= Solution
In the ideal point-nucleus Dirac-Coulomb problem, hydrogen levels depend on $n$ and $j$, so $2S_{1/2}$ and $2P_{1/2}$ are degenerate despite their different orbital angular momenta. The <Lamb shift> removes this degeneracy: the $2S_{1/2}$ level lies about \b[$h\times1.06\,\mathrm{GHz}$ above $2P_{1/2}$], an <energy> of roughly $4.4\,\mathrm{\mu eV}$. This is distinct from the much larger $2P_{3/2}-2P_{1/2}$ fine-structure splitting.
<quantum electrodynamics> changes the electron's interaction with the Coulomb field. <electron self-energy> and <vacuum polarization> give state-dependent <radiative corrections>; a complete prediction also keeps specified <recoil> and nuclear-size contributions. An $S$ wave samples the nucleus differently from a $P$ wave, so equal $n,j$ no longer ensures equal <energy>. A leading-log estimate for an $nS$ radiative contribution scales as $\alpha(Z\alpha)^4m_ec^2\ln[(Z\alpha)^{-2}]/n^3$, with a numerical prefactor and further state-dependent terms. This is a scale estimate, not a replacement for renormalized bound-state QED. The measurable shift is a difference relative to a specified reference <Hamiltonian>, rather than an unregulated absolute vacuum <energy>.<Theory of the Lamb shift in hydrogen and light hydrogen-like ions by Vladimir Yerokhin and others>[§II, definition of level contributions relative to the point-nucleus Dirac value and bound-state QED expansion]{ref}
The original microwave experiment used a tungsten oven to dissociate hydrogen, an electron beam to populate metastable $2S$ atoms, and electron ejection from a metal target to measure surviving metastable atoms. Microwaves transferred population to rapidly decaying $2P$, reducing detector current. At a fixed microwave <frequency>, interruption of the radiation and a scan of static <magnetic field> revealed Zeeman-tuned resonances. Comparing resonance locations with the Dirac-level prediction showed the $S$ level shifted upward. Field calibration and homogeneity, excessive microwave intensity, unresolved <hyperfine structure> and stray-electric-field quenching limit the measurement; the signal is not direct absorption by a large equilibrium excited population.<Fine Structure of the Hydrogen Atom by a Microwave Method by Willis Eugene Lamb Jr. and Robert Curtis Retherford>[pp. 241–243, apparatus, Fig. 1 caption and resonance-field comparison]{ref}
\Image[/past-exam-of-the-atomic-and-optical-physics-course-of-the-university-of-cambridge/2015/paper-1-lamb-two-photon.png]
{title=Original schematics of metastable-beam microwave quenching and Doppler-free two-photon excitation}
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