Hydrogen fine-structure correction ($\Delta E_{nj}=-m_e{c}^2{\alpha }_{\mathrm{fs}}^4[1/(j+1/2)-3/(4n)]/(2n^3)$)

For a point, fixed proton, the sum of the relativistic kinetic-energy correction, spin-orbit interaction including the Thomas precession factor, and Darwin term has the displayed leading correction. It depends on n and total angular momentum j, but not separately on l. Thus $2S_{1/2}$ and $2P_{1/2}$ remain degenerate while $2P_{3/2}$ lies above them. Radiative corrections, recoil, nuclear size and hyperfine structure require additional terms. The fine-structure constant is ${\alpha }_{\mathrm{fs}}=e^2/(4\pi {ϵ}_0\hslash c)$.