The divergence of the bound self-energy

SI units throughout, with energies in eV and shifts quoted as frequencies δE/h; the electron stays non-relativistic, so the course metric (+, −, −, −) never enters. Topic 13.01 derived δEₙ = (2α/(3πm²c²)) ∫₀^K dE E Σₘ |pₘₙ|²/(Eₙ − Eₘ − E), quoted here, with E the photon energy. Follow one term. For E ≪ Δₘₙ = Eₘ − Eₙ it is −E|pₘₙ|²/Δₘₙ, vanishing with E: soft photons cost nothing. For E ≫ Δₘₙ it tends to −|pₘₙ|², a constant, derived. The reason is counting. The mode density grows as ω² and each mode's squared amplitude ħ/(2ε₀ωV) falls as 1/ω, so the numerator grows as E, while the energy denominator grows as E too. Each further keV of photon energy then adds the same amount, and a constant integrated to K grows like K. For 2s→3p, Δ = 1.890 eV, and the term is within 1 per cent of its plateau from E = 99Δ = 187 eV on, 1/2700 of the way to mc².

The algebra is one identity, derived: E/(Eₙ − Eₘ − E) = −1 + Δₘₙ/(E + Δₘₙ). It is not an expansion; it holds at every E. The first piece is blind to the spectrum. The second falls as Δₘₙ/E once E passes the atomic interval, and it carries everything the atom knows. Integrated from 0 to K they give −K and Δₘₙ ln((K + Δₘₙ)/Δₘₙ), so for K ≫ Δₘₙ each term is |pₘₙ|² (−K + Δₘₙ ln(K/Δₘₙ) + Δₘₙ²/K + …). For the 2s→3p term at K = mc² = 511.0 keV the second piece is 1.890 eV × 12.51 = 23.6 eV against 5.110 × 10⁵ eV, a ratio of 4.6 × 10⁻⁵. Double K and the first piece doubles, while the second gains only Δ ln 2 = 1.31 eV. Dimensions check term by term: K, Δ and Δ ln(K/Δ) are all energies, and |pₘₙ|²/(m²c²) is a pure number. Topic 13.04's subtraction will rest on this same split.

Summed over m, the −1 pieces give −K Σₘ |pₘₙ|², and closure, Σₘ |m⟩⟨m| = 1 over the complete set, turns that sum into ⟨n|p²|n⟩. So the leading part is −(2α/(3πm²c²)) K ⟨n|p²|n⟩ = −(4αK/(3πmc²)) ⟨T⟩ₙ, derived. Dimensions: α and K/mc² are pure numbers and ⟨T⟩ₙ is an energy. Limits: it vanishes as K → 0, and in the free-field limit α → 0, where it is first order: one emission, one reabsorption. Closure needs every state, continuum included. For 2s the bound np states supply only 31.4 per cent of ⟨p²⟩, computed from hydrogen's dipole matrix elements, and the ionisation continuum the other 68.6, so a sum over bound states alone would miss the coefficient by a factor of 3.2. The virial theorem, quoted from Quantum Mechanics I, gives ⟨T⟩ₙ = −Eₙ = 13.606 eV/n², so for hydrogen the linear part is −(2α³/(3πn²)) K: at K = mc², −42.1 meV (−10.19 THz) for 1s and −10.5 meV (−2.547 THz) for 2s. The A² term of topic 13.01 adds a constant common to all levels; this term is not common, because ⟨T⟩ₙ is not.

The remainder is Rₙ(K) = (2α/(3πm²c²)) Σₘ |pₘₙ|² Δₘₙ ln((K + Δₘₙ)/|Δₘₙ|), derived. Once K lies far above the intervals that carry the weight, each logarithm becomes ln K − ln|Δₘₙ|, so Rₙ grows as ln K times one coefficient, Σₘ |pₘₙ|² Δₘₙ, plus a finite part. That coefficient is finite; topic 13.05 evaluates it with a commutator sum rule and finds it nonzero only for s states. Its sign is already visible for 2s. The 2p states are degenerate with it, so their pₘₙ = i m ωₘₙ rₘₙ vanishes, and 1s is not dipole-connected to it; every state with weight lies above 2s, every Δₘₙ is positive, and every term of R₂ₛ raises the level, the direction Lamb and Retherford found. A state below, as 1s is for 2p, puts a pole inside the integration range that topic 13.04 treats as a principal value. Now compare growth: raising K from mc² to 2mc² lowers 2s by another 2.547 THz through the linear part, while the logarithm adds ln 2 times its coefficient, 94 MHz with topic 13.05's value, quoted.

The linear part depends on the state through ⟨T⟩ₙ alone, and that has consequences you can check. In non-relativistic hydrogen ⟨T⟩ = 3.401 eV for 2s and 2p alike, so both are lowered by the same 2.547 THz at K = mc², and the term cannot split them: Lamb's 2S₁/₂–2P₁/₂ interval does not see it. Between shells it does not cancel. It widens the 1S–2S interval by (2α³/(3π))(1 − ¼)K = (α³/(2π))K, which is 7.64 THz at K = mc², 0.31 per cent of the 2 466 061 413 187 kHz measured by two-photon spectroscopy to about 10 Hz, and about a thousand times the 7.1 GHz by which the measured 1s and 2s Lamb shifts change it. A correction proportional to ⟨p²/2m⟩ is also exactly what a small change in the electron's mass would make in its kinetic energy. That clue is followed in topic 13.03, which puts a free electron through the same integral.

Two things stand between this result and a measured number. First, every interval still depends on K: 1S–2S linearly, and 2S–2P through the ln K in R₂ₛ, which 2p does not share. Nothing in the non-relativistic theory fixes K. Second, the photons that build the linear part lie outside the theory that computed it. Set the energies on a logarithmic ladder: 3.40 eV binds 2s; αmc² = ħc/a₀ = 3.73 keV is where a photon's reduced wavelength shrinks to the Bohr radius and e^(ik⋅r) ≈ 1 fails; and near mc² = 511 keV the electron recoils relativistically and pairs appear. With K = mc², photons above αmc² supply 1 − α = 99.3 per cent of the linear part. The dominant term is set entirely by physics the dipole Hamiltonian cannot describe, so it must be interpreted before it can be evaluated. The comparison with 1057.8 MHz waits for topic 13.07.