Introduction: Beyond the Bohr Formula
Fine structure is the small splitting of hydrogen energy levels due to relativistic effects and spin-orbit coupling. It lifts degeneracies and gives hydrogen's spectrum its detailed structure.
Three Contributions
- Relativistic kinetic energy: correction to \(T = p^2/2m\)
- Spin-orbit coupling: interaction between \(\vec{L}\) and \(\vec{S}\)
- Darwin term: relativistic correction at the nucleus
The Fine Structure Formula
\[E_{n,j} = -\frac{13.6 \text{ eV}}{n^2}\left[1 + \frac{\alpha^2}{n^2}\left(\frac{n}{j + 1/2} - \frac{3}{4}\right)\right]\]where \(\alpha = e^2/4\pi\epsilon_0\hbar c \approx 1/137\) is the fine structure constant.
Effect on Degeneracy
Now energy depends on \(n\) and \(j\), not just \(n\):
- 2s (\(j = 1/2\)) and 2p (\(j = 1/2\)) are degenerate
- 2p (\(j = 3/2\)) is slightly higher
The Quantum Connection
Fine structure corrections are of order \(\alpha^2 \approx 10^{-4}\), giving splittings of ~0.001 eV. The fine structure constant \(\alpha\) measures the strength of electromagnetic interaction—its value of ~1/137 is one of the fundamental mysteries of physics.