Schrödinger Journal of Physics Education

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

Ibrahima Sakho

Published Aug 30, 2026• 0 citations

Abstract

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.

DOI: 10.37251/sjpe.v7i4.3496

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