The Einstein relations and Planck's law as outputs

SI units throughout, with k for Boltzmann's constant; no four-vector is needed, but where one appears the metric is (+, −, −, −). Quoted from topic 12.01: in the dipole and rotating-wave approximations, emission takes |2; n⟩ to |1; n + 1⟩ in one mode with squared matrix element (ħω/(2ε₀V))|ê*⋅d₁₂|²(n + 1), and absorption takes |1; n⟩ to |2; n − 1⟩ with n in place of n + 1. Summed over final field states these become ⟨aa†⟩ = n̄ + 1 and ⟨a†a⟩ = n̄, for any state of the field: derived. Let the field be isotropic and unpolarised, so every mode near ω₀ holds the same n̄ whatever its direction and polarisation, and broadband, so n̄ barely changes across the line. Then n̄(ω₀) leaves the mode sum, and what remains is topic 12.02's sum, A = ω₀³|d₁₂|²/(3πε₀ħc³), quoted. For non-degenerate levels each upper atom emits at A(n̄ + 1) and each lower atom absorbs at A n̄: derived. Stimulated and spontaneous emission are one matrix element, its n̄ and its 1, and n̄ → 0 returns the vacuum rate A.

Einstein wrote the induced rates as B times the field's spectral energy density; take it here as u(ω), the energy per unit volume per unit angular frequency. Unit 11's mode count, ω²/(π²c³) per unit volume per unit ω with both polarisations (quoted), times ħω per photon gives u(ω) = ħω³n̄/(π²c³). Setting A n̄ = B u defines B₁₂ = B₂₁ = π²c³A/(ħω³) = π|d₁₂|²/(3ε₀ħ²), so A/B₂₁ = ħω³/(π²c³): derived. Dimensions: ħω³/c³ is J s × s⁻³ ÷ (m³ s⁻³) = J s m⁻³, which is s⁻¹ ÷ (m³ J⁻¹ s⁻²), the units of A/B. No matrix element survives, because the ratio only converts photons per mode into energy density: A/B is the u(ω) of a field with one photon per mode, where stimulated and spontaneous emission are equal. Per unit frequency ν, u_ν = 2πu(ω) turns it into 8πhν³/c³. Quantum Mechanics II's semiclassical golden rule with a random classical field found this same B and an A of zero: a classical field carries the n̄ but not the 1. For Lyman-α, ħω³/(π²c³) = 1.475 × 10⁻¹² J s m⁻³.

Give the lower level g₁ magnetic substates and the upper level g₂. In an isotropic, unpolarised field the direction and polarisation sums weight every Cartesian component of d alike, so each pair of substates contributes in proportion to |⟨1, m₁|d|2, m₂⟩|². By the Wigner-Eckart theorem (quoted), the sum of these over m₁ is the same for every m₂, namely S/g₂, and the sum over m₂ is the same for every m₁, namely S/g₁, where the line strength S is the sum over both. So every upper substate decays at A = ω³S/(3πε₀ħc³g₂), every lower substate absorbs at (g₂/g₁)A n̄, and B₁₂ = (g₂/g₁)B₂₁: derived, which is g₁B₁₂ = g₂B₂₁ = πS/(3ε₀ħ²), with A/B₂₁ unchanged. For Lyman-α, g₁ = 2 and g₂ = 6 with spin, or 1 and 3 without; the ratio is 3 either way, because d does not act on spin, and B₁₂ = 1.27 × 10²¹ m³ J⁻¹ s⁻². With one substate in each level the relation returns B₁₂ = B₂₁.

Now hold the atoms at temperature T, by collisions or contact with the walls, so that N₂/N₁ = (g₂/g₁)e^(−ħω/kT): the canonical ensemble, quoted. For radiation on this line to leave that ratio steady, the downward flow N₂A(n̄ + 1) must equal the upward flow N₁(g₂/g₁)A n̄. The g's cancel against the Boltzmann factor and A cancels outright, leaving (n̄ + 1)/n̄ = e^(ħω/kT), so n̄ = 1/(e^(ħω/kT) − 1): derived. Times ħω³/(π²c³) it is Planck's spectrum. Nothing about the atom survives, so every line of every atom demands the same n̄, and the field must be Planckian wherever matter can absorb. Quantum Mechanics II ran this the other way, Planck in and A/B out, and Einstein in 1917 needed equilibrium itself, with Wien's displacement law and the classical high-temperature limit, to fix both relations; here they came from the photon factors and Planck comes out. The step assumes Boltzmann populations and balance line by line, which for one line is just steady state. Cross-check: e^(ħω/kT) is also p(n)/p(n + 1) for a mode in its own canonical ensemble, which gives the same n̄. Drop the 1 and N₂n̄ = N₁(g₂/g₁)n̄ balances at finite T only if n̄ = 0.

Limits first. For ħω ≪ kT, n̄ = kT/(ħω) − ½ + …, so u → ω²kT/(π²c³), the classical Rayleigh-Jeans law with kT per mode, and stimulated emission outruns spontaneous by n̄ ≫ 1. For ħω ≫ kT, n̄ ≈ e^(−ħω/kT), Wien's law, and spontaneous emission wins. The two are equal at n̄ = 1, where ħω = kT ln 2: at 300 K that is a wavelength of 69.2 µm, so at every shorter wavelength a room-temperature body emits mostly spontaneously, with stimulated emission under 1% of it at 10 µm, and equality at 500 nm would need 41 500 K. Then measured numbers. Integrating u over ω with ∫x³/(eˣ − 1)dx = π⁴/15 (quoted) gives U = π²k⁴T⁴/(15ħ³c³), and the flux cU/4 through a surface gives σ = π²k⁴/(60ħ³c²) = 5.670 × 10⁻⁸ W m⁻² K⁻⁴, the Stefan-Boltzmann constant that absolute radiometry measures. Per unit frequency the spectrum peaks where hν/kT = 3(1 − e^(−hν/kT)), at hν = 2.821kT, which is 160.2 GHz at the 2.7255 K of the cosmic microwave background, where COBE's FIRAS instrument found the spectrum Planckian to within 50 parts per million of its peak.

Each step leaned on the golden rule, quoted from Quantum Mechanics II and justified for a continuum by the Markov step of topics 12.05 and 12.06, and on a field isotropic, unpolarised and broadband across the line. Remove each in turn. A collimated, polarised beam drives each pair of substates at a rate set by |ê⋅d|² for its own direction and polarisation; only when the substates are equally populated does the average restore the isotropic B. A source narrower than the line no longer samples n̄ at ω₀ alone, and the rate becomes an overlap of its spectrum with the line shape of topic 12.07; once the Rabi frequency exceeds both the line width and the source's bandwidth, the populations oscillate coherently instead of relaxing, and no B describes them. A cavity changes the mode density, so A and A/B change with it (topic 12.08). And Boltzmann populations were an input: a laser medium with N₂/g₂ above N₁/g₁ obeys the same A and B but no longer forces Planck's n̄. Unit 14 writes these rates as the population part of a master equation that also evolves coherences.