The ultraviolet catastrophe

Heat any object and it glows: first dull red, then orange, then, if it's hot enough, blue-white. A body that absorbs and re-emits radiation perfectly at every wavelength — an idealized 'blackbody' — turns out to have a glow whose color and brightness depend on nothing but its temperature, not on what it's made of. By the 1890s, careful measurements had mapped this blackbody spectrum precisely: intensity rises from zero at very long wavelengths, peaks at a wavelength that shifts shorter as temperature climbs (Wien's displacement law), and falls back to zero at very short wavelengths. Any theory of radiation had to reproduce that curve, peak and all.

Classical physics had a theory, and it failed spectacularly at short wavelengths. Treat the electromagnetic field inside a hot cavity as a collection of standing-wave modes, and treat each mode as a simple harmonic oscillator obeying the equipartition theorem — the same rule that hands every classical degree of freedom an average energy of ½kT. The trouble is that a cavity supports far more modes at short wavelengths than long ones (mode density grows as 1/λ⁴), so if each mode really carries ½kT of energy regardless of its frequency, the total radiated energy — summed over modes — diverges as wavelength shrinks toward the ultraviolet and beyond. The Rayleigh–Jeans law, built exactly this way, matches the observed spectrum beautifully at long wavelengths and then shoots to infinity at short ones — a divergence later nicknamed the 'ultraviolet catastrophe.' A blackbody, by this reasoning, should radiate infinite power. It doesn't.

Max Planck found the fix in 1900, more or less by curve-fitting before he understood why it worked. Instead of letting each cavity mode exchange energy with matter continuously, he assumed the exchange happens only in discrete lumps — a mode of frequency f can only gain or lose energy in whole multiples of a fixed quantum hf, where h is a new constant of nature. That single assumption changes everything at high frequency: modes with hf ≫ kT need a full quantum's worth of energy just to be excited at all, and at ordinary temperatures that's a vanishingly unlikely fluctuation, so high-frequency modes barely radiate — exactly the cutoff the data demanded and equipartition couldn't produce. Planck's formula reproduces the entire measured spectrum, peak and both tails, with one fitted constant: h ≈ 6.626×10⁻³⁴ J·s. Planck treated the quantization as a mathematical trick he wasn't sure he believed; it took Einstein, five years later, to argue that light itself — not just the oscillators that emit it — is quantized. That's the subject of the next lesson.