The 2s shift computed: about 1040 MHz

Units are SI, with α = e²/(4πε₀ħc) = 1/137.036; everything happens in the atom's rest frame, so the course metric (+, −, −, −) never enters. The previous topics reduced the mass-subtracted self-energy of an s level to ΔEₙ = (4α⁵mc²/(3πn³)) ln(K/K₀), derived there and used here. For n = 2 the numerical factor is 4/(3π × 8) = 1/(6π). With α⁵ = 2.069 × 10⁻¹¹ and mc² = 510 999 eV, α⁵mc² = 1.057 × 10⁻⁵ eV; dividing by 6π gives 5.61 × 10⁻⁷ eV, and dividing by h = 4.136 × 10⁻¹⁵ eV s gives 135.6 MHz; from here on a shift quoted in MHz means ΔE/h, the frequency Lamb measured. The dimensions check at once: α, n and a logarithm are pure numbers, so ΔE has the dimensions of mc². What the derivation leaves open are the two energies inside the logarithm. K₀ is a property of the atom, the sum-rule-weighted logarithmic mean of hydrogen's excitation energies, computed in the previous topic. K is a property of the theory: the photon energy above which the non-relativistic calculation stops being believable. The 2s shift is 135.6 MHz times the number of factors of e between them.

Lay the energies on one logarithmic ladder: the 2s binding, 3.40 eV; K₀ = 16.64 Ry = 226 eV, 67 times higher; αmc² = 3.73 keV, where a photon's wavelength equals 2πa₀; and mc² = 511 keV. A photon of energy ħω carries momentum ħω/c, and Schrödinger's electron absorbing it recoils with (ħω)²/(2mc²). That is small beside ħω only while ħω ≪ 2mc². Near mc² the recoil is relativistic, so the non-relativistic kinetic energy is wrong, and from 2mc² the intermediate states include electron-positron pairs the theory has no room for. So mc² is where the model must be cut off: this is Bethe's argument, not a derivation. Photons above αmc² have wavelengths below 2πa₀, shorter than the atom, yet they supply ln 137 = 4.92 of the 7.72, 64 per cent of the logarithm; the defence is that after the subtraction they reach the electron only through the contact term at the nucleus, a region small beside their wavelength until ħω nears mc². The covariant calculation quoted in the unit's closing topic is equivalent to his formula with K ≈ 0.79 mc² before the anomalous moment is added, inside a factor of two.

Since Ry = α²mc²/2, the cutoff in rydbergs is mc²/Ry = 2/α² = 37 558. Bethe's own K₀ = 17.8 Ry, the estimate he made in 1947 from approximate bound and continuum sums and quoted here, gives mc²/K₀ = 2110 and ln 2110 = 7.654; then ΔE(2s) = 135.6 MHz × 7.654 = 1038 MHz, his result of about 1040 MHz. The modern K₀ = 16.64 Ry, from ln(K₀/Ry) = 2.8118 as computed in the previous topic, gives 2257, a logarithm of 7.72 and 1047 MHz. The measured 2S₁/₂–2P₁/₂ interval is 1057.8 MHz; Lamb and Retherford's first figure in 1947 was about 1000 MHz. Bethe sits 1.8 per cent low and the modern evaluation 1.0 per cent low. The logarithm is forgiving: K₀ moved by 7 per cent between the two evaluations, yet the shift moved by only 9 MHz, 0.9 per cent. In energy, 1047 MHz is 4.33 μeV, about 1.3 × 10⁻⁶ of the 2s binding energy, and its wavelength c/ν = 28.6 cm puts it in the microwave band, which is why the measurement needed microwave techniques and not an optical spectrometer.

Both factors of the result are positive. The logarithm is positive because K ≫ K₀, and the sum rule turned its coefficient into ⟨∇²V⟩ = (e²/ε₀)|ψ(0)|², positive for an attractive Coulomb potential, so the level rises. The picture behind the sign: the vacuum field jiggles the electron over a small region, and averaging −e²/(4πε₀r) over that region near the nucleus makes it shallower, so an electron found at the nucleus is bound less. Only s electrons are found there. A 2p wavefunction vanishes at r = 0, so its coefficient of ln K is zero and at the order of the logarithm it does not move; full QED (quoted) moves 2P₁/₂ by about −13 MHz, 1 per cent of the interval, through terms with no ln K. The Dirac degeneracy of 2S₁/₂ and 2P₁/₂, exact to all orders in Zα in Unit 7, is lifted with 2S₁/₂ on top, the order Lamb and Retherford observed. The limits are right too: the shift falls as n⁻³ with |ψ(0)|², so high Rydberg s states barely move, and it vanishes as α → 0, the free-field limit, faster than the fine structure by a factor of order α ln(1/α²).

Nothing in the argument picks mc² over mc²/2 or 2mc². Each factor of two moves the shift by 135.6 MHz × ln 2 = 94 MHz, so the honest statement is 1047 ± 94 MHz, the range 953 to 1141 MHz, a 9 per cent band. The measured 1057.8 MHz lies 10 MHz from the centre, about a ninth of the half-width. Solving for the cutoff that fits exactly gives K = 1.08 mc² with the modern K₀, but any K from about 0.55 to 2.15 mc² keeps the measurement within the band, so that fit is a choice made by the data, not a prediction. The remaining per cent is real physics this theory omits, quoted here and accounted for in the unit's closing topic: the anomalous moment raises 2s by 51 MHz, vacuum polarisation lowers it by 27 MHz, the covariant treatment of the hardest photons removes 32 MHz, and 2P₁/₂ moving down widens the interval by 13 MHz. Each is 1 to 5 per cent of the total; they net to about +5 MHz, and smaller higher-order terms supply the rest. Bethe's near-agreement is partly a cancellation among pieces he did not have.