Cavity modes and the Planck law

Inside a perfectly conducting box each Cartesian field component obeys the wave equation, and the walls impose E_∥ = 0. Separating out time leaves the Helmholtz equation −∇²u = (ω/c)²u on [0, L]³, and modelling one component with Dirichlet conditions — the same operator, basis and boundary-condition language as every Schrödinger problem to come — gives eigenfunctions sin(nₓπx/L) sin(n_yπy/L) sin(n_zπz/L) with nᵢ = 1, 2, 3, …, so the allowed wavevectors form the lattice k = (π/L)(nₓ, n_y, n_z) in the positive octant, each carrying two transverse polarisations. The full vector problem with n̂ × E = 0 swaps sines for cosines component by component and admits modes with a single zero index, but it lives on the same lattice and changes only the sub-leading terms of the count. Substitute one mode back into Maxwell's equations and its amplitude obeys q̈ = −ω²q: a harmonic oscillator of frequency ω = c|k|. Black-body radiation is therefore a countable family of independent oscillators indexed by three integers and a polarisation, and this whole topic is statistical mechanics applied to that basis.

How many lattice points lie within |k| ≤ k? Each owns a cell of volume (π/L)³ and only the positive octant counts, so N(k) = (1/8)(4πk³/3)/(π/L)³ = Vk³/6π², doubled to Vk³/3π² for polarisation. With k = 2πν/c and one derivative, g(ν) = 8πVν²/c³ modes per unit frequency. Weyl's theorem is what licenses the sphere approximation: the leading term of the Laplacian's eigenvalue count depends only on the volume, with corrections of order the surface area times k² — the box could be a kettle and the ν² would survive. The count is enormous where it matters: a 1 cm³ cavity holds 8π×10⁻⁶ × (6×10¹⁴)³/(3c³) ≈ 7×10¹³ modes below the frequency of green light, and the density keeps growing as ν². Whatever mean energy each mode is assigned multiplies this same relentless count.

Each mode's energy is quadratic in its amplitude and its conjugate momentum, so the classical canonical ensemble assigns it ⟨E⟩ = 2 × kT/2 = kT, independent of frequency. Multiplying by the count gives the Rayleigh–Jeans law, u(ν, T) = 8πν²kT/c³. It is genuinely correct at low frequency — radio engineers still quote antenna noise as a temperature because the long-wavelength tail of every thermal spectrum obeys it — but the total energy density ∫ 8πν²kT/c³ dν diverges as ν³. Ehrenfest later named this the ultraviolet catastrophe. Note where the failure is allowed to be: the ν² count is a theorem about the Dirichlet Laplacian, and the multiplication is bookkeeping. The only physical assumption in the chain is that a mode may hold any energy from a continuum — so that is what has to give.

Planck's move: let a mode of frequency ν hold only Eₙ = nhν, n = 0, 1, 2, …. With β = 1/kT the partition function collapses to a geometric series, Z = Σ e^(−nβhν) = 1/(1 − e^(−βhν)), and ⟨E⟩ = −∂ ln Z/∂β = hν/(e^(hν/kT) − 1). Write x = hν/kT and read it as ⟨E⟩ = n̄hν with n̄ = 1/(eˣ − 1) — the Bose–Einstein occupancy, derived from nothing but Boltzmann weights on a ladder. The numbers show the crossover: at x = 0.1 the mode gets 0.95 kT, essentially the classical answer; at x = 3 it gets 0.16 kT; at x = 10 it gets 4.5×10⁻⁴ kT. Nothing removed the high-frequency modes — their first rung simply costs more than the thermal budget, so the Boltzmann factor prices them out. Multiplying by the unchanged count gives Planck's law, u(ν, T) = (8πhν³/c³)/(eˣ − 1).

A new law must contain the old ones. For hν ≪ kT, eˣ − 1 ≈ x and ⟨E⟩ → kT: Rayleigh–Jeans returns as the low-frequency limit, so everything it explained stays explained. For hν ≫ kT the −1 is negligible and u → (8πhν³/c³)e^(−hν/kT): Wien's empirical exponential falls out. The full integral now converges: substituting x = hν/kT gives u = (8πk⁴T⁴/h³c³)∫x³/(eˣ − 1)dx, the integral is π⁴/15, and u = aT⁴ with a = 7.566×10⁻¹⁶ J m⁻³ K⁻⁴ — Stefan–Boltzmann with its constant now built from h, k and c (radiated flux σT⁴, σ = ac/4 = 5.670×10⁻⁸ W m⁻² K⁻⁴). Maximising ν³/(eˣ − 1) gives 3(1 − e⁻ˣ) = x, so xₚₑₐₖ = 2.821 and νₘₐₓ = 58.8 GHz K⁻¹ × T. The cosmic microwave background at 2.725 K peaks at 160 GHz and follows the whole curve to better than one part in 10⁴ — the best-measured Planck spectrum in nature.

Read the 1900 derivation closely and the quantum sits in the walls, not the field: Planck restricted the energy exchanged by the material resonators in the cavity walls, treated hν as a bookkeeping device he hoped to remove, and left the field itself classical. Einstein's 1905 light quantum moved the restriction into the radiation; Bose's 1924 counting derived the same law from photon statistics; and quantum electrodynamics closes the loop by promoting each cavity mode to a quantum harmonic oscillator, Ĥ = ℏω(â†â + ½), whose number states are exactly the ladder Eₙ = nhν Planck guessed. The zero-point ½ℏω adds a temperature-independent constant per mode, so it cancels from every thermal average here — but it is not fiction: summed over the same Dirichlet modes it returns as the Casimir force between the cavity's own walls.