Selection rules, intensities and lifetimes

Time-dependent perturbation theory turns the atom–field coupling into a transition rate. With a continuum of final photon modes available, first order gives Fermi's golden rule, R = (2π/ħ)|⟨f|Ĥ′|i⟩|²ρ(E_f): a constant probability per unit time rather than an oscillation, because the density of final states smears the energy-conserving δ function. Everything else in this topic is bookkeeping on that one matrix element. A selection rule is therefore not an extra postulate — it is the observation that ⟨f|Ĥ′|i⟩ is identically zero for certain pairs of states, so the rate vanishes at this order and the line is missing. That framing tells you where to look when a forbidden line turns up anyway: either Ĥ′ was truncated too early, or the labels i and f were never exact quantum numbers in the first place.

The coupling to a plane-wave field carries a factor e^(ik⋅r), and r ranges over the atom, so the natural expansion is e^(ik⋅r) = 1 + ik⋅r + …. The small parameter is ka ≈ 2πa/λ, the atom's size divided by the wavelength. For Lyman-α, 2π(0.0529 nm)/(121.6 nm) = 2.7×10⁻³; for the sodium D lines at 589 nm, whose valence electron spreads over about 4a₀, it is 2.3×10⁻³. Rates go as the square of the matrix element, so the next terms — magnetic dipole and electric quadrupole — arrive suppressed by (ka)², roughly 10⁻⁵. Keeping only the leading 1 is the electric-dipole approximation, and it works precisely because atoms are tiny compared with optical light. Push to the hard X-rays of heavy atoms, where the photon energy grows as Z² while the inner shell shrinks only as a₀/Z, so ka grows in proportion to Zα and reaches about 0.2 for lead, and higher multipoles stop being a footnote.

With the leading term kept, the operator is the electric dipole −er. Under r → −r it changes sign, while a one-electron state has parity (−1)^l. The integrand ψ_f* r ψᵢ is then odd unless the two states have opposite parity, and an odd integrand over all space gives zero — Laporte's rule, which already deletes half of every level diagram. The angular integral sharpens it. The three Cartesian components of r are proportional to the l = 1 spherical harmonics, so ∫Y*(l_f m_f) Y(1q) Y(lᵢ mᵢ) dΩ survives only for l_f = lᵢ ± 1 and m_f = mᵢ + q with q = 0, ±1. Physically the photon carries one unit of angular momentum and the atom must absorb exactly that. The q = 0 case is light linearly polarised along z, q = ±1 the two circular polarisations — which is why a Zeeman pattern viewed along the field loses its π component.

In a multi-electron atom the dipole operator is a sum over electrons, each term acting on one coordinate, so a single electron jumps and the configuration's parity must change. Written in term symbols under LS coupling the rules become ΔS = 0, because −er does not touch spin at all; ΔL = 0, ±1 with L = 0 → L = 0 barred; and ΔJ = 0, ±1 with J = 0 → J = 0 strictly barred, since one unit of photon angular momentum cannot be taken up by two states that both have none. Note what is absent: there is no rule on n, so hydrogen's 5p → 1s is as allowed as 2p → 1s. Note also which rule is softest. ΔS = 0 relies on S being a good quantum number, and spin-orbit coupling grows steeply with nuclear charge (as Z⁴ in a hydrogen-like ion, more slowly for a screened valence electron), so in mercury the 6s6p ³P₁ level reaches the ¹S₀ ground state at 254 nm with a lifetime near 120 ns — about 10² times slower than an allowed line, not 10⁸ times.

Carrying the golden rule through to spontaneous emission gives A₂₁ = ω³|d₂₁|²/(3πε₀ħc³). Two things set the rate: the matrix element, and the cube of the photon frequency. In practical units A ≈ 1.06×10⁶ (ħω/eV)³ (d/ea₀)² s⁻¹. For hydrogen 2p → 1s, |⟨1s|r|2p⟩| = 0.745 a₀ and ħω = 10.20 eV, giving A = 6.3×10⁸ s⁻¹. For the sodium D₂ line, ħω = 2.105 eV and the measured A is 6.2×10⁷ s⁻¹ — only ten times smaller, though the ω³ factor alone would predict 114 times. The difference is the matrix element: sodium's valence orbitals spread over several Bohr radii, so |d| is about 3.3 times hydrogen's. Absorption and stimulated emission share that same element through g₁B₁₂ = g₂B₂₁ and A₂₁/B₂₁ = 8πhν³/c³, which is why at 500 K a thermal field drives stimulated emission on the sodium line at only 6×10⁻²² of the spontaneous rate: inversions must be built, never waited for.

A level decays through every channel open to it, so τ = 1/Σ_f A_if, and the branching ratios are the individual A values divided by that sum. A finite lifetime is also a finite energy width: ΔE ≈ ħ/τ, or Δν = 1/(2πτ). Lyman-α's 1.6 ns gives 99 MHz on a 2.47×10¹⁵ Hz line, a fractional width of 4×10⁻⁸ — usually buried under Doppler broadening. Now close the dipole channel. Hydrogen's 2s cannot reach 1s by one electric-dipole photon, so it decays by two-photon emission at 8.2 s⁻¹: a 0.12 s lifetime and a 1.3 Hz natural width. Atomic oxygen's ¹D₂ level radiates at 630.0 nm by a magnetic-dipole route with a lifetime near 110 s, and lights the red aurora only because the upper atmosphere is thin enough that collisions do not de-excite it first. The [O III] 500.7 nm nebular line is the same story. Forbidden lines are missing from a laboratory discharge for a density reason, not a rate-zero reason.