Semi-inverse variational approach to solve the Klein-Gordon equation for harmonic- and perturbed Coulomb potentials


  • Khalid Reggab
  • Ahmed Gueddim
  • Abdelkrim Naas



Klein-Gordon equation, energy eigenvalues, perturbed-Coulomb potential, harmonic potential, semi-inverse variational approach


The events observable at the atomic scale cannot be adequately explained by classical physics, hence a new conceptual framework for physics must be developed. "Quantum mechanics" is the name given to this new theory of the physical cosmos. The basic formula of relativistic quantum mechanics is the problem of Klein-Gordon. In relativistic quantum physics, the wave function is crucial since it contains all the data describing a quantum system, in relativity quantum physics, they are crucial. How to solve Klein Gordon's equation has been the topic of intense debate in recent years. To determine the system's spectrum using numerous potentials and methodologies. To obtain the Klein-Gordon issues solution Through determining the harmonic and perturbed Coulomb potentials' energies, The recent study employed an entirely distinct methodology. using the Semi-Inverse Variational Technique to aid in the process, we were able to identify the eigen energies for Harmonic- and Perturbed Coulomb Potentials for a variety of quantum numbers. A further benefit of the technique was that it enabled us to obtain the wave eigenfunctions for various potentials, demonstrating the viability of this approach. We are excited to deal with additional challenges in our upcoming works, the first being the use of nonlinear potentials to solve the Dirac equation due to the effectiveness of this approach.


Motavalli, H, Akbarieh, “Klein–Gordon equation for the coulomb potential in noncommutative space”, Modern Physics Letters A.vol 25, no. 29, pp.2523-2528,2010.

Yasuk, F., Durmus, A., Boztosun, “Exact analytical solution to the relativistic Klein-Gordon equation with noncentral equal scalar and vector potentials”, Journal of Mathematical Physics,Vol. 47,2006.

B. K. Bagchi, “Supersymmetry in Quantum and Classical Mechanics”, 1st ed, London, UK,2000.

F. Cooper, A. Khare and U. Sukhatme, “Supersymmetry and quantum mechanics,” Physics Reports, vol. 251, no. 5-6, pp. 267–385, 1995.

G. Junker, Supersymmetric Methods in Quantum and Statistical Physics, 1st ed, Berlin, Germany: Springer ,1996.

A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics, 1st ed, Basel, Switzerland: Springer Basel AG ,1988.

B.M.Karnakov,V.P.Krainov,WKB Approximation in Atomic Physics ,1st ed, Berlin, Heidelberg,Germany: Springer,2013.

A. Zerarka and K. Libarir, “A semi-inverse variational method for generating the bound state energy eigenvalues in a quantum system: the schrödinger equation” Commun Nonlinear sci Numer Simulat,vol 14,pp.3195-3199, 2009.

K. Libarir and A. Zerarka “A semi-inverse variational method for generating the bound state energy eigenvalues in a quantum system: the dirac coulomb type-equation”, Journal of Modern Optics, no.0950-0340,2017.

B. Nine, O. Haif-Khaif, A. Zerarka, “The eigenenergies of the wave function through the non-variational galerkin-b-spline approach”, Applied Mathematics and Computation, vol.178, pp.486–492,2006.

Xin-Wei Zhou, “Variational approach to the broer–kaup–kupershmidt equation”,Physics Letters A,no 363,no.108–109,2007.

Xin-Wei Zhou, Lin Wang, “A variational principle for coupled nonlinear Schrödinger equations with variable coefficients and high nonlinearity”, Computers and Mathematics with Applications, vol. 61, pp.2035–2038,2011.

Hong-Mei Liu, “Generalized variational principles for ion acoustic plasma waves by he’s semi-inverse method hong-mei liu chaos,” Solitons and Fractals, vol.23, pp. 573–576,2005.

Hong-Mei Liu, “Variational approach to nonlinear electrochemical system”, International Journal of Nonlinear Sciences and Numerical Simulation, vol.5, pp.95-96, 2004.

T. Özis, A.Yıldırım, ,“Application of he’s semi-inverse method to the nonlinear schrödinger equation” Computers and Mathematics with Applications,vol.54 ,pp.1039–1042,2007.

J.Manafian, P.Bolghar, A.Mohammadalian, “Abundant soliton solutions of the resonant nonlinear Schrodinger equation with time-dependent coefficients by item and he’s semi-inverse method”, Optical and Quantum Electronics, vo.49,pp.322,2017.

M.Najaf and S.Arbab, “Exact solutions of five complex nonlinear Schrodinger equations by semi-inverse variational principal”, Communications in Theoretical Physics, Vol. 62,no.3, pp.301–307,2014.

Gangwei Wang and Tianzhou Xu, “Optical soliton of time fractional Schrödinger equations with he’s semi-inverse method”, Laser Physics.vol.25, no.5,2015.

Ji-Huan He, “A modified Li-He’s variational principle for plasma”, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 31, no. 5,2021.

Anjan Biswas, “Soliton solutions of the perturbed resonant nonlinear Schrodinger’s equation with full nonlinearity by semi-inverse variational principle”, Quantum Physics Letters.vol.1, no.2, pp.79-83,2012.

P.D. Green, «Optical solitons with higher order dispersion by semi-inverse variational principal”, Progress in Electromagnetics Research, vol.102, pp.337–350, 2010.

Xue-Wei LI, Ya LI, Ji-Huan HE, «On the semi-inverse method and variational principle”, Thermal Science, vol. 17, no. 5, pp. 1565-1568,2013.

A. Biswas, D. Milovic, M. Savescu, M. F. Mahmood, K. R. Khan and Ru. Kohl, “Optical soliton perturbation in nanofibers with improved nonlinear Schrödinger’s equation by semi-inverse variational principle”, Journal of Nonlinear Optical Physics &Materials, vol. 21, no. 4 ,2012.

A. Biswas, Q. Zhou, H. Triki, M. Zaka Ullah, M. Asma, S. P.

Moshokoa and M. Belic, “Resonant optical solitons with parabolic and dualpower laws by semi-inverse variational principle”, Journal of Modern Optics, vol. 65, no. 2, pp. 179–184,2018.

A. Jabbari, H. Khairi, A. Bekir, “Exact solutions of the coupled Higgs equation and the maccari system using he’s semi-inverse method and (G^'/G) expansion method”, Computers and Mathematics with Applications, vol. 62, no. 5 pp. 2177–2186,2011.

R. W. Kohl, A. Biswas, M. Ekici, Q. Zhou, S. Khan,Ali S. Alshomrani, M. R. Belic, “Highly dispersive optical soliton perturbation with kerr law by semi-inverse variational principle” , International Journal for Light and Electron Optics,vol. 199 ,2019.

Tapan K. Sarkar, “A note on the variational method (Rayleigh-Ritz), Galerkin's Method, and the method of least squares”, Advancing earth and space sciences, vol.18,No.6,1983.

M. Benaissa, “Introduction de quelques approches variationnelles récentes”, Master, Department of Material Sciences, University of Mohamed Khider, Biskra, Algeria,2014.

L. Girgis and A. Biswas, “Study of solitary waves by He's semi-inverse variational principal», Waves in Random and Complex Media, Vol. 21, No. 1, pp. 96–104,2011.

T. Barakat, M. Odeh and O. Mustafa, “Perturbed coulomb potentials in the Klein-Gordan equation via the shifted – l expansion technique”, Journal of Physics A: Mathematical and General, Vol. 31, no. 15,1998.

Alhaidari A A D, Bahlouli H and Al-Hasan A 2006 Phys. Lett. A 349 87.

Om Prakash Agrawal, “Generalized Variational Problems and Euler–Lagrange equations», Computers and Mathematics with Applications, vol. 59, no.5, pp.1852–1864,2010.

Mehmet Tekkoyun, “On para-Euler–Lagrange and para-Hamiltonian equations”, Physics Letters A, vol. 340, pp. 7–11,2005.

Mathematica,2020, Wolfram.

Nagalakshmi A Rao and B A Kagali, “Energy profile of the one -dimensional Klein-Gordan oscillator”, PHYSICA SCRIPTA,vol. 77 (2008).

B. R. McQuarrie and E. R. Vrscay, “Rayleigh-Schrödinger perturbation theory at large order for radial Klein-Gordon equations” Physical Review A. vol. 47, no. 2,1993.

Khalid Reggab and Houssam Eddine Hailouf, “Study of bound states for diatomic molecules by resolution of Schrödinger equation with pseudo-harmonic and Mie potentials via Nikiforov–Uvarov (NU) method”, International Journal of Geometric Methods in Modern Physics,vol 20.No. 5,2023.

Ahmed Gueddim, and Nadir Bouarissa, Electronic structure, and optical properties of dilute InAs1-xNx: pseudopotential calculations, Physica Scripta 79 (2009) 015701.

A. Gueddim, S. Zerroug, N. Bouarissa, N. Fakroun, Study of the elastic properties and wave velocities of rocksalt Mg1-xFexO: ab initio calculations, Chinese Journal of Physics, Volume 55, Issue 4, August 2017, Pages 1423.

N. Drissi, A. Gueddim, N. Bouarissa, First-principles study of rocksalt MgxZn1- xO: band structure and optical spectra, Philosophical Magazine 100 (12) (2020) 1620-1635.

S. Djaili, A. Gueddim, D. Guibadj, N. Bouarissa, Temperature dependence of the optical properties of MgO: ab initio molecular dynamics calculations, Optik 200 (2020) 163421(1-5).

A. Gueddim, N. Bouarissa, L. Gacem, A. Villesuzanne, Structural phase stability, elastic parameters, and thermal properties of YN from first-principles calculation, Chinese Journal of Physics 56 (5) (2018) 1816-1825.

S. Zerroug, A. Gueddim, N. Bouarissa, Composition dependence of fundamental properties of Cd1-xCoxTe magnetic semiconductor alloys, Journal of Computational Electronics 15 (2) (2016) 473.

Houssam Hailouf,Lakhdar Gacem,Ahmed Gueddim and Kingsley Onyebuchi Obodo, “ DFT studies on the structural, electronic, and optical properties of Na2ZnP2O7 compound”, Materials Today Communications,vol.29,2021.

Houssam Hailouf,L. Gacem ,Ahmed Gueddim and Ali H Reshak, “ Structural, electronic, magnetic, and optical properties of Fe-doped Na2ZnP2O7 host: ab-initio calculation”, Physica B Condensed Matter,vol. 650,2022.




How to Cite

Reggab, K., Gueddim, A., & Naas, A. (2023). Semi-inverse variational approach to solve the Klein-Gordon equation for harmonic- and perturbed Coulomb potentials . STUDIES IN ENGINEERING AND EXACT SCIENCES, 4(1), 248–267.