The atom-field Hamiltonian ordered in powers of e

SI units throughout, with metric (+, −, −, −) wherever a four-vector appears. Quoted from Unit 6: for an electron of charge q = −e, H = [σ⋅(p − qA)]²/2m + qφ. The identity (σ⋅a)(σ⋅b) = a⋅b + iσ⋅(a × b), with p × A + A × p = −iħ∇ × A, turns it into (p − qA)²/2m − (qħ/2m)σ⋅B + qφ, derived, with g = 2 built in. In Coulomb gauge (topic 11.01) qφ is the instantaneous Coulomb attraction V_C, and A is topic 11.03's A(r) = Σₖₛ √(ħ/(2ε₀ωₖV)) [εₖₛaₖₛe^(ik⋅r) + h.c.], taken at the electron's position operator. Expanding the square, p⋅A = A⋅p − iħ∇⋅A = A⋅p, since every εₖₛ is perpendicular to k. Adding H_rad = Σₖₛ ħωₖ(a†ₖₛaₖₛ + ½) gives H = Hₐₜₒₘ + H_rad − (q/m)A⋅p − (qħ/2m)σ⋅B + (q²/2m)A², derived. Dimensions: V s m⁻¹ times C is kg m s⁻¹, so qA is a momentum and every term an energy. Limit: as e → 0 only Hₐₜₒₘ + H_rad remain, with product eigenstates |j⟩|n₁, n₂, …⟩, the basis everything after expands in.

A is linear in a and a†, and so is B = ∇ × A, so both order-e terms change the total photon number by exactly one: they absorb or emit. A² is quadratic, and its products aₖaₖ′, a†ₖa†ₖ′, a†ₖ′aₖ and aₖa†ₖ′ change the number by −2, +2 or 0. A process is therefore classified by its power of e, not by how many times the interaction acts. At order e, one photon is absorbed or emitted. At order e², scattering, two-photon absorption and level shifts each have two routes, A² once or an order-e term twice, and a calculation at that order is wrong unless it keeps both. For the vacuum field, e and ε₀ enter only through α = e²/(4πε₀ħc) = 1/137.036, so vacuum effects are ordered in α. Measured check: hydrogen's 2p level, whose decay rate is the square of an order-e amplitude, lives 1.60 ns, so its rate is only Γ = 4.0 × 10⁻⁸ ω for Lyman-α: effects of order α are small fractions of the energies they modify.

Take an atom in a classical field E₀ cos ωt along z, so A₀ = E₀/ω. At the field's peak the order-e orbital term acting on a state has size (e/m)A₀√⟨p_z²⟩; for hydrogen ns the virial theorem gives ⟨p²⟩ = ħ²/(n²a₀²), so this is (e/m)A₀ħ/(√3na₀). The order-e² term is (e²/2m)A₀², and the ratio eA₀/(2√⟨p_z²⟩) grows as E₀λ: the strength of the applied field, not the smallness of e, decides whether a driven atom can be expanded. The spin term is also of order e, but B₀ = kA₀, so it trails A⋅p by (ħk/2)/√⟨p_z²⟩ = (√3/2)kna₀, an atom-size-over-wavelength factor, the same one that weakens magnetic-dipole and electric-quadrupole lines. For Lyman-α, ka₀ = 2.7 × 10⁻³. The spin term wins only where A⋅p is forbidden outright: hydrogen's 1420 MHz line is a pure spin flip, driven by σ⋅B alone, with a rate of 2.9 × 10⁻¹⁵ s⁻¹ (quoted).

When ka ≪ 1 the phase e^(ik⋅r) is 1 across the atom and A(r) can be taken at the nucleus. Then (q²/2m)A(0)² contains no atomic operator: its matrix elements are δⱼⱼ′⟨F′|(q²/2m)A(0)²|F⟩, so at first order it shifts every level j by the same amount and moves no transition frequency, derived. In a classical field that shift is Uₚ = q²E₀²/(4mω²), 0.0106 eV at 1.0 × 10¹⁵ W m⁻² and 1064 nm. Order e² does not stop there. Second-order A⋅p shifts 1s by −Uₚ − ¼α_d(ω)E₀²: writing ⟨n|p|1s⟩ = imωₙ₁⟨n|r|1s⟩, the Thomas–Reiche–Kuhn sum rule (quoted) splits off the −Uₚ exactly. So 1s moves by only −¼α_d(ω)E₀² ≈ −9 × 10⁻⁵ eV, with polarisability α_d ≈ 4.5 × 4πε₀a₀³. A free electron, whose p commutes with A(0), has no second-order A⋅p shift and keeps all of Uₚ. Rydberg levels, nearly free, rise with the threshold, which short-pulse above-threshold ionisation observes (Freeman et al., 1987).

Beyond that uniform shift, A² leads two familiar effects. Its a†ₖ′ₛ′aₖₛ part moves a photon from mode ks to k′s′ with amplitude (2πrₑħc²/(V√(ωω′)))ε*ₖ′ₛ′⋅εₖₛ whatever the electron's state, derived: Thomson scattering, which Unit 12 turns into σ_T = (8π/3)rₑ² = 6.652 × 10⁻²⁹ m². For a free electron it is the whole answer, since in the dipole approximation A⋅p cannot change a free electron's state, and it dominates for any electron once ħω far exceeds the binding energy. Its 2a†a part retunes every mode, raising ω by e²/(2mε₀ωV) for one electron in volume V; diagonalised exactly together with the a² and a†² parts, that becomes e²/(mε₀V) added to ω², and a density nₑ gives ω² = c²k² + ωₚ², the dispersion relation of a plasma. For nₑ = 1.0 × 10¹² m⁻³, fₚ = ωₚ/2π = 8.98 MHz, and the ionosphere's reflection of short-wave radio below such critical frequencies is this term measured. Both results are non-relativistic: Klein–Nishina corrections to Thomson scattering (quoted) have relative size 2ħω/mc², already 3% for 8.05 keV copper Kα X-rays.

Let one mode start in topic 11.07's coherent state |α⟩ = D(α)|0⟩, with D(β) = exp(βa† − β*a) and D†aD = a + β. Write |Ψ(t)⟩ = D(β(t))|Φ(t)⟩ with β(t) = αe^(−iωt), the classical motion. Then iħ∂ₜ|Φ⟩ = [D†HD − iħD†∂ₜD]|Φ⟩. In D†HD every a becomes a + β, so A becomes A + A_cl, the classical wave the coherent state carries, and −iħD†∂ₜD = −ħω(βa† + β*a + |β|²) cancels exactly the linear and constant pieces that a + β adds to ħωa†a. What remains is H_sc(t) + H_rad + V(t): Quantum Mechanics II's semiclassical Hamiltonian H_sc = (p − qA_cl)²/2m + V_C − (qħ/2m)σ⋅B_cl, plus the quantised field and its coupling V, which includes (q²/m)A_cl⋅A. And |Φ(0)⟩ = |ψ⟩|0⟩: the quantised field starts in its vacuum. Nothing was approximated, so semiclassical theory is exact except for V acting on an empty field, the source of spontaneous emission (Unit 12), the Lamb shift (Unit 13) and topic 11.09's photon-number effects.