose of x, ad the radial quatum umber is equal to the iteratio umber k for this case. However, for otrivial potetials that have o exact solutios ad for a specific pricipal quatum umber, we choose a suitable x poit, determied geerally as the maximum value of the asymptotic wave fuctio or the miimum value of the potetial [-5]. The, the approximate eergy eigevalues are obtaied from the roots of q. (6) for sufficietly large umber of iteratios k, where k is always greater tha. The stability of the results i AIM depeds o differet factors: First; a appropriate choice of coordiate trasformatios ad fuctioal trasformatios to trasfer Schrödiger equatio ito suitable form, because of the fact that choosig usuitable coordiate trasformatios lead to istability results. Secod; the stability of results depeds o choosig a suitable x poit which leads to correct ad stability results, ad if we chage x poit to aother value, we might reach to istability results, that is why it is critical to choose the right iitial value x very carefully. Third; whe choosig suitable x poit ad substitutig it i equatio (6), the as much as we icrease the iteratios we ca reach to covergece ad stability results. 3. The Potetial igevalues by AIM I order to overcome the covergece problem oted above, we make a chage of variables as follows x= tah( λr) After makig the coveiet chage of variable, a straightforward calculatio shows that equatio (4) becomes

3 America Joural of Computatioal ad Applied Mathematics 8, 8(): h d d λ ϕ ω + ( λϕ χ) ψ+ m d (7) ll ( + ) ( λ + V ) ψ = η with ω = ( x+ ), (8) ϕ = ( x ) (9) χ = 3x+ () η = arctah ( x+ ) () ϕ V = [ V + Vω + Vω ] () ω 4 the, we obtai the secod-order homogeeous liear differetial equatio i the followig form: ad d ψ d ψ = λ + s ψ (3) χ λ = (4) ϕ ω m s = [4 ϕ η( xv ) 8 h λϕ ω η + ϕ ω η V+ ϕ ω η V + 4 ω η( x ) λ ll ( + ) ω] (5) It is ow possible to calculate λ k, ad sk ( x ) applyig equatio (). Fially, oe fids the eergy eigevalues of the potetial i () by usig the quatizatio coditio give i (6) ad choosig the proper iitial poit x. I order to improve the eergy eigevalues, the iteratio umber has to be icreased withi the rage of covergece of the method. 4. Results ad Discussios The AIM method should satisfy the coditio i equatio (6), ad the results i Table prove the covergece of the method, our results for show very good agreemet with the results of = which obtaied from (). Covergece seems to take place with icreased the umber of iteratios k. I Table, a compariso betwee AIM results ad umerical results obtaied by the tridiagoal represetatio approach (TRA) for the potetial () with V =, V = 5, V =, λ =, l =, ad = m =. It is foud that the results obtaied by AIM are i good agreemet with the results of the tridiagoal represetatio approach ad up to 8 sigificat digits i the groud state. I Table 3, we preseted the eigevalues for the potetial () whe V = which correspods to the well-kow ad exactly solvable hyperbolic Pöschl-Teller potetial. This is to illustrate the accuracy of the AIM as compared to the TRA. We took V =, V = 5, V =, λ =, l =, ad = m =. The Table shows that the eigevalues calculated by AIM, are i a good agreemet with the exact values obtaied from equatio (). I Table 4, we compare the boud states eergies obtaied by the AIM ad those obtaied by the complex scalig method i [, 3]. The parameters are take as V =, V = 8, V =, λ = ad for various values of the agular mometum l. Our results are i good agreemet with those listed i [, 3]. I Table 5, we preseted the eigevalues for the potetial (), the results obtaied with V =, V = 7, V =, λ =, ad various values of the agular mometum L. The results i this Table cosidered as ew results which have ot bee addressed before, ad it cofirm that the AIM is valid for arbitrarily values of parameters. I our calculatio we have chose several values for x ad foud the uique oe that does ot produce istability ad that value was x =. Fially, we poit out that the accuracy of the results for the higher excited states could be icreased if the umber of iteratios were icreased limited oly by the stability ad covergece properties of the AIM. Table. The rate of covergece of AIM for groud state, with V = 8, V = 8, V =, λ =, l = = m =,, computed with the umber of iteratios k k Table. A compariso of the eergy eigevalues with ad = m = by TR Approach of the potetial () V =, V = 5, V =, λ =, l = by AIM

4 3 A. J. Sous: l -State Solutios of a New Four-Parameter /r^ Sigular Radial No-Covetioal Potetial via Asymptotic Iteratio Method Table 3. A compariso of the exact eergy eigevalues of the potetial () with V = as obtaied by the exact formula of q. () ad those obtaied umerically by the AIM ad the TRA. We took = m = V =, V = 5, V =, λ =, l =, ad xact(q. 3) by AIM by TRA ACKNOWLDGMNTS I would like to thak Prof. A. D. Alhaidari for his commets, ad for his fruitful discussios ad valuable suggestios o this writte work. May thaks also to Dr. g. Islam Amro Director of (ICTC), ad to Dr. Bassam Al Turk ad all techical staff i (ICTC) for their assistat durig the research Table 4. A compariso of the eergy eigevalues of the potetial () obtaied here by the AIM ad compared to those obtaied by the complex scalig Method (CSM) i [3]. We took V =, V = 8, V =, λ = ad for various values of the agular mometum L by CSM i [5] by AIM Table 5. The eergy eigevalues agular mometum L of the potetial () with V =, V = 7, V =, λ =, ad various values of the L= L= L= L= Coclusios We calculated the eergy eigevalues of the Schrödiger equatio for a ew four-parameter /r^ sigular o-covetioal potetial usig the AIM. Our method is easy to apply ad leads to a good agreemet with the complex scalig method (TRA, also its agreemet with the results obtaied by tridiagoal represetatio approach (TRA). To the best of our kowledge, this paper is the first to study the eigevalues associated with this ew four-parameter radial o-covetioal potetial. RFRNCS [] H. Bahlouli ad A. D. Alhaidari, xtedig the class of solvable potetials: III. The hyperbolic sigle wave, Phys. Scr (). [] A. D. Alhaidari, Solutio of the orelativistic wave equatio usig the tridiagoal represetatio approach, J. Math. Phys. Vol. 58, 74 (7). [3] A. D. Alhaidari, xtedig the class of solvable potetials. IV Iverse square potetial with a rich spectrum, [4] A D Alhaidari, H. Bahlouli, Two New Solvable Potetials, Joural of Physics A: Mathematical ad Theoretical, 4, No. 6 (9). [5] AbdallahJ. Sous, The Asymptotic Iteratio Method for the igeeergies of the a Novel Hyperbolic Sigle Wave Potetial, Joural of Applied Mathematics ad Physics, Vol.3 No. (5). [6] D. Agboola, Solutios to the Modified Pöschl Teller Potetial i D-Dimesios, Chiese Physics Letters 7(4), 43 (). [7] Chu-Sheg Jia, Tao Che, Li-Gog Cui, Approximate aalytical solutios of the Dirac equatio with the geeralized Pöschl Teller potetial icludig the pseudo-cetrifugal term, Physics Letters A, 6, (9). [8] Gao-Feg Wei, Shi-Hai Dog, The spi symmetry for deformed geeralized Pöschl Teller potetial, Physics Letters A, 373, 48 (9). [9] Miloslav Zojil, pt -symmetrically regularized ckart, Pöschl-Teller ad Hulthé potetials, Joural of Physics A: Mathematical ad Geeral, 33, (9). [] A. D. Alhaidari, Solutio of the orelativistic wave equatio usig the tridiagoal represetatio approach, Joural of Mathematical Physics, 58, 74 (7). [] Hassaabadi Hassa, Yazarloo Betol Hoda, Liag Liag Lu, Approximate Aalytical Solutios to the Geeralized Pöschl Teller Potetial i D Dimesios, Chi. Phys. Lett. 9 33, (). [] Haka Ciftci, Richard L. Hall, Nasser Saad, Asymptotic iteratio method for eigevalue problems, J. Phys. A 36, 87 (3). [3] T. Barakat, The asymptotic iteratio method for the eigeeergies of the aharmoic oscillator potetial, Physics Letters A, 4 (5).

5 America Joural of Computatioal ad Applied Mathematics 8, 8(): [4] Abdallah. J. Sous, The Asymptotic Iteratio Method for the igeeergies of the a Novel Hyperbolic Sigle Wave Potetial, Joural of Applied Mathematics ad Physics, 3, 46 (5). [5] Abdullah. J. Sous, Abdulaziz D. Alhaidari, ergy Spectrum for a Short-Rage /r Sigular Potetial with a No-Orbital barrier Usig the Asymptotic Iteratio Method, Joural of Applied Mathematics ad Physics, 4, (6).