The A coefficient from the golden rule over empty modes

SI units throughout; no four-vector appears, and where one does in this course the metric is (+, −, −, −). Quoted from topic 12.01: in the dipole approximation −d⋅E takes |e, 0⟩ to |g, 1ₖₛ⟩ with squared matrix element (ħωₖ/(2ε₀V))|êₖₛ*⋅d|², where d = q⟨g|r|e⟩ and êₖₛ, s = 1, 2, are unit polarisation vectors perpendicular to k. Keep only one resonant mode and Unit 11's Jaynes-Cummings solution, quoted, applies: the excitation swings between atom and mode at the vacuum Rabi frequency 2g, with ħg = √(ħω₀/(2ε₀V))|d| for a mode polarised along d, and never leaves for good. For hydrogen 2p in a 1 cm³ box, g = 1.8 × 10⁴ s⁻¹. What turns the swing into a decay is how many modes the atom can reach. The same box holds on average one mode per 1.1 s⁻¹ of angular frequency near Lyman α, so 5.7 × 10⁸ of them lie inside the natural width. Each is coupled feebly, their amplitudes dephase long before any one of them could swing back, and probability leaves at a constant rate: the golden rule's regime, in which the mode density, not any single coupling, sets A.

In a periodic box of side L the allowed wavevectors are k = (2π/L)(n₁, n₂, n₃), one per volume (2π)³/V of k-space. A shell of radius k and thickness dk, opened to solid angle dΩ, has volume k² dk dΩ, so it holds Vk² dk dΩ/(2π)³ wavevectors for each polarisation. With k = ω/c that is dN = Vω² dω dΩ/(8π³c³): derived, and a pure number, as Vω² dω/c³ must be. The golden rule's energy δ-function, δ(ħω₀ − ħωₖ), equals δ(ω₀ − ωₖ)/ħ, which turns 2π/ħ into 2π/ħ² and consumes the frequency integral, leaving dΓ = (2π/ħ²)(ħω₀/(2ε₀V))(Vω₀²/(8π³c³)) Σₛ|êₖₛ*⋅d|² dΩ. The V in the mode count meets the 1/V in the squared field per photon and cancels: the rate belongs to the atom and to free space, not to the box. Nothing thermal has appeared either. The vacuum supplied the 1 in n + 1 for every mode, no mode needed a temperature, and Planck's law was never used.

The two êₖₛ and k̂ form an orthonormal triad, so Σₛ (êₖₛ)ᵢ(êₖₛ*)ⱼ = δᵢⱼ − k̂ᵢk̂ⱼ: derived from completeness, with the transversality k̂⋅êₖₛ = 0 that the Coulomb gauge built into Unit 11's field. Contracting with dᵢ*dⱼ gives Σₛ|êₖₛ*⋅d|² = |d|² − |k̂⋅d|², the squared length of the part of d perpendicular to the photon's direction; a photon has no longitudinal polarisation to receive the rest. For a Δm = 0 line, d = d ẑ is real and the sum is |d|² sin²θ, with θ the angle between k and z: nothing leaves along the dipole, most leaves at right angles, and the light is linearly polarised in the plane of z and k. For Δm = ±1, d ∝ (x̂ ± iŷ)/√2 rotates in the xy plane; then |k̂⋅d|² = |d|² sin²θ/2 and the sum is |d|²(1 + cos²θ)/2. It is largest along z, where the light is circularly polarised, and half as large at θ = 90°, where only the component perpendicular to both k and z survives and the light is linearly polarised.

Integrate over directions with ∫k̂ᵢk̂ⱼ dΩ = (4π/3)δᵢⱼ, fixed by rotational symmetry and the trace ∫dΩ = 4π. Then ∫(|d|² − |k̂⋅d|²) dΩ = (4π − 4π/3)|d|² = (8π/3)|d|² for any d, linear or circular. Collecting factors, A = (2π/ħ²)(ħω₀/(2ε₀V))(Vω₀²/(8π³c³))(8π/3)|d|² = ω₀³|d|²/(3πε₀ħc³): derived. Dimensions: ω₀³|d|² is s⁻³ C² m², and ε₀ħc³ is (C² J⁻¹ m⁻¹)(J s)(m³ s⁻³) = C² m² s⁻², so A is in s⁻¹. Correspondence check: ħω₀A = ω₀⁴(2|d|)²/(12πε₀c³), exactly the Larmor power of a dipole p₀ cos ω₀t with p₀ = 2|d|, as it must be, since d is the coefficient of e^(−iω₀t), half of a real oscillation. Free-field limit: as e → 0, A → 0 as |d|² ∝ e², the first order assumed. The ω₀³ is worth reading: two powers from the mode density and one from the field per photon. At fixed |d|, moving a line from 10.2 eV down to 10.2 µeV multiplies its lifetime by 10¹⁸, from 1.6 ns to about 50 years.

For 2p → 1s, d = −e⟨1s|r|2p, m⟩, and Quantum Mechanics II's matrix element, quoted, gives |⟨1s|r|2p, m⟩|² = (2¹⁵/3¹⁰)a₀² = 0.5549a₀² for every m, so |d| = 6.316 × 10⁻³⁰ C m. With ħω₀ = (3/4) × 13.606 eV = 10.204 eV, ω₀ = 1.5503 × 10¹⁶ s⁻¹, and the derived formula gives A = 6.27 × 10⁸ s⁻¹ and τ = 1/A = 1.60 ns: the numbers Quantum Mechanics II reached through Einstein's detailed balance, now with no thermodynamics in the argument. In closed form A = (2/3)⁸α⁵mₑc²/ħ, and the reduced mass multiplies it by μ/mₑ = 0.99946, giving 6.265 × 10⁸ s⁻¹ and 1.596 ns. Beam-foil timing of fast hydrogen atoms measures 1.6 ns. Each approximation can now be priced. The dipole approximation needs ka₀ = (3/8)α = 2.7 × 10⁻³ ≪ 1. The golden rule needs A ≪ ω₀, and A/ω₀ = 4.0 × 10⁻⁸: the line is 2.5 × 10⁷ times narrower than its frequency. Relativistic corrections from the Dirac equation enter at order (Zα)² = 5 × 10⁻⁵.

Divide dΓ/dΩ by A and a Δm = 0 line sends a fraction (3/(8π))sin²θ of its photons into each steradian, a Δm = ±1 line (3/(16π))(1 + cos²θ): derived, and each integrates to 1, because the 8π/3 does not care which way d points. That is why all three 2p sublevels decay at the same A. It also fixes what a level with its sublevels equally filled must do: sin²θ + 2 × (1 + cos²θ)/2 = 2 at every θ, so it radiates isotropically and unpolarised, as it must with no axis to prefer. Single out an axis with a weak magnetic field B ∥ z and the patterns become visible. Viewed along B, only the two σ lines appear, circularly polarised in opposite senses. Viewed across B, the π line is linearly polarised along B and the σ lines across it, and with equal populations the π line is twice as bright as each σ line, since sin²90° = 1 against (1 + 0)/2. That 2:1:1 transverse triplet with those polarisations is the normal Zeeman pattern measured on singlet lines such as cadmium's 643.8 nm, and the quantised-field sum reproduces it.