Solution of Five-dimensional Schrodinger equation for Kratzer ’ s potential and trigonometric tangent squared potential with asymptotic iteration method ( AIM )

Non-relativistic bound-energy of diatomic molecules determined by non-central potentials in five dimensional solution using AIM. Potential in five dimensional space consist of Kratzer’s potential for radial part and Tangent squared potential for angular part. By varying nr, n1, n2, n3, dan n4 quantum number on CO, NO, dan I2 diatomic molecules affect bounding energy values. It knows from its numerical data. Keyword: Schrodinger equation, D-dimension, Kratzer’s potential, Tangent squared potential, AIM


INTRODUCTION
Non-relativistic energy solutions can be obtained from the Schrodinger equation.The Schrodinger equation is the basis for describing a physical event related to quantum mechanics.There are many types of equations that can be used to solve the case of quantum mechanics, such as: Dirac equation (Barakat & Alhendi, 2013), Klein-Gordon equation (Barakat, 2009), and Schrodinger equation (Arbabi, 2016).In this equation can be used interference of potentials for certain system, e.g.Poschl-Teller potential (Yahya & Oyewumi, 2015), Deng-Fan potential and Hulthen potential (Hassanabadi et al., 2013).These equations can be applied to a higher dimension space (Dong, 2011) with the interference of a potential.Higher dimension is a space that has more than three dimensional space components.
This study was to look for energy eigenvalues on Schrodinger equations in fivedimensional space with Kratzer's potential interference and trigonometric tangen squared potential.Kratzer's potential (Bayrak et al., 2006) describes dissociation events in diatomic molecules.Kratzer's potential is used in the radial part with variable r.Thus, Kratzer's potential is used for the radial part ( where D e is the dissociation energy, r is the distance between diatomic molecular nuclei and a is the distance between the nuclei in equilibrium.Potential tangent squared (Ciftchi et al., 2013) describes a potential that is affected by the angular change.

 
with V 0 is the initial potential used for each of the equally considered corner components and θ 1 , θ 2 , θ 3 , and θ 4 are the elevation angle.This study was completed by AIM method (Falaye, 2012).In order to be solved with AIM, the Schrodinger equation is reduced to a hypergeometric type of two-order differential equation.Then the potentials in equations (1-5) are combined in form

Asymptotic Iteration Method
From the second-order differential equation of the hypergeometric type, then a part of the differential equation which is of type AIM is taken.Two-order differential equations with type AIM is represented in equation ( 7) (Sari et al.,2015).
From equation ( 7) we get the values of χ 0 and s 0 , and then iteration is done with the pattern as in equations (8-9).) From equations (8-9) can be used to find the eigenvalues by equation ( 10 with k = 1, 2, 3, ... Then to determine the eigen function of equation ( 7), we can use equation ( 11)
Where n is a quantum number, C 2 is the normalization constant, 2 F 1 is a hypergeometric function.While the other parameters in equations (14-15) are obtained by comparing equation ( 7) with equation ( 16) as follows (16)

Variable Separation
Schrodinger equation by using natural unit (ħ  c 1) and in five-dimensional space, can be written by equation ( 17) In equation ( 17) there is a Laplacian at five-dimensional coordinates.The Laplacian can be written to be In equation ( 17) there is also a variable Ψ applied in five dimensions to Then equations (18-19) are substituted into equation ( 17) to derive a Schrodinger equation for the radial, angular θ 1 , θ 2 , θ 3 , and θ 4 .
From combining equations (17-19), we get the five dimensional Schrodinger equation.

Solution of Radial Part
The general solution for five dimensional Schrodinger equation in radial part is given by equation ( 20).Then use the equations (25-29) to reduce equation ( 20): Equations (25-29) are substituted into equation ( 20), so that equation ( 30) is obtained.
Equation ( 30) to be applicable in AIM, must use transformation in equation (31).
By comparing equation (32) with equation ( 7), we get the values of χ 0 and s 0 for the radial part.
Equations (33-34) are iterated by using equations (8-9) to obtain the parameter values χ 1 , χ 2 , χ 3 , ...χ k and s 1 , s 2 , s 3 , ... s k .These parameter values are used to find  k . k is used to find energy, so we get the energy equation The energy eigenvalue depends on the parameters of all components of the composed potential and also depend on the quantum number n r .It can be explained by equation ( 35).Then we find the wave function for radial part by using equations (11)(12).So, we can resulting the ground state wave function that shown in equation ( 36).

 
and by raising parameter k in equation ( 12), we can get the first excited wave function
The lowest total wavefunction from equation (36), equation ( 43), and equation (60-62) is given as which is normalization factor of the total lowest wave function.

Discussion
Eigen value of energy in equation ( 35), and constants of variabel separation in equation ( 44) and equations (63-65) can calculated by using computational method.We can calculate numerical solution of the energy for diatomic molekul are listed in Table 1.From Table 1 it can be seen if CO has dissociation energy than NO and I 2 .But CO has the lowest mass and equilibrium distance of nucleus.From parameters in Table 1, numerical solution of energy are shown in Table 2.The negative value show that the energy is repulsive.Repulsive energy of CO and NO molecules decrease caused by the increase of quantum number n 4 is more significant than the increase in energy caused by the increase of the quantum number n r and the decrease of energy due to the increase in n 1 , n 2 , n 3 .But for the diatomic molecule I 2 , the increase of all quantum numbers causes the value of the repulsive energy getting smaller.

CONCLUSION
In this paper, we have presented the solution of five dimensional Schrodinger for some diatomic molecule with disturbance Kratzer's potential combined with trigonometric tangent squared potential used AIM.We have gotten eigen value of energy total from solution radial part.In eigen value energy, there are constants of variabel separation.It is resulted from all angular part.So, the eigen value energy depended on all quantum number.The radial wavefunctions and angular wavefunctions was obtained using wave function generator in equation ( 11) or equation ( 13).After we have had an eigen value energy and all wave function, we can result numerical solution for eigen value energy and plotting wave function in spherical coordinates.

FUTURE WORKS
Suggestions can be given for further research: a.The variable equations used to reduce Schrodinger equations at the angle are made equal, so there is no difference in the shape of the probability of finding a particle or a diatomic molecule.b.More understanding of quantum mechanics, particle physics and the concept of plotting real and imaginary waves.c.Need to do research with the same case, but different methods to be able to compare energy value and wave function.

Figure 1 .
Figure 1.Radial wave function with variation n r : (a) n r = 0, (b) n r = 1, (c) n r = 2 Figure (1) is ilustration of equation (36).At certain ranges the wave function looks constant and then at certain values the wave function toward infinity.When at a certain r value, n r = 2 has larger wave function than n r = 0 and n r = 1.If quantum number n r is getting greater, the wave function getting greater too.

Figure 2 .
Figure 2. Wave function P 1 on spherical coordinates with varying n 1 : (a) n 1 = 0, (b)n 1 = 1, (c) n 1 = 2.Equation (43) is portrayed by Figure2.It shows that the increase of the quantum number n 1 affected to wave function.However the wave function of the quantum number n 1 = 0 is greater than wave function of the quantum number n 1 = 1, wave function of the quantum number n 1 = 2 has the greatest of them.The shape of wave function in Figure (2) is faced-cone.The upper cone is bigger than a lower cone.This shape is represented a probability of diatomic molecule distribution.

Table 1 .
Mass and Spectroscopic Properties of Diatomic Molecular Variations

Table 2 .
The Energy Spectrum of Particles