Blackbody Radiation & Planck's Hypothesis

Cut a small hole in a closed cavity whose walls sit at temperature T, and the radiation leaking out has a spectrum fixed by T alone. Kirchhoff established this in 1860 from thermodynamics: at every wavelength a good absorber must be an equally good emitter, or you could drive heat from a cold body to a hot one. So the curve carries no fingerprint of iron, or soot, or ceramic. That universality is exactly what makes it a hard test — there is nothing left to tune. By the 1890s the curve was measured well enough to argue over: intensity falling to zero at short wavelengths, one peak that moves left as T rises, a long tail, and a total emitted flux going as T⁴. A hob at 1000 K peaks at 2.9 μm; the Sun's photosphere at 5772 K peaks at 502 nm. Wien's displacement law and Stefan's fourth power were both fitted to data before anyone could derive the shape lying between them.

The classical calculation has two honest steps. First, count the standing electromagnetic waves that fit inside the cavity: between ν and ν + dν there are 8πν²/c³ per unit volume, the 8π already carrying both polarizations. Second, apply equipartition. Each mode is a harmonic oscillator with two quadratic degrees of freedom, so its average energy is kT, the same for every frequency. Multiply: u(ν) = 8πν²kT/c³, or in wavelength u(λ) = 8πkT/λ⁴. At long wavelengths that matches the measurements. Everywhere else it is a disaster. The mode count keeps climbing as ν², so the predicted density has no peak at all, and ∫u dν diverges — a cavity at any temperature would hold infinite energy and drain itself into the ultraviolet. Ehrenfest named this the ultraviolet catastrophe in 1911. There is no arithmetic slip to blame: the mode count is geometry, and equipartition is standard statistical mechanics.

Planck's move was to deny each oscillator a continuum of energies. It may hold only Eₙ = nhν with n = 0, 1, 2, …, where h is a new constant carrying units of J s. Redo the average with Boltzmann weights exp(−nhν/kT). Both sums are geometric, and writing x = hν/kT they collapse to ⟨E⟩ = hν/(eˣ − 1). Read the two limits. When x ≪ 1 the rungs sit far closer together than kT, the staircase looks smooth, and ⟨E⟩ → kT: equipartition restored, not contradicted. When x ≫ 1 even the first rung costs more than a thermal kick usually delivers, e⁻ˣ buries the mode in its ground state, and ⟨E⟩ collapses. Multiply the mode count by ⟨E⟩ rather than kT and you have u(λ, T) = (8πhc/λ⁵)/(exp(hc/λkT) − 1). Note carefully what was quantized: the energy exchanged by the cavity's material oscillators, not the electromagnetic field itself. Treating the quantum as a particle of light was Einstein's step in 1905, and it waits for the photoelectric effect later in this unit.

Everything quantitative follows from x = hν/kT = hc/(λkT), and hc/k = 1.4388 × 10⁻² m K makes it a one-line calculation: x = (1.4388 × 10⁻² m K)/(λT). Planck's prediction divided by the classical one is exactly x/(eˣ − 1). Three cases. Microwaves in a 300 K room at λ = 1.0 mm: x = 0.048, ratio 0.976, so the classical law is 2.4% high and entirely usable. Visible light at the Sun's surface, λ = 500 nm and T = 5800 K: x = 4.96, ratio 0.035, and the mean occupancy ⟨n⟩ = 1/(eˣ − 1) is 0.0071, so such a mode is empty 99.3% of the time even there. Visible light from a 300 K wall: x = 95.9, ratio 2 × 10⁻⁴⁰. That last number is why a room-temperature object is not merely dim to the eye but utterly dark, and why the classical prediction of a fierce ultraviolet glow is not a small correction to be patched.

Two laws that had been fitted to data now become consequences. Differentiate u(λ, T) with respect to λ and set the result to zero: the condition reduces to x = 5(1 − e⁻ˣ), whose non-zero root is 4.9651, so λmax T = hc/(4.9651k) = 2.898 × 10⁻³ m K. Wien's constant stops being measured and becomes h, c and k. Integrate u over all wavelengths and the divergence is gone: the total is finite and proportional to T⁴, with emitted flux M = σT⁴ and σ = 2π⁵k⁴/(15h³c²) = 5.670 × 10⁻⁸ W m⁻² K⁻⁴. One new constant, fitted once, delivers both. Planck's own 1901 fit to the measured curves returned h = 6.55 × 10⁻³⁴ J s — 1.1% below the value that now defines the kilogram — and k = 1.346 × 10⁻²³ J K⁻¹, 2.5% low; dividing the gas constant by that k handed him Avogadro's number to within 3%, out of a radiation spectrum.

The classical result was never erased. It is the x → 0 limit of the correct one, and radio astronomers still use it: the cosmic microwave background is a 2.725 K blackbody peaking at 1.06 mm, but at the 21 cm line x = 0.025, the Rayleigh–Jeans form is within 1.3%, and that is why microwave sky maps are quoted as brightness temperatures at all. This is the pattern the rest of the unit repeats. Newtonian mechanics is the v/c → 0 limit of relativity; ray optics is the λ → 0 limit of wave optics; classical trajectories are the ħ → 0 limit of quantum mechanics. A theory that survived real measurement keeps its domain, and any replacement is obliged to reproduce it there — the correspondence requirement, which is a constraint on the new theory rather than a concession to the old one. The working question is never whether classical physics is true. Identify the characteristic ratio, evaluate it, then choose the model.