= Solution
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 is
$$
\Delta E_{nS}\sim\frac{\alpha(Z\alpha)^4}{\pi n^3}m_ec^2
\ln[(Z\alpha)^{-2}],
$$
not 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.<Theory of the Lamb shift in hydrogen and light hydrogen-like ions by Vladimir Yerokhin and others>[§III A, equations (7)–(8), one-loop self-energy and vacuum-polarization scale; introduction, separation of radiative, recoil and nuclear effects]{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. \b[Microwave quenching converted a small <energy> difference into a measurable loss of a long-lived population], avoiding optical Doppler widths that obscure the splitting.<Fine Structure of the Hydrogen Atom by a Microwave Method by Willis Eugene Lamb Jr. and Robert Curtis Retherford>[1947, pp. 241–243, metastable beam, target-current readout and microwave resonance]{ref}
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