Analytical solution of Klein Gordon equation Trigonometric Pӧschl-Teller potential using asymptotic iteration method

Radial part of Klein Gordon equation for trigonometric PӧschlTeller potential was obtained within framework of a centrifugal term approximation. The relativistic energy spectrum and wave functions was obtain by using asymptotic iteration method. The value of relativistic energy was calculated numerically using Matlab 2013. The results showed that the relativistic energy is increasing due to the increase of potential constant and quantum number.

In this research, the relativistic solution of spin-less particle with Trigonometric Pӧschl-Teller Potential is studied.The method used is asymptotic iteration method.Asymptotic Iteration Methods (AIM) is an alternative method which has accuracy and high efficiency to determine eigen energies and eigenfunctions for analytically.Asymptotic Iteration Methods is also giving solution for exactly solvable problem.(Suparmi, Cari, Deta, Husein, & Yuliani, 2014).

Klein Gordon Equation
The equation of time independent KG with vector () Vr and scalar () Sr potentials, where spinless particle is describing with r  r which can be written as (Bayrak &  Boztosun, 2006; Qiang & Dong, 2007) and E and M express the Klein Gordon energy and mass.And then x is vector dimensional of position.The radial part wave function () Rris define as

   
, Fr Rr r  (3) For limit non-trivial nonrelativistic result for case SV  , we get a potential function 2V and not .V in nonrelativistic limit, the interaction of potential is become V and not 2V.That is why equation ( 1) can be reduced into radial part of 3-Dimensional Klein Gordon equation.(in the relativistic atomic units 1 c ) (Alhaidari, Bahlouli, & Al- Hasan, 2006),

Asymptotic Iteration Method (AIM)
Asymptotic Iteration Method (AIM) is one of alternative method which has high efficiency and accuracy to determine eigenenergy and eigenfunction.Asymptotic Iteration Method is also giving exactly solvable problem solution (Pramono, Suparmi, & Cari, 2016). where In this case 0 ( ) 0 x   and 0 () Asymptotic Iteration Method can be applied exactly for different problem if the wave function is known and fulfill the boundaries of condition zero (0) and infinite (∞).
Equation ( 5) can be simply iterated until (k+1) and (k+2), k = 1, 2, 3, ... then the result become where Which call as recurrence relation.Eigen value can be found using the following equation is iteration number and n is the quantum number for radial part.
Equation ( 5) is the linear homogenous equation in the second orde which can be solved by comparing it with the second order of linear equation as follows (Sari, et al., 2015) where (15) The wave function of equation ( 4) is determined using the follows equation

Result
The Pӧschl Teller Potential is define as (Xu-Yang Liu, 2010) Equation ( 5) is substitute with equation (2 -4, 19), and the result is By using the centrifugal term approximation for Equation ( 23) can be solve by letting Equation ( 24) is hypergeometric differential equation which have to differentiate to second orde differential equation in AIM form.Let equation ( 24) is substitute with equation ( 25), and the result is Equation ( 27) is inserted into equation ( 26), we get The AIM type equation ( 28) is compared with equation ( 5), thus From equation (11 -13), we get the general eigen value from equation (28) as follows where n E in equation ( 32) is the eigen value and n is the radial quantum number, 0,1, 2, n  . By inserting equation (20 -21, 32 -34) into equation ( 31) the relativistic energy equation is determined as l is the quantum orbital number for radial part solution.From equation ( 35) can be calculated the relativistic energy numerically by using Matlab 2013.The result is shown in Table 1.Based on Table 1.The energy value is affected by potential constant of Trigonometric Pӧschl-Teller potential.The energy value is increasing due to the increase of the potential constant.The value of relativistic energy is increasing due to the increase of the quantum number.By comparing equation ( 6) and ( 29), then by using equation ( 16) and inserting it into equation ( 17) we get By inserting equation (37) to where ' C is normalization constant and 21 ( , , , )  F a b c z is the hypergeometric function.Table 2.is presented the unnormalized wave function for the radial part in any states solved from equation (38).

Conclusion
Klein Gordon equation for Trigonometric Pӧschl-Teller potential was obtained by using centrifugal term approximation.This equation was solved by using asymptotic iteration method.The relativistic energy spectrum and the wave function were obtained by using asymptotic iteration method.Based on the result, the relativistic energy is increasing due to the increase of potential constant and the quantum number value.
sx are differentiable.Equation (4) is differentiated to x to get the solution, and we get 11