The Weisskopf-Wigner ansatz and its exact equations

SI units throughout; no four-vector appears, and where the course writes one its metric is (+, −, −, −). From topic 11.08, quoted, build a model: two atomic levels |g⟩ and |e⟩ split by ħω₀, the dipole approximation, and the coupling −(q/m)A⋅p, with A² dropped, since in the dipole approximation it does not act on the atom. With real polarisation vectors the coupling is ħΣₖ(gₖσ₊ + gₖ*σ₋)(aₖ + aₖ†), where σ₊ = |e⟩⟨g|, σ₋ = σ₊† and k labels a wavevector and a polarisation. Two of its four products, σ₊aₖ and σ₋aₖ†, trade one photon for one atomic excitation. The other two, σ₊aₖ† and σ₋aₖ, raise or lower both at once. The rotating-wave approximation drops those, and then N̂ = σ₊σ₋ + Σₖaₖ†aₖ commutes with H, derived: each kept term moves one quantum across, each dropped one changes N̂ by ±2. In the free-field limit, every gₖ = 0, N̂ is conserved trivially; the content is that the coupling keeps it. Everything below is exact for this model until topic 12.06 takes the Markov step.

The excited atom in the vacuum, |e, 0⟩, has N̂ = 1, so it can reach only itself and the one-photon states |g, 1ₖ⟩. Hence |ψ(t)⟩ = cₑ(t)e^(−iω₀t)|e, 0⟩ + Σₖ cₖ(t)e^(−iωₖt)|g, 1ₖ⟩ is not a trial function but the general state; the phases remove the free evolution, with energies counted from |g, 0⟩. Put it into iħ∂|ψ⟩/∂t = H|ψ⟩ and project on ⟨e, 0| and ⟨g, 1ₖ|: dcₑ/dt = −iΣₖ gₖcₖe^{(i(ω₀−ωₖ)t)} and dcₖ/dt = −igₖ*cₑe^{(−i(ω₀−ωₖ)t)}, derived, with cₑ(0) = 1 and every cₖ(0) = 0. Checks: gₖ is in s⁻¹, as both sides need. In d/dt(|cₑ|² + Σₖ|cₖ|²) the photon terms are minus the complex conjugates of the atomic ones, so the real part vanishes and probability is conserved. With every gₖ = 0, cₑ stays 1. The price is the number of equations, one per mode: a cubic metre holds about 5 × 10²¹ modes below the Lyman-α frequency alone.

Each cₖ is driven by cₑ alone, so integrate it: cₖ(t) = −igₖ*∫₀ᵗ cₑ(t′)e^{(−i(ω₀−ωₖ)t′)} dt′, still exact. Substituting folds the whole set into one equation, dcₑ/dt = −∫₀ᵗ K(t − t′)cₑ(t′) dt′, with K(τ) = Σₖ|gₖ|²e^{(i(ω₀−ωₖ)τ)}, derived. Reabsorption is included: the right side is the photon amplitude emitted at t′ acting back on the atom. K is in s⁻², so the right side is in s⁻¹. Two limits check it. One resonant mode alone gives K = g², so d²cₑ/dt² = −g²cₑ and cₑ = cos(gt): topic 11.09's vacuum Rabi oscillation at 2g, observed for a Rydberg atom in a microwave cavity near 2g/2π = 50 kHz (Brune et al., 1996, quoted). For any set of modes, differentiating at t = 0 gives d²cₑ/dt² = −K(0), so Pₑ = |cₑ|² = 1 − K(0)t² + O(t⁴), the t³ term vanishing because dK/dτ at τ = 0 is imaginary: Quantum Mechanics II's quadratic start, now exact, with K(0) = Σₖ|gₖ|².

Write K(τ) = ∫₀^∞ J(ω)e^{(i(ω₀−ω)τ)} dω, with J(ω) = Σₖ|gₖ|²δ(ω − ωₖ), the coupling-weighted density of modes. The coupling −(q/m)A⋅p with p_eg = imω₀r_eg gives |gₖ|² = ω₀²|êₖ⋅d|²/(2ε₀ħωₖV), d = q⟨e|r|g⟩. Topic 12.02's sum, Vω²/(8π³c³) modes per unit solid angle and angular frequency for each polarisation, over both polarisations and all directions, gives J(ω) = ω₀²ω|d|²/(6π²ε₀ħc³) = (A/2π)(ω/ω₀), derived. Units: ε₀ħc³ is C² m² s⁻², so J is in s⁻¹. V has cancelled, and J(ω₀) = A/2π = 9.98 × 10⁷ s⁻¹ for 2p. Off resonance the two-level model is gauge-sensitive: the length form −d⋅E gives |gₖ|² ∝ ωₖ and J ∝ ω³, equal at ω₀, where the rate is set, and different elsewhere, where the shift is. The integral replaces the box sum only while no photon has come back from a wall: light covers c/A = 0.48 m in one 2p lifetime, so in a smaller mirrored box the kernel would show echoes.

Model the dipole approximation's failure near ω_c = c/a₀ by a factor e^(−ω/ω_c) on J, which lowers J(ω₀) by only 0.3%. Then K(τ) = (A/(2πω₀))e^(iω₀τ)/(ω_c⁻¹ + iτ)², derived, with |K| = (A/(2πω₀))/(ω_c⁻² + τ²). It has two parts. A spike of width 1/ω_c = 1.8 × 10⁻¹⁹ s and height K(0) = Aω_c²/(2πω₀) = 2.1 × 10²⁹ s⁻² sets the quadratic start; to leading order its integral is −iAω_c/(2πω₀), a phase rate linear in the cutoff and no decay at all, the divergent shift that topic 12.06 meets and Unit 13 treats. Then a tail falling as τ⁻² and rotating at ω₀, which cancels itself against a slowly varying cₑ: integrating by parts, lags beyond T add a fraction (ω₀T)⁻²/π of the A/2 the decay needs, 0.3% at T = 10/ω₀. So the past acts on cₑ over about 1/ω₀ = 6.45 × 10⁻¹⁷ s, while cₑ changes over 1/A = 1.595 ns: a ratio A/ω₀ = 4.0 × 10⁻⁸. Topic 12.06 uses that small number.

Restore σ₊aₖ† and σ₋aₖ and the sector leaks: |g, 1ₖ⟩ is driven to |e, 1ₖ1ₖ′⟩ with N̂ = 3, and |g, 0⟩ itself mixes with every |e, 1ₖ⟩, so the true ground state carries virtual photons. Since ħω₀ + ħωₖ is never zero, the dropped terms cannot conserve energy and add nothing to the rate at order e². They do move the energies. By second-order perturbation theory, used as Quantum Mechanics II derived it, the shift of |e, 0⟩ in the two-level model comes only from kept terms, Σₖ|gₖ|²/(ω₀ − ωₖ) in angular frequency, and that of |g, 0⟩ only from dropped ones, −Σₖ|gₖ|²/(ω₀ + ωₖ). Their difference carries 1/(ω₀ − ω) + 1/(ω₀ + ω) = 2ω₀/(ω₀² − ω²), falling as ω⁻² where each term falls as ω⁻¹, so the counter-rotating terms turn a transition shift linear in the cutoff into one logarithmic in it. Dropping them keeps the decay right and gets the shift wrong. They return with A², which changes photon number by zero or two, with the atom's other levels and with the cutoff, in Unit 13, where Bethe's calculation sums over every level.