Past exam of the atomic and Optical Physics course of the University of Cambridge / 2013 / Paper 1 / 3 / a

The Lamb shift is a radiative displacement of atomic energy levels beyond the values from the Dirac equation in a Coulomb potential. In hydrogen, the $2S_{1/2}$ and $2P_{1/2}$ levels are degenerate for a point Coulomb potential in the Dirac equation, yet the $2S_{1/2}$ level lies higher by roughly a gigahertz. This is distinct from the fine structure splitting between different $J$ levels and from nuclear-spin hyperfine structure.

Electron self-energy in the quantized electromagnetic field is the dominant contribution to the familiar hydrogen splitting; vacuum polarization contributes too. Renormalization separates the physical electron mass from its divergent free-electron self-energy. A useful leading-log scale for a hydrogenic $nS$ level isnot an exact numerical formula. The enhanced sensitivity of S states follows from their nonzero wavefunction at the nucleus; leading short-distance corrections act like contact perturbations. A precision prediction also includes finite nuclear size and recoil, and must consistently separate those effects from the radiative contribution.^[ref]

The original Lamb-Retherford experiment used a beam of hydrogen atoms, electron-impact excitation into metastable $2S$, and microwave-induced transfer into short-lived $2P$ states. Those atoms decayed before reaching the detector. Metastable atoms surviving to a metal target ejected electrons, so a drop in the measured current signalled a resonance. Varying the microwave excitation and magnetic field established the displacement from the Dirac equation expectation. Microwave quenching converted a small energy difference into a measurable loss of a long-lived population, avoiding optical Doppler widths that obscure the splitting.^[ref]