Bound-State Solution of s-Wave Klein-Gordon Equation for ...inspirehep.net/record/1382864/files/923076.pdf · Bound-State Solution of s-Wave Klein-Gordon Equation for Woods-Saxon

Research ArticleBound-State Solution of s-Wave Klein-Gordon Equation forWoods-Saxon Potential

Eser OlLar and Haydar Mutaf

Engineering of Physics Department University of Gaziantep Gaziantep Turkey

Correspondence should be addressed to Eser Olgar olgargantepedutr

Received 25 March 2015 Revised 20 June 2015 Accepted 25 June 2015

Academic Editor Yao-Zhong Zhang

Copyright copy 2015 E Olgar and H Mutaf This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The bound-state solution of s-wave Klein-Gordon equation is calculated for Woods-Saxon potential by using the asymptoticiteration method (AIM) The energy eigenvalues and eigenfunctions are obtained for the required condition of bound-statesolutions

1 Introduction

In relativistic quantum mechanics the Klein-Gordon (KG)equation has a wide range of applications Recently therehave been many studies on KG equation with various typesof potentials by using different methods to describe thecorresponding relativistic physical systems namely asymp-totic iteration method [1] formal variable separation method[2] supersymmetric quantum mechanics [3ndash7] algebraicmethod [8ndash13] and Nikiforov-Uvarov method [14] Most ofthese studies have been considered for equal or pure scalar119878(119909) and vector potentials 119881(119909) cases In this study to havea bound-state solution we use the proposed transformationin [15] with parameters 120573 which adjust the relation betweenscalar and vector Woods-Saxon potentials Woods-Saxonpotential [16] is a short range nuclear potential and it isimportant in all branches of physicsThebound-state solutionof KG equation forWoods-Saxon potential has been of a greatinterest in the last decades with different methods [17ndash21]It has been solved with Nikiforov-Uvarov method by usingthe Pekeris approximation to the centrifugal potential forany l-states for constant mass [22 23] and for effective mass[24 25]

In this paper we use an alternativemethodwhich is calledthe asymptotic iteration method (AIM) [26] for obtainingthe whole spectra of Woods-Saxon potentials AIM has beenused in many physical systems to obtain the correspondingeigenvalues and eigenfunctions [26ndash31] Still there are a few

applications on KG equation with AIM [32ndash34] with ageneral transformation between vector and scalar potentialsTo obtain the corresponding potential the vector potentialis chosen in exponential potential form Therefore the KGequation reduces to Schrodinger-like equation with Woods-Saxon potential

The organization of this paper is as follows Section 2deals with the formalism of the one-dimensional KG equa-tion with scalar potential greater than vector potential Ageneral description of AIM is outlined in Section 3 Thesolution of Woods-Saxon potential in KG equation is treatedin the subsequent section bymeans of AIM Finally Section 5is devoted to conclusions

2 Formalism of the Klein-Gordon Equation

Generally the s-wave Klein-Gordon equation with scalarpotential 119878(119909) and vector potential119881(119909) can be written as [9]

1198892

1198891199092+

1ℏ21198882

[(119864 minus119881 (119909))2minus (119898119888

2+ 119878 (119909))

2]119891 (119909)

= 0

(1)

where 119864 represents the energy and 119898 represents the massof the particle In this equation the radial wave functionis expressed as 119877(119909) = 119891(119909)119909 We know that when therelation between vector and scalar potentials 119878(119909) ge 119881(119909)

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 923076 5 pageshttpdxdoiorg1011552015923076

2 Advances in Mathematical Physics

it yields real bound-state solutions In this work we considerthe relation between potentials in the condition for bound-state solution which was introduced in [15] as

119878 (119909) = 119881 (119909) (120573 minus 1) 120573 ge 0 (2)

where 120573 parameter is arbitrary constant When 120573 = 0 1 2the scalar potential leads to case of 119878(119909) = minus119881(119909) 119878(119909) = 0(purely vector potential) and 119878(119909) = 119881(119909) (equal potentials)respectively When 120573 gt 2 we get the required condition forthe case of 119878(119909) gt 119881(119909) Substituting (2) into (1) yields

1198892

1198891199092minus

1ℏ21198882

[119881eff (119909) minus (1198642minus119898

2)]119891 (119909) = 0 (3)

where the effective potential is

119881eff = 119881 (119909) [2119864+ 21198882119898(120573minus 1) +119881 (119909) (120573 minus 2) 120573] (4)

Now the final equation is the transformed KG equationin the condition of unequal vector and scalar potentials Thefollowing section deals with the corresponding method tosolve the KG equation

3 Asymptotic Iteration Method

The AIM is proposed to solve the second-order differentialequations and the details can be found in [26] First we con-sider the following second-order homogeneous differentialequation of the form

11991010158401015840

(119909) = 12058201199101015840

(119909) + 1199040119910 (119909) (5)

where 1205820 and 1199040 are functions and 1199101015840(119909) and 11991010158401015840(119909) denote

derivatives of 119910 with respect to 119909 It is easy to show that (119899 +2)th derivative of the function 119910(119909) can be written as

119910(119899+2)

(119909) = 1205821198991199101015840

(119909) + 119904119899119910 (119909) (6)

The functions 120582119899and 119904119899are given by the recurrence relations

120582119899= 1205821015840

119899minus1 + 119904119899minus1 +120582119899minus11205820

119904119899= 1199041015840

119899minus1 +120582119899minus11199040(7)

For sufficiently large 119899 the ratio of the functions satisfies thefollowing equations

120582119899

119904119899

=120582119899minus1119904119899minus1

= 120572 (119909) (8)

then the solution of (5) can be written as [26]

119910 (119909) = exp(minusint119909

120572119889119905)

sdot [1198621 +1198622 int119909

exp(int119904

(1205820 + 2120572) 119889119905) 119889119904] (9)

In calculating the parameters in (7) for 119899 = 0 we take theinitial conditions as 120582

minus1 = 1 and 119904minus1 = 0 [29] and Δ

119899(119909) = 0

for

Δ119899(119909) = 120582

119899(119909) 119904119899minus1 (119909) minus 120582119899minus1 (119909) 119904119899 (119909) (10)

where Δ119899(119909) is the termination condition in (8)

If the second-order homogeneous linear differentialequation can be written in the form of [27]

11991010158401015840

(119909) = 2( 119886119909119873+1

1 minus 119887119909119873+2minus(119898 + 1)119909

)1199101015840

(119909)

minus119908119909119873

1 minus 119887119909119873+2

(11)

using the function generator of AIM it has an exact solutionfor 119910119899(119909) [27] as

119910119899(119909) = (minus1)119899 1198621 (120590)119899 (119873+ 2)119899 (120590)

119899

sdot 21198651 (minus119899 120588 + 119899 120590 119887119909119873+2

)

(12)

with the defined parameters (120590)119899= Γ(120590 + 119899)120590 120590 = (2119898 +

119873 + 3)(119873 + 2) and 120588 = ((2119898 + 1)119887 + 2119886)(119873 + 2)119887

4 Solution for Woods-Saxon Potential

Let us consider the Woods-Saxon vector potential

119881 (119909) =minus1198810

1 + 119890(119909minus119877)120572and then 119878 (119909) =

1198810 (1 minus 120573)1 + 119890(119909minus119877)120572

(13)

After substituting these potentials into (3) and changing thevariables 119910 = 1(1 + 119890(119909minus119877)120572) we obtain the second-orderdifferential equation

1198892

1198891199102+(1 minus 2119910)119910 (1 minus 119910)

119889

119889119910minus

11199102 (1 minus 119910)2

[120585 +119860+119861]

sdot 119891 (119910) = 0

(14)

with the corresponding parameters

1205852= minus(

1198642minus 119898

21198884

ℏ21198882)

119860 = (21198641198810120572

2+ 2 (120573 minus 1) 11988821198981198810120572

2

ℏ21198882)

119861 = minus((120573 minus 2) 1205731198812

0 1205722

ℏ21198882)

(15)

As we expected (13) is transformed to the suitable form toapply the AIM Therefore (14) should have a solution in theform of ldquonormalizedrdquo wave functions

119891 (119910) = 119910120585(1minus119910)V 120594 (119910) (16)

with V = radic120585 + 119860 + 119861 Substituting (16) into (14) one obtains

12059410158401015840(119910) = [

2120585 (119910 minus 1) + 2 (1 + V) 119910 minus 1119910 (1 minus 119910)

] 1205941015840(119910)

+ [(120585 + V) (1 + 2120585) + 119860

119910 (1 minus 119910)] 120594 (119910)

(17)

Within the framework of procedures of AIM the functions1205820(119910) and 1199040(119910) can be written by comparing (17) with (5)It can be seen that 1205820(119910) and 1199040(119910) functions are obtained byusing the termination relation in (10) as

Advances in Mathematical Physics 3

1205820 =1 minus 2120585 (119910 minus 1) minus 2 (V + 1) 119910

119910 (119910 minus 1)

1199040 =2 + (119860 minus 3V minus 6) 119910 + (4V2 + 9V minus 119860 + 6) 1199102 + 3120585 (119910 minus 1) (minus2 + (3 + 2V) 119910) + 21205852 (1199102 minus 3119910 + 2)

1199102 (119910 minus 1)2

(18)

By substituting these functions into the quantization condi-tion we get

11990401205820

=11990411205821

997904rArr 1205850 +radic120585 + 119860 + 119861 = minus12minus12radic1 + 4119860 119899 = 0

11990411205821

=11990421205822


11990411205821

=11990421205822


(19)

Generalizing the above expressions it is possible to write thegeneral formula of 120585 as

120585119899+radic120585 + 119860 + 119861 = minus

(2119899 + 1)2

minus12radic1 + 4119860 (20)

Using the definitions of parameters 120585 and 120576 it can be shownthat

1198642119899minus119898

21198884= minus ℏ

21198882(minus

(2119899 + 1)2

minus12radic1 + 4

211986411988101205722+ 2 (120573 minus 1) 11988821198981198810120572

2

ℏ21198882minus V)

(21)

Rearranging this expressionwe get amore explicit expressionfor eigenvalues

119864119899= plusmn[

[

11989821198884minus ℏ

21198882(minus

(2119899 + 1)2

minus12radic1 + 4

211986411988101205722+ 2 (120573 minus 1) 11988821198981198810120572

2

ℏ21198882

minus V)]

]

12

(22)

which is exactly the same as with the eigenvalue equationobtained in [21] through a proper choice of parameters

41 Calculation of Eigenfunction To get the normalized wavefunction we have to compare (17) with the differential

equation defined in (11) After some algebraic calculations weget the required parameters as

119886 =2V + 12

119887 = 1

119873 = minus 1

119898 =2119864 minus 1

2

120590 = 2120585 + 1

120588 = 2120585 minus 2V+ 1

(23)

Thus these parameters yield a solution in the form of

120594119899(119910) = (minus1)119899 1198622

sdotΓ (2120585119899+ 119899 + 1) 21198651 (minus119899 2 (120585119899 minus V) + 1 + 119899 2120585

119899+ 1 119910)

Γ (2120585119899+ 1)

(24)

with the Gamma function Γ and the Gauss hypergeometricfunction 21198651With the aid of (24) the corresponding radialfunction becomes

119877119899(119909) = (minus1)119899119873

119899119909 (119910120585(1minus119910)V)

sdot 21198651 (minus119899 2 (120585119899 minus V) + 1+ 119899 2120585119899 + 1 119910) (25)

where the normalization constant is

119873119899= 1198622

Γ (2120585119899+ 119899 + 1)

Γ (2120585119899+ 1)

(26)

Substituting the values of 119910 in (25) the total radial differentialequation transforms to

119877119899(119910) = (minus1)119899119873

119899119909(

(119890(119909minus119877)120572

)V

(1 + 119890(119909minus119877)120572)V+120585)

sdot 21198651 (minus119899 2 (120585119899 minus V)

+ 1+ 119899 2120585119899+ 1 1

1 + 119890(119909minus119877)120572)

(27)

After rearranging the parameters this result is exactly thesame as with those of calculated for Woods-Saxon potentialsin literature [17ndash19 21]


5 Conclusion

In this study we have obtained the whole spectrum ofWoods-Saxon potential for s-wave KG equation by usingAIM method During the calculation we have used theadjusting parameter 120573 between scalar and vector potentialsto obey the bound-state solution condition The eigenvaluesand eigenfunction are calculated directly and exactly by usingthe procedure of AIM in simple way

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was supported by the Research Fund ofGaziantep University (BAP) and the Scientific and Techno-logical Research Council of Turkey (TUBITAK)

References

[1] E Olgar R Koc and H Tutunculer ldquoBound states of the s-wave equation with equal scalar and vector standard Eckartpotentialrdquo Chinese Physics Letters vol 23 no 3 pp 539ndash5412006

[2] C-F Hou X-D Sun Z-X Zhou and Y Li ldquoGeneral formulasfor radial matrix elements of isotropic harmonic oscillatorsrdquoActa Physica Sinica vol 48 no 3 pp 385ndash388 1999

[3] C Y Chen C L Liu F L Lu andD S Sun ldquoBound states of theKlein-Gordon equation with n-dimensional scalar and vectorhydrogen atom-type potentialsrdquoActa Physica Sinica vol 52 pp1579ndash1584 2003

[4] Y He Z Q Cao and Q S Shen ldquoBound-state spectra forsupersymmetric quantum mechanicsrdquo Physics Letters A vol326 no 5-6 pp 315ndash321 2004

[5] G Chen Z-D Chen and Z-M Lou ldquoExact bound state solu-tions of the 119904-wave Klein-Gordon equation with the generalizedHulthen potentialrdquoPhysics Letters A vol 331 no 6 pp 374ndash3772004

[6] W-C Qiang ldquoBound states of the Klein-Gordon and Diracequations for potential 119881(119903) = 119860119903

minus2minus 119861119903minus1rdquo Chinese Physics

vol 12 no 10 pp 1054ndash1058 2003[7] L-Z Yi Y-F Diao J-Y Liu and C-S Jia ldquoBound states of the

Klein-Gordon equation with vector and scalar Rosen-Morse-type potentialsrdquo Physics Letters A vol 333 no 3-4 pp 212ndash2172004

[8] C-S Jia Y Li Y Sun J-Y Liu and L-T Sun ldquoBound statesof the five-parameter exponential-type potentialmodelrdquoPhysicsLetters A vol 311 no 2-3 pp 115ndash125 2003

[9] F Domınguez-Adame ldquoBound states of the Klein-Gordonequation with vector and scalar Hulthen-type potentialsrdquoPhysics Letters A vol 136 no 4-5 pp 175ndash177 1989

[10] Z-Q Ma S-H Dong X-Y Gu J A Yu and M Lozada-Cassou ldquoThe Klein-Gordon equation with a coulomb plusscalar potential in d dimensionsrdquo International Journal ofModern Physics E vol 13 no 3 pp 597ndash610 2004

[11] C-Y Chen D-S Sun and F-L Lu ldquoRelativistic bound statesof Coulomb potential plus a new ring-shaped potentialrdquo ActaPhysica Sinica vol 55 no 8 pp 3875ndash3879 2006

[12] W-C Qiang and S-H Dong ldquoAnalytical approximations to thel-wave solutions of the Klein-Gordon equation for a secondPoschl-Teller like potentialrdquo Physics Letters A vol 372 no 27-28 pp 4789ndash4792 2008

[13] S SDong SHDongH Bahlouli andV B Bezerra ldquoAlgebraicapproach to the KleinndashGordon equation with hyperbolic Scarfpotentialrdquo International Journal ofModern Physics E vol 20 pp55ndash61 2011

[14] S M Ikhdair and R Sever ldquoExact solution of the Klein-Gordon equation for the PT-symmetric generalized Woods-Saxon potential by the Nikiforov-Uvarov methodrdquo Annalen derPhysik vol 16 no 3 pp 218ndash232 2007

[15] E Olgar ldquoExact solution of Klein-Gordon equation by asymp-totic iterationmethodrdquoChinese Physics Letters vol 25 no 6 pp1939ndash1942 2008

[16] R DWoods and D S Saxon ldquoDiffuse surface optical model fornucleon-nuclei scatteringrdquo Physical Review vol 95 no 2 pp577ndash578 1954

[17] J Y Guo X Z Fang and F X Xu ldquoSolution of the relativisticDirac-Woods-Saxon problemrdquo Physical Review A vol 66 Arti-cle ID 062105 3 pages 2002

[18] C F Hou Z X Zhou and Y Li ldquoBound states of theKlein-Gordon equation with vector and scalar Wood-Saxonpotentialsrdquo Acta Physica Sinica vol 8 no 8 pp 561ndash564 1999

[19] G Chen ldquoBound states for Dirac equation with wood-saxonpotentialrdquo Acta Physica Sinica vol 53 no 3 pp 680ndash683 2004

[20] V M Villalba and C Rojas ldquoBound states of the Klein-Gordonequation in the presence of short range potentialsrdquo InternationalJournal of Modern Physics A vol 21 no 2 pp 313ndash325 2006

[21] G Chen Z-D Chen and Z-M Lou ldquoExact bound state solu-tions of the 119904-wave Klein-Gordon equation with the generalizedHulthen potentialrdquo Physics Letters A vol 331 no 6 pp 374ndash3772004

[22] V H Badalov H I Ahmadov and S V Badalov ldquoAny l-state analytical solutions of the Klein-Gordon equation forthe Woods-Saxon potentialrdquo International Journal of ModernPhysics E vol 19 no 7 pp 1463ndash1475 2010

[23] A Altug and S Ramazan ldquoApproximate 119897-state solutions ofa spin-0 particle for Woods-Saxon potentialrdquo InternationalJournal of Modern Physics C vol 20 no 4 pp 651ndash665 2009

[24] A Altug and R Sever ldquoApproximate ℓ-state solutions to theKlein-Gordon equation for modified Woods-Saxon potentialwith position dependent massrdquo International Journal of ModernPhysics A vol 24 pp 3985ndash3999 2009

[25] A Altug and R Sever ldquoBound states of the KleinndashGordonequation for WoodsndashSaxon potential with position dependentmassrdquo International Journal of Modern Physics C vol 19 no 5pp 763ndash773 2008

[26] H Ciftci R L Hall and N Saad ldquoAsymptotic iteration methodfor eigenvalue problemsrdquo Journal of Physics A Mathematicaland General vol 36 no 47 pp 11807ndash11816 2003

[27] N Saad R L Hall andH Ciftci ldquoSextic anharmonic oscillatorsand orthogonal polynomialsrdquo Journal of Physics A Mathemati-cal and General vol 39 no 26 pp 8477ndash8486 2006

[28] T Barakat K Abodayeh and A Mukheimer ldquoThe asymptoticiteration method for the angular spheroidal eigenvaluesrdquo Jour-nal of Physics A Mathematical and General vol 38 no 6 pp1299ndash1304 2005

[29] F M Fernandez ldquoOn an iteration method for eigenvalueproblemsrdquo Journal of Physics A Mathematical and General vol37 no 23 pp 6173ndash6180 2004


[30] I BoztosunM Karakoc F Yasuk and A Durmus ldquoAsymptoticiteration method solutions to the relativistic Duffin-Kemmer-Petiau equationrdquo Journal of Mathematical Physics vol 47Article ID 062301 2006

[31] A Soylu O Bayrak and I Boztosun ldquoThe energy eigenvaluesof the two dimensional hydrogen atom in a magnetic fieldrdquoInternational Journal of Modern Physics E vol 15 no 6 pp1263ndash1271 2006

[32] E Olgar and H Mutaf ldquoAsymptotic iteration method forenergies of inversely linear potential with spatially dependentmassrdquo Communications in Theoretical Physics vol 53 no 6 pp1043ndash1045 2010

[33] E Olgar ldquoAn alternative method for calculating bound-state ofenergy eigenvalues of KleinndashGordon for Quasi-exactly solvablepotentialsrdquo Chinese Physics Letters vol 26 no 2 Article ID020302 2009

[34] E Olgar R Koc and H Tutunculer ldquoThe exact solution ofthe s-wave Klein-Gordon equation for the generalized Hulthenpotential by the asymptotic iteration methodrdquo Physica Scriptavol 78 no 1 Article ID 015011 2008

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


it yields real bound-state solutions In this work we considerthe relation between potentials in the condition for bound-state solution which was introduced in [15] as

119878 (119909) = 119881 (119909) (120573 minus 1) 120573 ge 0 (2)

where 120573 parameter is arbitrary constant When 120573 = 0 1 2the scalar potential leads to case of 119878(119909) = minus119881(119909) 119878(119909) = 0(purely vector potential) and 119878(119909) = 119881(119909) (equal potentials)respectively When 120573 gt 2 we get the required condition forthe case of 119878(119909) gt 119881(119909) Substituting (2) into (1) yields

1198892

1198891199092minus

1ℏ21198882

[119881eff (119909) minus (1198642minus119898

2)]119891 (119909) = 0 (3)

where the effective potential is

119881eff = 119881 (119909) [2119864+ 21198882119898(120573minus 1) +119881 (119909) (120573 minus 2) 120573] (4)

Now the final equation is the transformed KG equationin the condition of unequal vector and scalar potentials Thefollowing section deals with the corresponding method tosolve the KG equation

3 Asymptotic Iteration Method

The AIM is proposed to solve the second-order differentialequations and the details can be found in [26] First we con-sider the following second-order homogeneous differentialequation of the form

11991010158401015840

(119909) = 12058201199101015840

(119909) + 1199040119910 (119909) (5)

where 1205820 and 1199040 are functions and 1199101015840(119909) and 11991010158401015840(119909) denote

derivatives of 119910 with respect to 119909 It is easy to show that (119899 +2)th derivative of the function 119910(119909) can be written as

119910(119899+2)

(119909) = 1205821198991199101015840

(119909) + 119904119899119910 (119909) (6)

The functions 120582119899and 119904119899are given by the recurrence relations

120582119899= 1205821015840

119899minus1 + 119904119899minus1 +120582119899minus11205820

119904119899= 1199041015840

119899minus1 +120582119899minus11199040(7)

For sufficiently large 119899 the ratio of the functions satisfies thefollowing equations

120582119899

119904119899

=120582119899minus1119904119899minus1

= 120572 (119909) (8)

then the solution of (5) can be written as [26]

119910 (119909) = exp(minusint119909

120572119889119905)

sdot [1198621 +1198622 int119909

exp(int119904

(1205820 + 2120572) 119889119905) 119889119904] (9)

In calculating the parameters in (7) for 119899 = 0 we take theinitial conditions as 120582

minus1 = 1 and 119904minus1 = 0 [29] and Δ

119899(119909) = 0

for

Δ119899(119909) = 120582

119899(119909) 119904119899minus1 (119909) minus 120582119899minus1 (119909) 119904119899 (119909) (10)

where Δ119899(119909) is the termination condition in (8)

If the second-order homogeneous linear differentialequation can be written in the form of [27]

11991010158401015840

(119909) = 2( 119886119909119873+1

1 minus 119887119909119873+2minus(119898 + 1)119909

)1199101015840

(119909)

minus119908119909119873

1 minus 119887119909119873+2

(11)

using the function generator of AIM it has an exact solutionfor 119910119899(119909) [27] as

119910119899(119909) = (minus1)119899 1198621 (120590)119899 (119873+ 2)119899 (120590)

119899

sdot 21198651 (minus119899 120588 + 119899 120590 119887119909119873+2

)

(12)

with the defined parameters (120590)119899= Γ(120590 + 119899)120590 120590 = (2119898 +

119873 + 3)(119873 + 2) and 120588 = ((2119898 + 1)119887 + 2119886)(119873 + 2)119887

4 Solution for Woods-Saxon Potential

Let us consider the Woods-Saxon vector potential

119881 (119909) =minus1198810

1 + 119890(119909minus119877)120572and then 119878 (119909) =

1198810 (1 minus 120573)1 + 119890(119909minus119877)120572

(13)

After substituting these potentials into (3) and changing thevariables 119910 = 1(1 + 119890(119909minus119877)120572) we obtain the second-orderdifferential equation

1198892

1198891199102+(1 minus 2119910)119910 (1 minus 119910)

119889

119889119910minus

11199102 (1 minus 119910)2

[120585 +119860+119861]

sdot 119891 (119910) = 0

(14)

with the corresponding parameters

1205852= minus(

1198642minus 119898

21198884

ℏ21198882)

119860 = (21198641198810120572

2+ 2 (120573 minus 1) 11988821198981198810120572

2

ℏ21198882)

119861 = minus((120573 minus 2) 1205731198812

0 1205722

ℏ21198882)

(15)

As we expected (13) is transformed to the suitable form toapply the AIM Therefore (14) should have a solution in theform of ldquonormalizedrdquo wave functions

119891 (119910) = 119910120585(1minus119910)V 120594 (119910) (16)

with V = radic120585 + 119860 + 119861 Substituting (16) into (14) one obtains

12059410158401015840(119910) = [

2120585 (119910 minus 1) + 2 (1 + V) 119910 minus 1119910 (1 minus 119910)

] 1205941015840(119910)

+ [(120585 + V) (1 + 2120585) + 119860

119910 (1 minus 119910)] 120594 (119910)

(17)

Within the framework of procedures of AIM the functions1205820(119910) and 1199040(119910) can be written by comparing (17) with (5)It can be seen that 1205820(119910) and 1199040(119910) functions are obtained byusing the termination relation in (10) as



119910 (119910 minus 1)


1199102 (119910 minus 1)2

(18)


11990401205820

=11990411205821


11990411205821

=11990421205822


11990411205821

=11990421205822


(19)


120585119899+radic120585 + 119860 + 119861 = minus

(2119899 + 1)2

minus12radic1 + 4119860 (20)


1198642119899minus119898

21198884= minus ℏ

21198882(minus

(2119899 + 1)2

minus12radic1 + 4

211986411988101205722+ 2 (120573 minus 1) 11988821198981198810120572

2

ℏ21198882minus V)

(21)


119864119899= plusmn[

[

11989821198884minus ℏ

21198882(minus

(2119899 + 1)2

minus12radic1 + 4

211986411988101205722+ 2 (120573 minus 1) 11988821198981198810120572

2

ℏ21198882

minus V)]

]

12

(22)




119886 =2V + 12

119887 = 1

119873 = minus 1

119898 =2119864 minus 1

2

120590 = 2120585 + 1

120588 = 2120585 minus 2V+ 1

(23)


120594119899(119910) = (minus1)119899 1198622


119899+ 1 119910)

Γ (2120585119899+ 1)

(24)


119877119899(119909) = (minus1)119899119873

119899119909 (119910120585(1minus119910)V)



119873119899= 1198622

Γ (2120585119899+ 119899 + 1)

Γ (2120585119899+ 1)

(26)


119877119899(119910) = (minus1)119899119873

119899119909(

(119890(119909minus119877)120572

)V

(1 + 119890(119909minus119877)120572)V+120585)


+ 1+ 119899 2120585119899+ 1 1

1 + 119890(119909minus119877)120572)

(27)



5 Conclusion




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119910 (119910 minus 1)


1199102 (119910 minus 1)2

(18)


11990401205820

=11990411205821


11990411205821

=11990421205822


11990411205821

=11990421205822


(19)


120585119899+radic120585 + 119860 + 119861 = minus

(2119899 + 1)2

minus12radic1 + 4119860 (20)


1198642119899minus119898

21198884= minus ℏ

21198882(minus

(2119899 + 1)2

minus12radic1 + 4

211986411988101205722+ 2 (120573 minus 1) 11988821198981198810120572

2

ℏ21198882minus V)

(21)


119864119899= plusmn[

[

11989821198884minus ℏ

21198882(minus

(2119899 + 1)2

minus12radic1 + 4

211986411988101205722+ 2 (120573 minus 1) 11988821198981198810120572

2

ℏ21198882

minus V)]

]

12

(22)




119886 =2V + 12

119887 = 1

119873 = minus 1

119898 =2119864 minus 1

2

120590 = 2120585 + 1

120588 = 2120585 minus 2V+ 1

(23)


120594119899(119910) = (minus1)119899 1198622


119899+ 1 119910)

Γ (2120585119899+ 1)

(24)


119877119899(119909) = (minus1)119899119873

119899119909 (119910120585(1minus119910)V)



119873119899= 1198622

Γ (2120585119899+ 119899 + 1)

Γ (2120585119899+ 1)

(26)


119877119899(119910) = (minus1)119899119873

119899119909(

(119890(119909minus119877)120572

)V

(1 + 119890(119909minus119877)120572)V+120585)


+ 1+ 119899 2120585119899+ 1 1

1 + 119890(119909minus119877)120572)

(27)



5 Conclusion




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 Conclusion




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Bound-State Solution of s-Wave Klein-Gordon Equation for ...inspirehep.net/record/1382864/files/923076.pdf · Bound-State Solution of s-Wave Klein-Gordon Equation for Woods-Saxon