Schrödinger: Journal of Physics Education
Schrödinger: Journal of Physics Education

Advancing Physics and Physics Education Through Research and Innovation

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Schrödinger: Journal of Physics Education

Advancing Physics and Physics Education Through Research and Innovation

Eigenvalues of the Hamiltonian Fine structure of the Hydrogen Like Systems: Semi-Relativistic Theory of Sakho vs Relativistic Theory of  Dirac

Ibrahima Sakho
University Iba Der Thiam
Senegal
ibrahima.sakho@univ-thies.sn
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  • Purpose of the study: The goal of this work is to present a comparative study between the semi-relativistic theory of Sakho and the relativistic theory of Dirac applied to calculations of the eigenvalue of the Hamiltonian fine structure of the hydrogen like systems. The three small perturbative corrections (relativistic mass correction, spin-orbit coupling, and the Darwin term) of the Hamiltonian are considered.

    Methodology: This research is based on correction of the semi-classical Bohr’s theory on the hydrogen like systems considering the relativistic mass correction by ignoring the electron’s spin. This approach referred to as semi-relativistic theory allows one to express the eigenvalues of the Hamiltonian fine structure to be compared to that from the relativistic wave function of Dirac

    Main Findings: The research gives the first expression of the quantized energy of hydrogen like systems taking only into account, the relativistic mass correction.   This result provides exactly the eigenvalues of the Hamiltonian fine structure predicted from Dirac’s relativistic theory for all the non-degenerated quantum states n2Lj (j = l ± s) of the hydrogen like systems (i.e.  1 2s1/2, 2 2p3/2, 3 2d5/2, 42f7/2, and so on).

    Novelty/Originality of this study: Theoretical determination of the eigenvalue of the Hamiltonian fine structure predicted from Dirac’s relativistic theory for all the non-degenerated quantum states is obtained with a soft semi-relativistic theory without including in the calculations the spin-orbit coupling and the Darwin term corrections.

  • How to cite

    [1]
    I. Sakho, “Eigenvalues of the Hamiltonian Fine structure of the Hydrogen Like Systems: Semi-Relativistic Theory of Sakho vs Relativistic Theory of  Dirac”, Sch. Jo. Phs. Ed, vol. 7, no. 4, pp. 238–255, Aug. 2026, doi: 10.37251/sjpe.v7i4.3496.

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    1. I. Sakho, Introduction to Quantum Mechanics 1: Thermal Radiation and Experimental Facts Regarding the Quantization of Matter, vol. 1. London, U.K.: ISTE Ltd. and Hoboken, NJ, USA: John Wiley & Sons, 2019.
    2. I. Sakho, Physique Atomique: Systèmes Hydrogénoïdes et Systèmes Hélio¬moïdes—Cours & Exercices Corrigés. Paris, France: Éditions Ellipses, 2020.
    3. I. Neutelings, “Rutherford scattering and the discovery of the nucleus,” Philos. Mag., ser. 6, vol. 21, no. 125, pp. 669–688, 2017.
    4. N. Straumann, “On the first Solvay Congress in 1911,” Eur. Phys. J. H, vol. 36, pp. 379–399, 2011, doi: 10.1140/epjh/e2011-20043-9.
    5. N. Bohr, “On the constitution of atoms and molecules,” Philos. Mag., ser. 6, vol. 26, no. 151, pp. 1–25, 1913, doi: 10.1080/14786441308634955.
    6. N. Bohr, “On the constitution of atoms and molecules. Part II—Systems containing only a single nucleus,” Philos. Mag., ser. 6, vol. 26, no. 153, pp. 476–502, 1913, doi: 10.1080/14786441308634993.
    7. N. Bohr, “On the constitution of atoms and molecules. Part III—Systems containing several nuclei,” Philos. Mag., ser. 6, vol. 26, no. 155, pp. 857–875, 1913, doi: 10.1080/14786441308635031.
    8. D. D. Pinke, K. Ősz, and G. Lente, “The origin of the postulates in the Bohr model of the hydrogen atom,” ChemTexts, vol. 11, Art. no. 11, 2025, doi: 10.1007/s40828-025-00208-4.
    9. M. Niaz and L. Cardellini, “What can the Bohr–Sommerfeld model show students of chemistry in the 21st century?,” J. Chem. Educ., vol. 88, no. 2, 2011.
    10. V. M. Simulik, “The Dirac equation near centenary: A contemporary introduction to the Dirac equation consideration,” J. Phys. A: Math. Theor., vol. 58, no. 5, Art. no. 053001, 2025, doi: 10.1088/1751-8121/adab56.
    11. K. Barley, A. Ruffing, and S. K. Suslov, “Old quantum mechanics by Bohr and Sommerfeld from modern perspective,” Phys.-Uspekhi, vol. 69, no. 1, pp. 74–93, 2026, doi: 10.3367/UFNe.2025.08.040014.
    12. I. Sakho, “Relativistic theory of one- and two-electron systems: Valley of stability in the helium-like ions,” J. At. Mol. Sci., vol. 3, no. 1, 2012.
    13. L. V. de Broglie, On the Theory of Quanta: A Translation of Recherches sur la Théorie des Quanta, A. F. Kracklauer, Ed. Paris, France: AFK, 2004.
    14. F. T. Tehrani, “On De Broglie’s wave-particle theory,” Int. J. Theor. Math. Phys., vol. 15, no. 1, pp. 1–3, 2025, doi: 10.5923/j.ijtmp.20251501.01.
    15. E. Biémont, Spectroscopie Atomique: Instrumentation et Structures Atomiques—Cours. Bruxelles, Belgium: De Boeck Université, 2006.
    16. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique Quantique, vol. 2. Paris, France: Hermann Éditeurs des Sciences et des Arts, 1992.
    17. I. Sakho, “Nouvelle méthode de calcul des énergies de l’état fondamental et des états singulets et triplets doublement excités (Nlnl′, 2S+1Lp) des systèmes atomiques à deux électrons, basée sur la détermination de la constante d’écran par unité de charge nucléaire,” Ph.D. dissertation, Institut de Technologie Nucléaire Appliquée, Université Cheikh Anta Diop de Dakar, Dakar, Sénégal, 2007.
    18. I. Sakho, “Formalisme général de la méthode de la constante d’écran par unité de charge nucléaire appliqué à l’étude de la photoionisation résonante des systèmes atomiques à plusieurs électrons,” Doctoral dissertation, Institut de Technologie Nucléaire Appliquée, Université Cheikh Anta Diop de Dakar, Dakar, Sénégal, 2013.
    19. I. Sakho, The Screening Constant by Unit Nuclear Charge Method: Description & Application to the Photoionization of Atomic Systems. London, U.K.: ISTE Science Publishing Ltd. and Hoboken, NJ, USA: John Wiley & Sons, 2018.
    20. I. Sakho, “A modification of atomic orbital theory and its application to (1snl)1Lp and (nl2)1Lp excited states of He-like ions,” J. At. Mol. Sci., vol. 1, pp. 103–117, 2010.
    21. I. Sakho, “General formalism of the modified atomic orbital theory for the Rydberg series of atoms and ions: Application to the photoionization of Ne+,” J. At. Mol. Sci., vol. 5, pp. 206–216, 2014.
    22. I. Sakho, “Electrodynamics calculations of the unit nuclear radius in agreement with the constant density model,” AASCIT J. Phys., vol. 4, pp. 26–44, 2018.
    23. A. Adamu, “A new measurement of nuclear radius from the study of β+ decay energy of finite-sized nuclei,” J. Rad. Nucl. Appl., vol. 6, no. 1, pp. 45–49, 2021, doi: 10.18576/jrna/060107.
    24. I. Sakho, “New method of estimation of the speed of light in vacuum using Q-values of beta-decay transitions in mirror nuclei,” Schrödinger: J. Phys. Educ., vol. 6, no. 1, pp. 1–8, 2025, doi: 10.37251/sjpe.v6i1.1220.
    25. I. Sakho, “New Q-β-decay theory applied to the calculations of the rest mass energy m₀c² of the electron and of the Q-value for β+-decay transitions in mirror nuclei A = 2Z − 1,” J. At. Mol. Condens. Matter Nano Phys., vol. 12, no. 1, pp. 27–43, 2025, doi: 10.26713/jamcnp.v12i1.3173.
    26. L. G. de Peralta and H. Farooq, “A notable quasi-relativistic wave equation and its relation to the Schrödinger, Klein-Gordon, and Dirac equations,” J. Mod. Phys., vol. 12, pp. 1145–1159, 2021, doi: 10.4236/jmp.2021.128068.
    27. S. Wu, “Exact solutions of Dirac equation for hydrogen atom using the linear combination of orthogonal Laguerre basis functions,” Sci. Rep., vol. 15, Art. no. 16004, 2025, doi: 10.1038/s41598-025-01139-0.
    28. S. R. Lundeen and F. M. Pipkin, “Separated-oscillatory-fields measurement of the Lamb shift in H, n = 2,” Phys. Rev. Lett., vol. 34, pp. 1368–1371, 1975.
    29. G. W. Erickson, “Energy levels of one-electron atoms,” J. Phys. Chem. Ref. Data, vol. 6, pp. 831–850, 1977.
    30. P. J. Mohr, “Lamb shift in a strong Coulomb potential,” Phys. Rev. Lett., vol. 34, pp. 1050–1053, 1975.