The Study of One-Dimensional Relativistic Shannon Entropy with Hulthen–Coulomb Interaction Using Klein-Gordon Equation

The study of quantum information entropies is significant in quantum computation and signal analysis. Much research work has been carried out in quantum information entropies under nonrelativistic wave equation. However, not much research has been reported in evaluating quantum information entropies using relativistic wave equation. In this work, we evaluate Shannon information entropy in one-dimensional space with generalised Hulthen plus Coulomb potential (GHPCP) in Klein-Gordon equation using parametric Nikiforov-Uvarov method. The numerical computations obtained both for position and momentum spaces as well as the Shannon sum obeys Bialynicki-Birula and Mycieslski (BBM) uncertainty inequality in order to ascertain the degree of accuracy of the analytical and numerical calculation. This research has relevance in atomic and molecular physics especially in localisation and delocalisation of electrons or particle in atomic orbitals.

Klein-Gordon Equation, Nikiforov-Uvarov Method Generalised Hulthen Plus Coulomb potential, Shannon information entropy.

1. Okon, I. B, Onate, C. A, William, E.S, Chen, W. L, Isonguyo, C. N, Akpan, D. N, Purohit, K. R, Antia, A. D, Araujo, J. P, Ukpong, A. M, Okorie, U. S, Inyang, E. P, Eyube, E. S, Emeje, K. O , Akpabio, L. E, Ituen, E. E, Aly, A. E., Essien, K. E. Analytical and computational study of Fisher and Shannon information entropies in one and three dimensional spaces for exponential type potential. Eur. Phys. J. Plus, 140. (2025) 250. https://doi.org/10.1140/epjp/s13360-025-06151-0. [Google Scholar] [Crossref]

2. Chen, W. L, Okon, , I. B, Onate, C. A, Isonguyo , C.N, Ekong , S.A. Araujo , , J.P, Aly, A. E, Anna A.O, William , E.S , Inyang, E.P, Emeje , K.O., Purohit, K.R. Fisher and Shannon information entropies in 2-dimension for a non-central potential as applied to hydrogen molecules. Results in Physics 75(2025)108303, https://doi.org/10.1016/j.rinp.2025.108303 [Google Scholar] [Crossref]

3. Shannon, C. E. A mathematical theory of communication. Bell Syst Tech J 27(1948)379. [Google Scholar] [Crossref]

4. Dehesa, J. S, Gonzalez-Ferez R, Sanchez-Moreno P. The Fisher-information-based uncertainty relation, Cramer–Rao inequality and kinetic energy for the D-dimensional central problem. J Phys A: Math Theor. 40(2007)1845–56. http://dx.doi.org/10.1088/1751-8113/40/8/011. [Google Scholar] [Crossref]

5. Fisher, R. A. Theory of statistical estimation. Math Proc Camb Philos 22(1925)700–25. http://dx.doi.org/10.1017/S0305004100009580. [Google Scholar] [Crossref]

6. Boumali A, Labidi M. Shannon entropy and Fisher information of the one-dimensional Klein-Godon oscillator with energy-dependent potential. Mod Phys Lett A. 33(2018)1850033. http://dx.doi.org/10.1142/S0217732318500335. [Google Scholar] [Crossref]

7. Hohenberg P, Kohn W. Inhomogeneous electron gas. Phys Rev B 136(1964)864–71. http://dx.doi.org/10.1103/PhysRev.136.B864. [Google Scholar] [Crossref]

8. Panahi H, Najafizade A, Hassanabadi H. Study of the Shannon entropy in the quantum model obtained from SO (2,2). J Korean Phys Soc;75(2019)87–93. http://dx.doi.org/10.3938/jkps.75.87. [Google Scholar] [Crossref]

9. Onate, C. A, Oyewumi, K. J, Falaye , B. J, Okon, I. B, Omugbe E, Wen-Li Chen. Fisher information of a modified trigonometric inversely quadratic potential. Chinese J Phys. 80 (2022) 1–11. http://dx.doi.org/10.1016/j.cjph.2022.09.019. [Google Scholar] [Crossref]

10. Sears, S. B, Parr, R. G, Dinur U. On the quantum mechanical kinetic energy as a measure of the information in a distribution. Isr J Chem. 19(1980)165–73. [Google Scholar] [Crossref]

11. Bercher, J. F, Vignat C. On minimum Fisher information distribution with restricted support and fixed variance. Inf.Sci.179(2009)3832 http://dx.doi.org/10.1016/j.ins.2009.07.013. [Google Scholar] [Crossref]

12. Yahya, W. A, Oyewumi, K. J, Sen, K. D. Position and momentum information-theoretic measures of the pseudoharmonic potential. Int J Quantum Chem. 115(2014)1543–52. http://dx.doi.org/10.1002/qua.24971. [Google Scholar] [Crossref]

13. Martin, M. T, Pennini F, Plastino A. Fisher’s information and the analysis of complex signals. Phys Lett A. 256(1999)173–6. http://dx.doi.org/10.1016/S0375-9601(99)00211-X. [Google Scholar] [Crossref]

14. Nagy, A. Fisher information in a two-electron entangled artificial atom. Chem Phys Lett. 425(2006)15. http://dx.doi.org/10.1016/j.cplett.2006.05.013. [Google Scholar] [Crossref]

15. Sun, G. H, Popov, D., Camacho-Nieto O, Dong, S. H. Shannon information entropies for position-dependent mass Schrödinger problem with a hyperbolic well. Chin Phys B. 24(2015)100303. http://dx.doi.org/10.1088/1674-1056/24/10/100303. [Google Scholar] [Crossref]

16. Sun, G. H, Avila, A. M, Dong, S. H. Quantum information entropies of the eigenstates for the Pöschl–Teller-like potential. Chin Phys B. 22(2013)050302. http://dx.doi.org/10.1088/1674-1056/22/5/050302. [Google Scholar] [Crossref]

17. Bakke, K, Furtado, C. On the Klein-Gordon oscillator subject to a Coulomb-Type potential. Annals of Physics 355(2015)48. https://doi.org/10.1016/j.aop.2015.01.028. [Google Scholar] [Crossref]

18. Qiang, W. C, Dong, S. H. Exact bound state solutions of the s-wave Klein–Gordon equation with the generalized Hulthén potential. Phys Lett A. 331 (2004) 374–377. http://dx.doi.org/10.1016/j.physleta.2004.09.032. [Google Scholar] [Crossref]

19. Simsek, M, Egrifes, H. The Klein-Gordon equation of generalized Hulthén potential in complex quantum mechanics. arXiv preprint. (2002). https://arxiv.org/abs/quant-ph/0211025. [Google Scholar] [Crossref]

20. Egrifes, H, Sever, R. Bound-state solutions of the Klein-Gordon equation for the generalized PT-symmetric Hulthén potential. Int J Theor Phys. 46 (2007) 935–950. http://dx.doi.org/10.1007/s10773-006-9251-8. [Google Scholar] [Crossref]

21. Antia, A. D, Ikot, A. N. Relativistic treatment of spinless particles subject to the Hulthén plus Yukawa potential with arbitrary states. Adv. Phys .Theor Appl. 35 (2014) 1–9. [Google Scholar] [Crossref]

22. Karayer, H, Demirhan, D, Buyukkilic, F. Analytical solution of local fractional Klein-Gordon equation for the generalized Hulthén potential. arXiv preprint. (2017). https://arxiv.org/abs/1703.03234. [Google Scholar] [Crossref]

23. Derezinski, J., Richard, S. On radial Schrödinger operators with a Coulomb potential. Annales Henri Poincaré, 19 (2018).2869–2917. [Google Scholar] [Crossref]

24. Woolley, R. G. Non-Relativistic Quantum Electrodynamics and the Coulomb Interaction. Physics 8(2026)20. https://doi.org/10.3390/physics8010020. [Google Scholar] [Crossref]

25. Campos P., Martins, M. G. R, Vianna, J. D. M. Quantum Mechanics on Phase space and the Coulomb potential. Physics Letters A. 381(2017)1129. https://doi.org/10.1016/j.physleta.2017.02.005. [Google Scholar] [Crossref]

26. Onate, C. A, Okon, I. B, Omugbe, E., Jude, G. O and D. B. Olanrewaju, D. B Analytical treatment of the non-relativistic system for diatomic molecules. Annals of the University of Craiova, Physics AUC. 32(2022) 103-113. [Google Scholar] [Crossref]

27. Okon, I. B, Isonguyo, C. N., Onate, C. A. Antia, A. D, Purohit, K. R, Ekott, E. E. , Essien , K. E, William, E. S and Asuquo, N. E.Eigensolutions and Thermodynamic properties of Manning-Rosen Plus exponential Yukawa potential. World Journal of Applied Science and Technology (WOJAST) 14(2022) 91-99. https://dx.doi.org/10.4314/WOJAST.v14i2.91. [Google Scholar] [Crossref]

28. Isonguyo, C. N, Oyewumi, K. J, Oyun, O.S, Okon, I. B. Entropic Uncertainty Relations and the Fisher Information For the Generalised Radial Yukawa Potential. World Journal of Applied Science and Technology (WOJAST) 14 (2022)44-50. https://dx.doi.org/10.4314/WOJAST.v14i1b.44 (Ajol) [Google Scholar] [Crossref]

29. Chen, W. L, Okon, I. B, Onate, C. A, Omugbe, E, Okorie, U.S. Obasuyi, A.R, Ikot, A. N. Diatomic molecular energy spectra and thermodynamic properties of exponential inversely quadratic plus improved deformed exponential-type potential. Results in Physics 51 (2023) 106686 . https://doi.org/10.1016/j.rinp.2023.106686. [Google Scholar] [Crossref]

30. Okon, I. B, Popoola, O, Ituen, E. E. Bound state solution to Schrodinger equation with Hulthen plus exponential Coulombic potential with centrifugal potential barrier using parametric Nikiforov-Uvarov method. International Journal of Recent Advances in Physics. 5 (2016) 1-15. Doi:10.14810/ijrap.2016.5101 [Google Scholar] [Crossref]

31. Okon, I. B, Popoola, O., Isonguyo, C. N. Approximate Solution of Schrodinger Equation with some Diatomic molecular interactions Using Nikiforov-Uvarov method. Advances in High Energy Physics. Article ID 9671816(2017). https://doi.org/10.1155/2017/9671816. [Google Scholar] [Crossref]

32. Antia, A. D, Okon, I. B . Applications of Hellmann-Feynman Theorem (HFT) to Angle Dependent Yukawa Potential of Diatomic Molecules. Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) 63(2017). 16-28 [Google Scholar] [Crossref]

33. Okon, I. B, Popoola, O., Isonguyo, C. N., Antia, A. D. Solutions of Schrodinger and Klein-Gordon Equations with Hulthen Plus Inversely Quadratic Exponential Mie-Type Potential. Physical Science International Journal. 19(2018) 1-27. DOI: 10.9734/PSIJ/2018/43610. [Google Scholar] [Crossref]

34. Tezcan, C., Sever, R. A General Approach for the Exact Solution of the Schrödinger Equation. Int J Theor Phys 48 (2009) 337–350 https://doi.org/10.1007/s10773-008-9806-y [Google Scholar] [Crossref]