Exponential decay and the radiative level shift derived

Topic 12.05 left one exact equation, quoted here: dcₑ/dt = −∫₀ᵗ K(t − t′) cₑ(t′) dt′, with kernel K(τ) = Σₖ |gₖ|² e^{(i(ω₀ − ωₖ)τ)}. Units are SI with ħ kept explicit; nothing here is relativistic, so no metric signature enters. The kernel is a sum of phasors that start in step at τ = 0 and fall out of step once ωₖ has spread over a range of order ω₀, so K is negligible after τ_c ≈ 1/ω₀ = 6.5 × 10⁻¹⁷ s for Lyman-α. Over that interval cₑ changes by a fraction Γτ_c/2 ≈ 2 × 10⁻⁸, because the 2p lifetime is 1.6 ns. Two moves follow, and together they are the Markov step. First, replace cₑ(t′) by cₑ(t) under the integral, at a cost of order Γτ_c. Second, with τ = t − t′, extend the upper limit from t to ∞; for t ≫ τ_c this adds only delays at which K has already died. The equation becomes dcₑ/dt = −cₑ(t) ∫₀^∞ K(τ) dτ, local in time: the atom's rate of change now depends on its present state alone.

With the kernel inserted, the mode sum and the time integral separate: ∫₀^∞ K(τ) dτ = Σₖ |gₖ|² ∫₀^∞ e^{(i(ω₀ − ωₖ)τ)} dτ. The time integral does not converge as written, so damp it with e^(−ετ) and let ε → 0⁺ at the end: ∫₀^∞ e^{(ixτ − ετ)} dτ = i/(x + iε) = ε/(x² + ε²) + i x/(x² + ε²). The real part is a Lorentzian of height 1/ε, width ε and area π, so it tends to π δ(x). The imaginary part equals 1/x except within about ε of x = 0, where it passes smoothly through zero; integrated against a smooth function it gives the principal value P(1/x). Hence ∫₀^∞ e^(ixτ) dτ = π δ(x) + i P(1/x), derived. The regulator has a physical reading: ε stands in for 1/t, the inverse of the time the integral has actually run. At finite t the real part is sin(xt)/x, half the time-derivative of QM-II's golden-rule peak sin²(xt/2)/(x/2)², and the Markov step is the statement that this peak has become narrower than any structure in the mode sum.

Collecting terms gives dcₑ/dt = −(Γ/2 + iΔ)cₑ, with Γ/2 = π Σₖ |gₖ|² δ(ω₀ − ωₖ) and Δ = P Σₖ |gₖ|²/(ω₀ − ωₖ). The rate Γ = 2π Σₖ |gₖ|² δ(ω₀ − ωₖ) is, term for term, topic 12.02's golden-rule sum over empty modes, polarisations and directions, so Γ = A = ω₀³|d|²/(3πε₀ħc³), quoted from there: 6.26 × 10⁸ s⁻¹ for hydrogen 2p. The solution is cₑ(t) = e^{(−Γt/2 − iΔt)}, so the population is e^(−Γt), with lifetime 1/Γ = 1.596 ns against a measured 1.60 ns. Four checks. Free field: with every gₖ → 0, Γ and Δ vanish and |e, 0⟩ is stationary. Dimensions: |gₖ|² is in s⁻² and a δ function of frequency in s, so Γ is in s⁻¹. Early times: e^(−Γt) ≈ 1 − Γt is QM-II's first-order golden rule, which this result carries past Γt ≈ 1 without ever letting the population go negative. One mode: replace the continuum by a single mode and K(τ) = |g|² e^{(i(ω₀ − ω₁)τ)} never dies, so there is no δ function and no decay, and the exact solution is Unit 11's vacuum Rabi oscillation.

The other half, δE = ħΔ = ħ P Σₖ |gₖ|²/(ω₀ − ωₖ), shifts the level: the amplitude of |e, 0⟩ now turns as e^{(−i(Eₑ + δE)t/ħ)}. Write ħgₖ = ⟨e, 0|H₁|g, 1ₖ⟩ and the shift becomes Σₖ |⟨g, 1ₖ|H₁|e, 0⟩|²/(Eₑ − E_g − ħωₖ), the second-order energy shift through the intermediate states |g, 1ₖ⟩. That is the result QM-II quoted, now derived, with one thing QM-II's formula cannot supply. The states |g, 1ₖ⟩ form a continuum that contains the energy Eₑ itself, so one denominator vanishes. The principal value says to omit that mode, and the δ function says where its share went: into the width. Together they make the energy complex, Eₑ + δE − iħΓ/2, where δE − iħΓ/2 = Σₖ |⟨g, 1ₖ|H₁|e, 0⟩|²/(Eₑ − E_g − ħωₖ + i0⁺). Signs can be read off directly. A mode above ω₀ makes its denominator negative and pushes the level down; a mode below pushes it up, as happens for any pair of coupled levels.

Topic 12.02's mode sum done at a general frequency, not just on resonance, gives the spectral density J(ω) = Σₖ |gₖ|² δ(ω − ωₖ). The mode density contributes ω², and the A⋅p vertex, with Unit 11's amplitude √(ħ/(2ε₀ωV)), contributes 1/ω. Fixing the constant by Γ = 2πJ(ω₀) gives J(ω) = (Γ/2π)(ω/ω₀). The rate needs J at one frequency; the shift needs all of it. Integrating J(ω)/(ω₀ − ω) by principal value from 0 to a highest mode ω_c gives δE/ħ = −(Γ/2π)[ω_c/ω₀ + ln(ω_c/ω₀ − 1)], derived. For large ω the integrand tends to the constant −Γ/(2πω₀), so every band of high modes adds the same amount and the shift grows linearly with ω_c. For hydrogen 2p, ω_c = c/a₀, where the dipole approximation fails, gives δE/h = −5.9 GHz; ω_c = mc²/ħ gives −0.80 THz. Measured radiative shifts are far smaller: the 1057.8 MHz Lamb shift between 2s₁/₂ and 2p₁/₂ is almost all an upward shift of 2s₁/₂, about 1.05 GHz, while 2p₁/₂ moves down by only about 13 MHz (quoted). Unit 13 restores the counter-rotating terms and the other levels, and identifies the linear piece with the electron's electromagnetic mass.

The exact and Markov equations disagree at both ends of time. At the start the memory integral has no length, so the exact equation gives dcₑ/dt = 0 at t = 0 and d²cₑ/dt² = −K(0), hence |cₑ|² ≈ 1 − K(0)t² with K(0) = Σₖ |gₖ|² = (ΔH/ħ)². That is QM-II's quadratic start, which no exponential has. With J(ω) = (Γ/2π)(ω/ω₀) up to ω_c, K(0) = Γω_c²/(4πω₀), and the quadratic law hands over to 1 − Γt near t = Γ/K(0) = 4πω₀/ω_c²: 6 × 10⁻²¹ s for hydrogen 2p with ω_c = c/a₀. At late times, the energy distribution of |e, 0⟩ has a floor, since no state lies below |g, 0⟩, so it cannot be an exact Lorentzian. Near that edge it rises as J(ω)/ω₀², and its Fourier transform adds to the survival amplitude ⟨e, 0|e^(−iHt/ħ)|e, 0⟩ a term −Γ/(2πω₀³t²), outlined here. Its square, (Γ/ω₀)⁶/(4π²(Γt)⁴), overtakes e^(−Γt) after about 125 lifetimes, when the survival is 10⁻⁵⁴. How J vanishes at the edge sets the power. Power-law tails were first seen in 2006 in organic molecules, whose broad spectra bring them forward (quoted).