Research Article  Open Access
The Application of Minimal Length in KleinGordon Equation with Hulthen Potential Using Asymptotic Iteration Method
Abstract
The application of minimal length formalism in KleinGordon equation with Hulthen potential was studied in the case of scalar potential that was equal to vector potential. The approximate solution was used to solve the KleinGordon equation within the minimal length formalism. The relativistic energy and wave functions of KleinGordon equation were obtained by using the Asymptotic Iteration Method. By using the Matlab software, the relativistic energies were calculated numerically. The unnormalized wave functions were expressed in hypergeometric terms. The results showed the relativistic energy increased by the increase of the minimal length parameter. The unnormalized wave function amplitude increased for the larger minimal length parameter.
1. Introduction
The relativistic effect gives the correction in the nonrelativistic quantum mechanics by applying the strong potential field in the particles dynamic. The particles dynamic in relativistic effect can be described by using KleinGordon equation, particularly for a zero spin particle. The KleinGordon equation is formed by two potentials coupling which are the fourvector potential and scalar potential . From these two potentials, the KleinGordon equation has two framework conditions, which are as follows: if the scalar potential was equal to vector potential and if the scalar potential was equal to minus vector potential . This condition appears in nuclear and high energy physics problem [1–3]. Some of researchers have investigated both these conditions with the certainty vector potential. The main case of that research was how to reduce the KleinGordon equation into the Schrodingerlike equation, so we can solve it by using the certainty suitable methods. The methods which are usually used are such as Supersymmetric Quantum Mechanics (SUSY) [4], NikiforovUvarov [1, 5], and Asymptotic Iteration Method [6, 7]. Various potentials are used to explain the dynamic particle, such as the harmonic potential [8], Makarov potential [2], Hulthen potential [1, 5], Kratzer potential [6], and Trigonometric PoschlTeller potential [7].
The particles dynamic in quantum mechanics corresponds to the position and momentum of particles. And, as we have known, the study of commutation relations between position and momentum operators is explained using Heisenberg uncertainty principle [9], which is given bywhere is position operator, is momentum operator, i is imaginer number, and with h being Planck constant. The presence of a quantum gravity on quantum mechanics has the consequence of the existence of a minimal observable distance on the order of the Planck length. Therefore the Heisenberg uncertainty principle gets additional correction due to the presence of a quantum gravity, which is well known as Generalized Uncertainty Principle (GUP) [10, 11], given bywhere is minimal length parameter that has value and P is magnitude of the momentum [11]. When the energy is much smaller than the Planck mass, goes to zero and we recover Heisenberg uncertainty principle [12].
In 2009, Jana and Roy have solved KleinGordon equation in the presence of minimal length for scalar potential was equal to vector potential using Algebraic Approach [13]. The KleinGordon equation in the presence of a minimal length is solved by using Feynman for scalar potential was equal to vector potential [14]. In addition, the hypergeometric method is used to solve KleinGordon equation using hyperbolic cotangent potential [15] and Asymptotic Iteration Method in trigonometric cotangent potential [16, 17].
In this paper, we solved the minimal length formalism in radial part of KleinGordon equation for the condition with Hulthen potential by using approximate solution. The approximate solution is used by Chabab et al. to solve the Bohr Mottelson Hamiltonian in the presence of a minimal length formalism by introducing the new wave function [11]. The minimal length formalism in the KleinGordon equation is reduced into secondorder differential equation. The relativistic energy equation and wave functions of KleinGordon equation are obtained by using Asymptotic Iteration Method.
The study is organized as follows. In Section 2, the approximate form of KleinGordon equation within minimal length formalism is presented. The Hulthen potential is introduced in Section 3. We describe Asymptotic Iteration Method in Section 4. The result and discussion are presented in Section 5. At last, conclusion is given in Section 6.
2. Approximate Form of KleinGordon Equation within Minimal Length Formalism
Generalized Uncertainty Principle is called the minimal length that is deformed from the commutation relations between position and momentum operators in quantum mechanics. In (2), the commutation relations can be rewritten as follows [9, 12]:where and are momentum operators at high and low energy, respectively. The magnitude of is expressed by p. The occurrence of minimal length is in string theory, black hole, quantum gravity, and noncommutative geometry, which yield new correction to Heisenberg uncertainty principle and imply a finite minimal uncertainty in position measurements, e.g., at the Planck scale [14].
The general KleinGordon equation with scalar potential and vector potential is given aswhere E is relativistic energy and M_{o} is rest mass. By setting in (5) and substituting (4) into (5), with and (natural unit), we haveand here we have set .
Accordingly, it would be natural to scale the potential term in (6), so that the nonrelativistic energy is reproduced [18]. The new wave function that is used to get the approximate form of KleinGordon equation is given [11]:Equation (7) is modification of in [11]. The modification is proposed to eliminate quadratic of Laplacian factor such that we get in [11]. By substituting (7) into (6), we obtainEquation (8) is the minimal length formalism in KleinGordon equation within the approximate form. The component is eliminated due to the value of which goes to zero, and value of is very small. Here, we have used the properties that is scalar differential operator. If operates to scalar fields at a point , will result in another scalar field. Here, is also called scalar Laplacian. Inserting (7) into (6) leads us to do the multiplication operation among Laplacian. By using the property that the multiplication operation is commutative for scalar differential operator with constant coefficient, then the multiplication operator among Laplacian also has commutative properties. Therefore, here we can eliminate when (7) is inserted into (6). To get simple solution of (8), binomial expansion is used for small ; then (8) becomesEquation (9) is obtained by setting which goes to zero, so is ignored. Applying spherical Laplacian operator, asinto (9), we use variable separable method by setting the new wave function , and we have a polar part and radial part of KleinGordon equation in the presence of minimal length. The polar part is given:and the radial part is as follows:where is a constant of variable separable method which corresponds to angular momentum (L). By applying and into (12), it yieldsEquation (13) is the minimal length formalism in KleinGordon equation with Hulthen potential in the form of onedimensional Schrodingerlike equation.
3. Asymptotic Iteration Method
Asymptotic Iteration Method is method to solve the secondorder differential equation in form [19–21] where and are the coefficients of a differential equation and n is a quantum number. To obtain solution, we derive (14).and hereThe eigenvalue is obtained from the quantization condition which is given byTo obtain the wave function, (14) is reduced into the formalism, as followsEquation (18) is onedimensional Schrodingerlike equation that has solution which is expressed in hypergeometric term, given aswhere is normalization constant and is a hypergeometric function. The unnormalized wave functions of KleinGordon equation are obtained by using (19)(20) [19–21].
4. Hulthen Potential
The Hulthen potential is one of short range potentials in physics. The Hulthen potential has been used in particle physics, atomic physics, nuclear physics, solidstate physics, and chemical physics. The Hulthen wave functions have been used in solidstate physics problems [22]. The Hulthenlike wave functions have been found to investigate atomic problems [22]. The general Hulthen potential is given by [23]where is a screening parameter and is potential depth. The value of screening potential is 0.025 for low screening and 0.15 for high screening [22]. To get simple solution, (21) was changed in hyperbolic trigonometric term [23], given asBy setting and , the visualization of Hulthen potential is expressed in Figure 1.
Figure 1 shows the visualization of Hulthen potential in r function. The Hulthen potential for different value of r is approximately from 0 until 0.05 (natural unit). The Hulthen potential has negative value in a very small value of r, while the Hulthen potential inclines to be constant for higher value of r.
5. Result and Discussion
Equation (13) can not be solved exactly unless we use the approximation to the function of . The approximation of function is given as [23]for small value of or . The visualization of that approximation is expressed in Figure 2.
Figure 2 shows the red line as a function of and light blue line as a function of . It is seen that the two lines overlap with each other; then the centrifugal term is approximated by also as in [23].To obtain the exact solution of (13) the approximate term of in (23) is inserted into (13) and together with (22); then we getEquation (24) is the KleinGordon equation with the minimal length for Hulthen potential which can be rewritten aswithEquation (25) is secondorder differential equation that will be reduced to hypergeometric differential equation type; by letting , we getand then we setin (29), so we haveBy setting , then (31) is reduced to AIMtype differential equation that is similar to (14).By comparing (14) and (32), we haveTo obtain eigenvalue, we use (15)(17) and (33)(34), which yieldsBy inserting (27)(28) and (30) into (35), we obtain the relativistic energy equation KleinGordon equation with minimal length for Hulthen potential, as follows: whereand n is quantum number. Equation (36) is relativistic energy equation of the minimal length formalism in KleinGordon equation with Hulthen potential. The relativistic energies were calculated numerically by using Matlab software. The results of relativistic energies are listed in Table 1.

Table 1 shows that the presence of minimal length and Hulthen potential in KleinGordon equation gives influence to the relativistic energy value. The relativistic energy value without minimal length parameter is lower than the relativistic energy value with minimal length parameter. Then, the relativistic energy value increases for the larger minimal length parameter and for the larger quantum number (n). The influence of Hulthen potential gives negative value in the relativistic energy value. If we set , without the presence of the minimal length parameter in relativistic energy equation (36), it was reduced to the relativistic energy equation which is in agreement with [5]. In [5] it was shown that the relativistic energy equation for the KleinGordon equation for Hulthen potential without the minimal length depends on the squared of quantum number (n).
To get the unnormalized wave function, we used (19)(20), so we haveInserting (40) into (19) yieldsEquation (41) is substituted into , so we have the unnormalized wave function given asBy applying in (42), we obtain the unnormalized wave function of minimal length formalism in KleinGordon equation, given asEquations (43), (44), and (45) are the unnormalized wave functions n=0 for ground state, n=1 for energy level 1, and n=2 for energy level 2, respectively. The visualization of the unnormalized wave functions is shown in Figures 3 and 4.
(a)
(b)
(c)
(a)
(b)
(c)
By inspecting Figures 3 and 4, we can see that the influence of minimal length parameter increases the amplitude of the unnormalized wave function.
6. Conclusion
The investigation of the minimal length formalism in the KleinGordon equation is obtained by approximate solution. The minimal length in KleinGordon equation for Hulthen potential is solved using Asymptotic Iteration Method. The Asymptotic Iteration Method is used to obtain the relativistic energy and unnormalized wave functions. The results show that the relativistic energies value increases for the larger minimal length parameter and for the larger quantum number (n). Then, the influence of minimal length parameter exerts effect in increasing the amplitude value of the unnormalized wave functions.
Data Availability
All data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was partly supported by Ministry of Research, Technology, and Higher Education with Contract no. 089/SP2H/LT/DRPM/2018.
Supplementary Materials
To solve the iteration equation and to do the numerical calculation, other software can be used such as Maple, Mathematica, and Octave. (Supplementary Materials)
References
 A. D. Antia, A. N. Ikot, E. E. Ituen, and I. O. Akpan, “Bound state solutions of the Klein  Gordon equation for deformed Hulthen potential with position dependent mass,” Sri Lankan Journal of Physics, vol. 13, no. 1, pp. 27–40, 2012. View at: Publisher Site  Google Scholar
 M. C. Zhang and Z. B. Wang, “Exact solutions of the KleinGordon equation with Makarov potential and a recurrence relation,” Chinese Physics, vol. 16, no. 7, article 1863, 2007. View at: Publisher Site  Google Scholar
 S. Ortakaya, “Exact solutions of the klein—Gordon equation with ringshaped oscillator potential by using the Laplace integral transform,” Chinese Physics B, vol. 21, no. 7, 2012. View at: Publisher Site  Google Scholar
 H. Hassanabadi, S. Zarrinkamar, and H. Rahimov, “Approximate solution of Ddimensional Klein  Gordon equation with hulthéntype potential via SUSYQM,” Communications in Theoretical Physics, vol. 56, no. 3, pp. 423–428, 2011. View at: Publisher Site  Google Scholar
 A. N. Ikot, L. E. Akpabio1, and E. J. Uwah, “Bound state solutions of the klein gordon equationwith the hulthen potential,” Electronic Journal of Theoretical Physics, vol. 8, no. 25, pp. 225–232, 2011. View at: Google Scholar
 D. A. Nugraha, A. Suparmi, C. Cari, and B. N. Pratiwi, “Asymptotic iteration method for solution of the Kratzer potential in Ddimensional KleinGordon equation,” Journal of Physics: Conference Series, vol. 820, 2017. View at: Publisher Site  Google Scholar
 D. A. Nugraha, A. Suparmi, and C. Cari, “Asymptotic Iteration Method foranalytical solution of Klein Gordon equation for Trigonometric PöschlTeller potential in D dimensions,” Journal of Theoretical and Applied Physics, vol. 795, no. 1, p. 42, 2017. View at: Publisher Site  Google Scholar
 A. Poszwa, “Relativistic generalizations of the quantum harmonic oscillator,” Acta Physica Polonica A, vol. 126, no. 6, pp. 1226–1234, 2014. View at: Publisher Site  Google Scholar
 L. Garay, “Quantum gravity and minimum length,” International Journal of Modern Physics A, vol. 10, no. 2, pp. 145–165, 1995. View at: Publisher Site  Google Scholar
 A. Tilbi, M. Merad, and T. Boudjedaa, “Particles of spin zero and 1/2 in electromagnetic field with confining scalar potential in modified heisenberg algebra,” FewBody Systems, vol. 56, no. 23, pp. 139–147, 2015. View at: Publisher Site  Google Scholar
 M. Chabab, A. El Batoul, A. Lahbas, and M. Oulne, “On γrigid regime of the BohrMottelson Hamiltonian in the presence of a minimal length,” Physics Letters B, vol. 758, pp. 212–218, 2016. View at: Google Scholar
 M. Alimohammadi and H. Hassanabadi, “Alternative solution of the gammarigid Bohr Hamiltonian in minimal length formalism,” Nuclear Physics A, vol. 957, pp. 439–449, 2017. View at: Publisher Site  Google Scholar
 T. K. Jana and P. Roy, “Exact solution of the KleinGordon equation in the presence of a minimal length,” Physics Letters A, vol. 373, no. 14, pp. 1239–1241, 2009. View at: Publisher Site  Google Scholar  MathSciNet
 M. Merad, F. Zeroual, and H. Benzair, “Spinless relativistic particle in the presence of a minimal length,” Electronic Journal of Theoretical Physics, vol. 6, no. 23, pp. 41–56, 2010. View at: Google Scholar
 I. L. Elviyanti, A. Suparmi, C. Cari, D. A. Nugraha, and B. N. Pratiwi, “Solution of Klein Gordon equation for hyperbolic cotangent potential in the presence of a minimal length using Hypergeometric method,” Journal of Physics: Conference Series, vol. 909, no. 1, 2017. View at: Google Scholar
 A. Suparmi, C. Cari, and I. L. Elviyanti, “Solution of Klein Gordon equation for trigonometric cotangent potential in the presence of a minimal length using Asymptotic Iteration Method,” Journal of Physics: Conference Series, vol. 909, no. 1, 2017. View at: Google Scholar
 C. Cari, A. Suparmi, and I. L. Elviyanti, “Approximate solution for the minimal length case of klein gordon equation for trigonometric cotangent potential using asymptotic iteration method,” Journal of Physics: Conf. Series, vol. 909, 2017. View at: Google Scholar
 S. M. Ikhdair and M. Hamzavi, “Effects of external fields on a twodimensional Klein—Gordon particle under pseudoharmonic oscillator interaction,” Chinese Physics B, vol. 21, no. 11, p. 110302, 2012. View at: Publisher Site  Google Scholar
 H. Ciftci, R. L. Hall, and N. Saad, “Construction of exact solutions to eigenvalue problems by theasymptotic iteration method,” Journal of Physics A: Mathematical and General, vol. 36, no. 47, pp. 11807–11816, 2003. View at: Publisher Site  Google Scholar  MathSciNet
 B. N. Pratiwi, A. Suparmi, C. Cari, and A. S. Husein, “Asymptotic iteration method for the modified poschlteller potential and trigonometric Scarf II noncentral potential in the Dirac equation spin symmetry,” Pramana—Journal of Physics, vol. 88, no. 2, 2017. View at: Google Scholar
 S. Pramono, A. Suparmi, and C. Cari, “Relativistic energy analysis of fivedimensional qdeformed radial rosenmorse potential combined with qdeformed trigonometric scarf noncentral potential using asymptotic iteration method,” Advances in High Energy Physics, vol. 2016, 2016. View at: Google Scholar
 Y. P. Varshni, “Eigenenergies and oscillator strengths for the Hulthén potential,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 41, no. 9, pp. 4682–4689, 1990. View at: Publisher Site  Google Scholar
 L. Naderi and H. Hassanabadi, “Bohr Hamiltonian with Eckart potential for triaxial nuclei,” The European Physical Journal Plus, vol. 131, no. 5, 2016. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2018 Isnaini Lilis Elviyanti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.