Dirac equation and its contribution to atomic fine structure

This article aims to establish the comprehensiveness of the Dirac equation as an effective modification of quantum mechanics for the analysis of electrons and atomic fine structure. The Dirac equation is applied to investigate two scenarios involving electron interactions with different potentials. In the case where =0, the Dirac equation aligns naturally with electron theories and yields an electron gyromagnetic factor of g_s=2. With additional radiative corrections, it is possible to bring this value into closer proximity to experimental results. On the other hand, when considering spin-orbit coupling within a central field of V=(-e^2)r, the spin-orbit Hamiltonian derived from the Dirac equation is shown to match calculations based on Larmor and Thomas interactions. These cases collectively demonstrate the superior utility of Diracs theory when dealing with spin-1/2 particles like electrons, underscoring Diracs historical success in addressing the complexities of the time. His achievements in elucidating atomic fine structure and spin-orbit coupling hold pivotal significance for advancing technologies rooted in these theories. However, it is important to note that the Dirac equation primarily remains valid in weak external field situations, as observed in electron orbital motion, while challenges persist in unifying it with general relativity in stronger external field contexts.