Planck's radiation law

Heat any object and it glows — dull red, then orange, then white as it gets hotter. What is the exact color spectrum of that glow? Nineteenth-century classical physics gave an absurd answer: the Rayleigh-Jeans formula predicted ever more energy at ever shorter wavelengths, an infinite total energy known as the ultraviolet catastrophe. In 1900 Max Planck found the correct spectrum by making the radical assumption that energy is exchanged in discrete quanta hbar omega — the birth of quantum theory.

Planck's law gives the spectral energy density of blackbody radiation. Per unit angular frequency, u(omega, T) = (hbar omega^3 / pi^2 c^3) / (exp(hbar omega/kT) - 1). The first factor, proportional to omega^2, is the classical count of electromagnetic modes; the second factor is the Bose-Einstein mean energy per mode, hbar omega / (exp(hbar omega/kT) - 1), which for a fair-sized frequency provides an exponential cutoff that classical physics lacked. Two limits recover older results: at low frequency (hbar omega << kT) the mean energy tends to kT and the spectrum reduces to Rayleigh-Jeans, u -> kT omega^2/pi^2 c^3; at high frequency it falls off exponentially (Wien's law). Integrating over all frequencies gives the Stefan-Boltzmann law u = a T^4, and locating the peak gives Wien's displacement law, lambda_max T = constant.

Planck's law is one of the most experimentally confirmed formulas in all of physics; the cosmic microwave background is a textbook fit to it. Its historical importance is hard to overstate: the quantum hbar entered physics here, and the ultraviolet catastrophe was cured not by any change to the mode count but by quantizing the energy each mode can hold. An honest note: Planck first introduced quantization as a formal trick and only later did Einstein interpret it as real photons; the quanta are physically genuine, not merely a bookkeeping device.