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## Relativistic Effects

If one considers the Bohr-Sommerfeld atomic model (Sommerfeld atomic model) of the hydrogen atom, the question arises whether relativistic effects such as the increase in mass with increasing speed have to be taken into account. The speed and the kinetic energy of the electron are small compared to the speed of light or energy of rest, but the effects are in the same order of magnitude as the fine structure caused by the spin. Sommerfeld has calculated these corrections.

In 1928 Paul Dirac set up the equation named after him, which takes these relativistic effects into account and at the same time describes the spin of the electron and the spin-orbit coupling. She also delivers ${G}_{s}=2\text{}$approximately the correct value for the g-factor of the electron. The same result is obtained with it as with a perturbation calculation based on the Schrödinger equation and taking into account the spin-orbit coupling and the relativistic effects.

- $$\text{\Delta}{\mathrm{E.}}_{\text{FS}}=\text{\Delta}{\mathrm{E.}}_{\text{SB}}+\text{\Delta}{\mathrm{E.}}_{\text{Rel}}=-\frac{m{c}^{2}{(Z\alpha )}^{4}}{2{n}^{3}}\left(\frac{1}{j+\frac{1}{2}}-\frac{3}{4n}\right)\phantom{\rule{2em}{0ex}}\begin{array}{l}\Delta {\mathrm{E.}}_{\mathrm{F.}\mathrm{B.}}=\text{Energy (fine structure)}\\ \Delta {\mathrm{E.}}_{\mathrm{S.}\mathrm{B.}}=\text{Energy (spin-orbit coupling)}\\ \Delta {\mathrm{E.}}_{\mathrm{Rel}}=\text{Energy (relativistic effects)}\\ m=\text{Dimensions}\\ c=\text{Speed of Light}\\ Z=\text{Nuclear load number}\\ n=\text{Principal quantum number}\\ \alpha =\text{Fine structure constant}\end{array}$$

Interestingly, it turns out that now the degeneracy according to the orbital angular momentum quantum number $l$ is canceled, but the levels according to the total angular momentum quantum number $j$ are degenerate.