Planck's quantised oscillator and the radiation law

Classically a wall oscillator of frequency ν may hold any energy, so its Boltzmann average is ⟨E⟩ = ∫E e^(−E/kT)dE / ∫e^(−E/kT)dE = kT — the same answer for every ν, which is exactly what makes the ultraviolet catastrophe unavoidable. Planck allowed only Eₙ = n hν. The two integrals become two sums, and with r = e^(−hν/kT) both come from the geometric series: Σ rⁿ = 1/(1 − r) and Σ n rⁿ = r/(1 − r)². Their ratio is ⟨E⟩ = hν r/(1 − r) = hν/(e^(hν/kT) − 1). Nothing else in the derivation moved. Notice where the frequency got in. In the classical integral the only energy scale is kT, set by the thermostat; in the sum the first available rung sits at hν, so the answer must depend on how that rung compares with kT. Write x = hν/kT and the result is ⟨E⟩ = kT · x/(e^x − 1), one dimensionless function that equals 1 at x = 0 and dies exponentially after.

The denominator is a probability statement. The chance of finding the oscillator on rung n is (1 − e^(−x))e^(−nx), so once hν exceeds kT even the first rung is expensive and every higher one is exponentially worse; the mean occupancy ⟨n⟩ = 1/(e^x − 1) collapses. Numbers say it better. At 300 K, kT = 25.9 meV, while a green mode at 500 nm needs hν = 2.48 eV, so x = 95.9 and ⟨n⟩ ≈ e^(−95.9) ≈ 2.3 × 10⁻⁴². A 30 m³ room holds about 6.0 × 10¹⁹ modes in a 1% frequency band around that line, so the expected number of green quanta in the room is around 1.4 × 10⁻²² — never, in any practical sense. Equipartition would have handed each of those modes a full kT and the room would glow. The classical mode count was never the problem: it still rises as ν², and it is the exponential that beats it down, which is why the integral over all frequencies now converges.

Assembling is one multiplication: u(ν, T) = (8πν²/c³)⟨E⟩ = (8πhν³/c³)/(e^(hν/kT) − 1), in J m⁻³ Hz⁻¹. Check it once against the coldest blackbody there is. The cosmic microwave background sits at T = 2.725 K, so kT = 3.763 × 10⁻²³ J, and at ν = 160 GHz a quantum is hν = 1.060 × 10⁻²² J: x = 2.817, e^x − 1 = 15.73, and ⟨E⟩ = 6.74 × 10⁻²⁴ J, only 0.179 kT. The classical density 8πν²kT/c³ comes to 8.99 × 10⁻²⁵ J m⁻³ Hz⁻¹, while Planck's value is 1.61 × 10⁻²⁵, a factor of 5.58 lower — and this is the very frequency at which u_ν peaks, since x = 2.82 there. Two conversions are worth keeping. Multiply by c/4 to get the exitance a black surface radiates, and change variable with u_λ dλ = u_ν dν, using |dν/dλ| = c/λ², to get u_λ = (8πhc/λ⁵)/(e^(hc/λkT) − 1).

Expand for small x: e^x − 1 = x(1 + x/2 + …), so ⟨E⟩ = kT/(1 + x/2 + …) ≈ kT(1 − x/2) and u → 8πν²kT/c³. Rayleigh–Jeans is not overturned, it is demoted to a limit with a computable error. A 30 GHz mode at 300 K has x = 4.80 × 10⁻³, so equipartition is high by x/2 = 0.24% — fine for a microwave engineer. For large x the −1 is negligible and u → (8πhν³/c³)e^(−hν/kT), which is Wien's exponential guess, its two fitted constants now identified as 8πh/c³ and h/k. The discipline worth having is to demand one percent rather than 'roughly'. At 300 K the Rayleigh–Jeans form holds to 1% only for λ > 2.4 mm (x < 0.02), and Wien's only for λ < 10.4 µm (x > 4.6). Between them lies a factor of 230 in wavelength, running from just past the 9.66 µm peak out to the millimetre band, where only the full expression will do.

Two constants that had been measured now become predictions. Differentiate the wavelength form and set du_λ/dλ = 0: the λ⁻⁵ prefactor and the exponential balance where y e^y/(e^y − 1) = 5, that is y = 5(1 − e^(−y)), whose root is y = 4.9651. Since y = hc/λkT, this says λₘₐₓ T = hc/(4.9651 k) = 2.898 × 10⁻³ m K. Integrating instead over all frequencies needs ∫₀^∞ x³dx/(e^x − 1) = π⁴/15 = 6.4939 and gives u = (8π⁵k⁴/15h³c³)T⁴; the emitted power is cu/4, so σ = 2π⁵k⁴/(15h³c²) = 5.670 × 10⁻⁸ W m⁻² K⁻⁴. Planck ran this backwards. From the measured σ and b he solved for h = 6.55 × 10⁻³⁴ J s and k = 1.346 × 10⁻²³ J K⁻¹, then took N_A = R/k ≈ 6.2 × 10²³ and e = F/N_A ≈ 1.56 × 10⁻¹⁹ C — both within a few percent of today's values, and in 1901 better than any direct measurement available.

The restriction Eₙ = n hν was imposed on the material oscillators in the cavity wall, not on the electromagnetic field. The field enters only through 8πν²/c³, which is classical standing-wave counting, and nothing in the derivation says a mode's energy travels in indivisible lumps. Planck treated ε = hν as a device for the entropy count and spent years trying to remove it; his 1912 'second theory' kept classical absorption and quantised emission alone. The modern derivation quantises the field mode itself, returns the same formula with an extra ½hν per mode, and the measured spectrum cannot see that term because it carries no temperature dependence. So take the modest claim: energy exchange between matter and radiation at frequency ν is grainy in units of hν. Whether the free field is itself quantised needs different evidence, and the experiments that settle it come later in this unit.