Exact Solution of Klein--Gordon Equation by Asymptotic Iteration Method

CHIN.PHYS.LETT.Vol.25,No.6(2008)1939 Exact Solution of Klein–Gordon Equation by Asymptotic Iteration Method∗Eser Ol˘g ar∗∗University of Gaziantep,Engineering of Physics Department,Gaziantep,Turkey(Received4March2008)Using the asymptotic iteration method(AIM)we obtain the spectrum of the Klein–Gordon equation for some choices of scalar and vector potentials.In particular,it is shown that the AIM exactly reproduces the spectrum of some solvable potentials.PACS:03.65.Ge,03.65.FdIn relativistic quantum mechanics,the particle in-teraction should be described by either the Klein–Gordon(KG)equation or Dirac equation.[1,2]Re-cently,there have been many studies on the KG equa-tion with various types of potentials by using different methods to describe the relativistic effects.[3−15]Inter-estingly,in most of the studies,equal scalar S(x)and vector V(x)potentials have been considered in order to obtain the bound-state solutions of the KG equa-tion.On the other hand,there have been very few studies with other choices of V(x)and S(x).[16−19]. Dutra et al.[19]defined a transformation between vec-tor and scalar potentials as V(x)=V0+βS(x),where V0andβare arbitrary constants.This choice is very restricted.Here,to have a bound state solution,βmust be less then1,if V0=0.In this study,to avoid this restriction,we define a more general transforma-tion between corresponding potentials.Recently,some methods have been developed in order to solve the Schr¨o dinger equation.[20−22]One of these methods is the asymptotic iteration method (AIM)which has been introduced by C¸it¸c i et al.[23] Subsequently,AIM has been used in many physi-cal systems to obtain the whole spectra.[23−32]This method reproduces exact solutions of many exactly solvable and also gives accurate result for the non-solvable potentials such as sextic oscillator,deformed Coulomb potential,etc.Although AIM has been ap-plied to solve the Schr¨o dinger equation for many po-tentials,there have been very few studies on its appli-cation to relativistic equations such as the KG equa-tion.In this study,we apply the AIM in order to obtain the bound state energy spectrum for the KG equa-tion with a general transformation between vector and scalar potentials.In order to construct exactly solv-able potentials for the KG equation,the vector po-tentials are chosen linear,exponential and inversely linear potentials.It is important that,when the vec-tor potentials are chosen in this form,the KG equation reduces to a Schr¨o dinger-like equation with harmonic oscillator,Morse oscillator and Kratzer potential,[33] respectively.In this Letter,we deal with the formalism of the one-dimensional KG equation with scalar potential greater than vector potential.A general description of AIM is outlined.Some exactly solvable example are treated by means of AIM.We construct a KG equation including some physical potentials.For this purpose we transform the KG equation in the form of the Schr¨o dinger-like equation,because there have been a large number of papers to tackle the problem in the framework of Schr¨o dinger equation.Generally,the s-wave Klein–Gordon equation with scalar potential S(x)and vector potential V(x)can be written as[11](¯h=1,c=1){d2dx2+[E−V(x)]2−[m+S(x)]2}f(x)=0,(1) where E is the energy,and m is the mass of the par-ticle.Indeed,the radial wave function is expressed as R(x)=f(x)/x.When the relation between the potentials is S(x)≥V(x),it is said that KG equa-tion has a real bound state solutions.When these two potentials is equal to each other,Eq.(1)reduces to the Schr¨o dinger-like equation whose solutions can be obtained with many methods developed in non-relativistic quantum mechanics.In this work,we try to consider the case of S(x)>V(x)for the KG equa-tion.When we rearrange Eq.(1),we obtain{d2dx2−[V eff(x)−(E2−m2)]}f(x)=0,(2) whereV eff=[S2(x)−V2(x)]+2[mS(x)+EV(x)]. Recently in Ref.[19],the relationship between the cor-responding potentials is considered to beV(x)=V0+βS(x),∗Supported by the Research Fund of Gaziantep University and the Scientific and Technological Research Council of Turkey (TUB˙ITAK).∗∗Email:olgar@.trc 2008Chinese Physical Society and IOP Publishing Ltd1940Eser Ol˘g ar Vol.25 where V0andβare arbitrary constants.In this con-straint,in order to have the condition S(x)>V(x),we must take the parameterβis less then1,if V0=0.To avoid this restriction,we define the relationshipbetween potentials asS(x)=V(x)(β−1),β≥0.(3)In this general description of scalar potential,bychoosingβ=0,1,and2,the scalar potentials lead tothe case of S(x)=−V(x),S(x)=0(purely vectorpotential),and S(x)=V(x)respectively.The otherchoices ofβleads to the required condition for thecase of S(x)>V(x).Substituting the expression inEq.(3)in the s-wave KG equation yields{d2 dx2−[V(x)(V(x)A2+B)−ε]}f(x)=0,(4)whereε=(E2−m2),A2=(β2−2β)and B= 2[E−m(1−β)].After some algebraic simplification, Eq.(4)can be written as{d2 dx2−[[AV(x)+B2A]2−δ]}f(x)=0,(5)whereδ=ε+(B/2A)2.Now,after this formalism of the KG equation for unequal vector and scalar potentials,the formulation of the AIM that can be used to obtain the spectra of KG is shown in the following.The AIM is proposed to solve the second-order dif-ferential equations and the details can be found in Ref.[23].First,we consider the following second or-der homogeneous differential equationy (x)=λ0y (x)+s0y(x),(6)whereλ0and s0are the functions,and y (x)and y (x) denotes derivative of y with respect to x.It is easy to show that the(n+2)th derivative of the function y(x) can be written asy(n+2)(x)=λn y (x)+s n y(x),(7)whereλn and s n are given by the recurrence relationsλn=λ n−1+s n−1+λn−1λ0,s n=s n−1+λn−1s0.(8)If we have,for sufficiently large n,λn s n =λn−1s n−1=α(x),(9)then the solution to Eq.(6)can be written as[23]y(x)=exp (−∫xαdt)[C1+C2∫xexp(∫s(λ0+2α)dt)ds].(10)In calculating the parameters in Eq.(8),for n=0,wetake the initial conditions asλ−1=1and s−1=0[26]and∆n(x)=0for∆n(x)=λn(x)s n−1(x)−λn−1(x)s n(x),(11)where∆n(x)is the termination condition in Eq.(9).We shall now obtain exact solutions to Eq.(2)fordifferent choices of S(x)and V(x).Wefirst consider the linear potential form of vec-tor potential.Let V(x)=x,then the scalar potentialbecomes S(x)=x(β−1).After substituting thesepotentials the corresponding KG equation becomes{d2dx2−[(Ax+B/2A)2−δ]}f(x)=0.(12)After changing of variables y=Ax+B/2A,thesecond-order differential Eq.(12)yields{d2dy2−[y2−δ]}f(y)=0,(13)which is the Klein–Gordon Harmonic oscillator po-tential.We note that Eq.(12)is in fact a Schr¨o dingerequation with an energy dependent potential.Now,at this point we apply the AIM.In the limitof large y,the asymptotic solutions to Eq.(13)canbe taken as any power of y times a decreasing Gaus-sian.Therefore,it should has a solution in the formof‘normalized’wavefunctionsf(y)=exp(−y2/2)χ(y),(14)where the functions f(y)are to be found by meansof the iteration procedure.Substituting Eq.(14)intoEq.(13)yieldsχ (y)=2yχ (y)+(1−δ)χ(y),(15)which is now amenable to an AIM solution.By com-paring Eq.(15)with Eq.(6),we can write theλ0(y)and s0(y)values and by means of Eq.(9),we may cal-culateλn(y)and s n(y).These functions for some nvalues areλ0=2y,s0=(1−δ),λ1=3−δ+4y2,s1=2(1−δ)y,λ2=8y+2(1−δ)y+2y(3−δ+4y2),s2=2(1−δ)+(1−δ)(3−δ+4y2),···Combining these results with the quantization condi-tion given by Eq.(13)yieldss0λ0=s1λ1=⇒δ0=1,s1λ1=s2λ2=⇒δ1=3,s2λ2=s3λ3=⇒δ2=5,···No.6Eser Ol˘g ar1941 If we generalize these expressions,wefindδn=2n+1,n=0,1,2,···If we go back to the definitions of parametersδ,ε,Aand B,it is easy to obtainE2−m2+[2(E−m(1−β))]24(2β−β2)=2n+1,(16)which is exactly the same as the eigenvalue equation obtained in Ref.[18].From Eq.(16),one canfind thatE±n=(−1+β)m±√γn−1+(−2+β)β,(17)whereγn=(−2+β)β(−1−2n+(−2+β)β(1+2m2+ 2n)).In this example,we deal with the exponential form of scalar potential.Let V(x)=−e−αx and then S(x)=−e−αx(1−β).After substituting these expres-sions into Eq.(5),we obtain the differential equation{d2 dx2−[(A2e−2αx−Be−αx)−ξ]}f(x)=0,(18)whereξ=−ε,and Eq.(18)is the equation form of the Klein–Gordon Morse potential.We note that as in the previous case,this can be interpreted as a Schr¨o dinger equation with an energy dependent potential.After changing of variables y=e−αx,we obtain{d2+1d+[B−A2+ξ]}f(y)=0.(19)Now,we reach a position that the differential equa-tion is suitable for applying AIM.Therefore,Eq.(19) should have a solution in the form of the normalized wavefunctions,f(y)=y √ξ/αexp(−Aαy)χ(y).(20)Substituting Eq.(20)into Eq.(19),one obtainsχ (y)=[−2√ξ−α+2Ayαy]χ (y)+[−B+2A√ξ+Aαα2y2]χ(y),(21)which is now amenable to an AIM solution.By com-paring Eq.(21)with Eq.(6),we can write theλ0(y) and s0(y)values and by means of Eq.(9),we may calculateλn(y)and s n(y).These functions for some low-lying values of n areλ0=−2√ξ+α−2Ayαy,s0=−B+A(2√ξ+α)α2y,λ1=1α2y2[4ξ−By+4A2y2−3Ayα+2α2+√ξ(−6Ay+6α)],s1=−1αy(√−Ay+α)[B−A(2√+α)],···Similar to the previous example,by combining theseresults with the quantization condition yieldss0λ0=s1λ1=⇒ξ0=(12α−B2A)2,s1λ1=s2λ2=⇒ξ1=(32α−B2A)2,s2λ2=s3λ3=⇒ξ2=(52α−B2A)2,···From the above expressions,it is possible to write thegeneral formula ofξasξn=((n+12)α−B2A)2,n<(Bα−12),(22)where the relation of n is obtained by comparing ourparameters with that of in Refs.[35,36].If we go backto the definitions of parametersξ,andε,it can beshown thatE2n−m2=−((n+1)α−B)2,(23)which is exactly the same as the eigenvalue equationobtained in Ref.[18]through a proper choice of pa-rameters.The last example is the Kratzer potential thathas an important role in quantum mechanical sys-tems.In order to obtain the required potential formin the KG equation,let us take V(x)=1/x and thenS(x)=1/x(1−β).Substituting them into Eq.(5),wereach the differential equation{d2dx2−[(A2x2−Bx)−ξ]}f(x)=0,(24)whereξ=−ε,and Eq.(24)is the equation form ofthe Klein–Gordon Kratzer potential.In this position,to obtain the form of differential equation(6),we pro-pose the wavefunctionf(x)=x A exp(−√ξx)χ(x).(25)Substituting Eq.(20)into Eq.(19)yieldsχ (x)=2[√ξx−Ax]χ (x)+[A+(B+2A√ξ)xx2]χ(x).(26)Comparing Eq.(26)with Eq.(6),we can write theλ0(x)and s0(x)values and we may calculateλn(x)and s n(x)from Eq.(9).Thefirst two terms areλ0=(2√−2A)/x,s0=(A+Bx+2A√ξx)/x2,1942Eser Ol˘g ar Vol.25λ1=1y 2[4A 2+A (3−6√ξx )+x (B +4ξx )],s 1=1x3[Bx (−1+2√ξx )−2A 2(1+2√ξx )−2A (1+Bx −2ξx 2)],···Similarly,from the quantization condition,we obtains 0λ0=s 1λ1=⇒ξ0=(B 2A)2,s 1λ1=s 2λ2=⇒ξ1=(B 2(1+A ))2,s 2λ2=s 3λ3=⇒ξ2=(B2(2+A ))2,···The general formula of ξfor n values can be written as ξn =(B2(n +A ))2,n =0,1,2, (27)The eigenvalues in Eq.(27)is exactly the same as the values for Schr¨o dinger equation of Kratzer potential with a suitable choice of parameters.[34]From the def-initions of parameters ξand ε,it is easy to obtain the energy expressionE 2n −m 2=−(B )2,(28)for the Klein–Gordon-Kratzer potential.In summary,we have obtained the whole spectrum of some exactly solvable potential in the KG equation by using the AIM method.We have found that the general transformation between vector and scalar po-tentials is more useful to show the relation between each other.By choosing appropriate vector potentials,most of the exactly solvable and quasi-exactly solvable poten-tials can be constructed in this way.We feel that it could be interesting to apply the AIM to other rela-tivistic equations,e.g.,the Dirac equation.The author is grateful to P.Roy for useful discus-sion.References[1]Dirac P A M 1927Proc.Roy.Soc.London A 114243[2]Ko¸c R and Koca M 2005Mod.Phys.Lett.A 20911[3]Hou C F,Sun X D,Zhou Z X and Li Y 1999Acta Phys.Sin.48385(in Chinese)Hou C F and Zhou Z X 1999Acta Phys.Sin.8561(inChinese)[4]Ol˘g ar E,Ko¸c R and T¨u t¨u nc¨u ler H 2006Chin.Phys.Lett.23539[5]Chen C 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