Bethe's logarithm: the mean excitation energy

Units are SI, with energies also given in eV and in rydbergs, Ry = 13.606 eV; no four-vector appears here, and where one does in this course the metric is (+, −, −, −). Topic 13.04 left the renormalised shift of level n, for K far above the atomic energies, as (2α/(3πm²c²)) Σₘ wₘ ln(K/|Δₘₙ|), with wₘ = |pₘₙ|²Δₘₙ, and topic 13.05 showed Σₘ wₘ = (ħ²/2)⟨∇²V⟩ₙ. Both are quoted. Now split each logarithm: ln(K/|Δₘₙ|) = ln(K/Ry) − ln(|Δₘₙ|/Ry). The first piece is the same for every m and multiplies Sₙ = Σₘ wₘ; the second is a weighted sum with no K in it. Calling that sum Sₙ ln(K₀/Ry) makes the result exact: δEₙ = (2α/(3πm²c²)) Sₙ ln(K/K₀). Dimensions check: Sₙ/(m²c²) is an energy and the logarithm a pure number. In the one-state limit, where a single m carries all the weight, K₀ is simply that state's excitation energy. For hydrogen's s states, |ψₙ(0)|² = 1/(πn³a₀³) turns the result into (4α⁵mc²/(3πn³)) ln(K/K₀), derived: h × 135.6 MHz × ln(K/K₀) for 2s.

The same dipole matrix elements can be averaged three ways, and each average is pinned by its own sum rule. Weight by fₘₙ alone and the Thomas-Reiche-Kuhn rule makes the weights add to 1. Weight by |pₘₙ|² = (3/2)m fₘₙΔₘₙ and closure makes them add to ⟨p²⟩, the coefficient of topic 13.02's linear divergence. Weight by |pₘₙ|²Δₘₙ, Bethe's choice, and topic 13.05's rule makes them add to (ħ²/2)⟨∇²V⟩. Each extra power of Δ drags the weighted mean upward. For hydrogen's 2s, computed here, the oscillator-strength mean is 3.20 eV, close to the 3.40 eV binding energy; the |p|² mean is 7.63 eV; Bethe's mean is 226 eV. The construction with fₘₙ alone is Bethe's 1930 mean excitation energy for stopping power, 15.0 eV for ground-state hydrogen when computed the same way, against 269 eV for the 1s Lamb-shift K₀: the same name and the same sum, but different weights and an energy eighteen times smaller. Only the Δ² weighting belongs in the self-energy.

Take 2s → 3p, the strongest line, with fₘₙ = 0.435 at Δ = 1.89 eV. It holds 43 per cent of the oscillator strength but only 1.26 per cent of Bethe's weight, and all the bound np states together hold 64.9 per cent of fₘₙ but 2.64 per cent of the weight. The reason lies at the nucleus. An s state has a cusp there, ψ ≈ ψ(0)(1 − r/a₀), and a cusp makes the momentum-space wavefunction fall only as k⁻⁴. For Δ ≫ Ry the continuum state is nearly a plane wave of wavenumber k, and the cusp makes its overlap with ψₙ tend to 8πψ(0)/(a₀k⁴). So |pₘₙ|² ∝ k² × k⁻⁸, Δ ∝ k², and the states between k and k + dk number ∝ k² dk: the weight per unit Δ falls as Δ⁻³⁄². Integrated above E and divided by Sₙ = (ħ²e²/(2ε₀))|ψ(0)|², the factors |ψ(0)|² cancel, and the fraction above E tends to (8/π)√(Ry/E) for every s state, derived: 1.31 per cent above mc². Lower down the computed spectrum puts 25 per cent of the weight above 1 keV and 57 per cent above 100 eV. A tail like this drags the log-mean deep into the continuum.

This is the unit's computational exercise, and it needs two kinds of state. The bound np states have closed-form radial integrals, R²(np, 2s) for 2s as quoted in topic 13.02; they are computed to n = 300 and continued as n⁻³ beyond. The continuum εp states are energy-normalised Coulomb functions, started from their power series at the origin and integrated outward by Numerov's method, then folded into the radial overlap ∫ R(εp) R(2s) r³ dr on a grid of electron wavenumbers up to k = 200/a₀, about 544 keV, beyond which the Δ⁻³⁄² tail is added analytically. Two checks decide whether the spectrum is complete before any logarithm is taken: Σₘ fₘₙ = 1, met to 4 × 10⁻⁵, and Σₘ fₘₙΔₘₙ² = (2/3)Sₙ/m, which for 2s is Eₕ²/6 with Eₕ = 27.211 eV, met to 2 × 10⁻⁶. Only then is ln(K₀/Ry) formed: 2.8118 for 2s and 2.9841 for 1s, against the quoted 2.81177 and 2.98413.

K₀ enters only through ln(K/K₀), and with K = mc² that logarithm is large: ln(511.0 keV/226.4 eV) = 7.722 for 2s. A factor of two in K₀ moves it by ln 2 = 0.693, nine per cent, or 94 MHz at 135.6 MHz per unit of logarithm. Bethe's 1947 estimate, K₀ = 17.8 Ry, was 7 per cent high, so with K = mc² his logarithm is 7.654 and the 2s shift about 1040 MHz, the value he reported, against 1047 MHz from the modern value. The cutoff is known no better, since a factor of two in K moves the result by the same 94 MHz. With the modern K₀ the 1s shift comes out at 8.19 GHz, and the two set beside the measured 1057.8 MHz 2S₁/₂–2P₁/₂ interval and 8.17 GHz 1s shift agree within one per cent, closer than the cutoff's uncertainty warrants, which topic 13.07 examines. What the logarithm cannot forgive is the wrong scale: put K₀ at the 3.40 eV binding energy and ln(K/K₀) becomes 11.92, 54 per cent too large.

Three limits bound the construction. First, Sₙ ln(K/K₀) stands in for Σₘ wₘ ln((K + Δₘₙ)/Δₘₙ), which needs K ≫ Δₘₙ wherever the weight lies. With 1.31 per cent of it above mc², keeping the exact logarithm raises the 2s value from 7.722 to 7.762, half a per cent, computed; and those states have relativistic electrons that the non-relativistic spectrum misdescribes, which topic 13.09's covariant theory repairs. Second, for a p state the weights add to zero, Sₙ = 0, so no mean exists: the shift becomes −(2α/(3πm²c²)) Σₘ wₘ ln(|Δₘₙ|/Ry), with no K in it at all, and the quoted ln k₀(2p) = −0.0300, defined with the 2s value of Sₙ as its scale, makes it about +4 MHz. Third, for a hydrogen-like ion every Δ scales as Z², so the 2s K₀ = Z² × 226 eV reaches mc² near Z = 48, where the logarithm has nothing left to measure.