Blackbody Radiation and Planck's Quantum Hypothesis

A blackbody is an ideal object that absorbs every bit of radiation that falls on it, reflecting none. Because it reflects nothing, all the light coming from it is emitted because of its temperature, so its spectrum depends only on temperature, not on what it is made of. A good laboratory version is a hollow box with a tiny hole: light that enters the hole bounces around inside and is almost never reflected back out, so the hole looks perfectly black when cold and glows with a pure thermal spectrum when hot. Stars, stove elements, lava and even your own skin are close approximations. Measuring the intensity at each wavelength gives a smooth, lopsided hump with a peak, and the hump changes in two clear ways as the temperature rises.

First, the peak shifts to shorter wavelengths as the temperature rises. This is Wien's displacement law: double the kelvin temperature and the peak wavelength halves. Your body, at about $310\text{K}$, peaks deep in the infrared near $9\mu \text{m}$, which is why thermal cameras are built to detect that range. A filament at $3000\text{K}$ peaks near $1\mu \text{m}$, still in the infrared, which is why old incandescent bulbs wasted most of their energy as heat. The Sun's surface, near $5800\text{K}$, peaks at about $500\text{nm}$, in the middle of the visible band. Second, the total power rises steeply, as $T^4$ (the Stefan-Boltzmann law). Doubling the temperature makes each square metre radiate 16 times as much power, which is why a hotter element feels so much more intense.

Classical physics treated the radiation inside the glowing box as electromagnetic standing waves, just like the harmonics on a guitar string, and assigned every possible standing wave the same average share of thermal energy. At long wavelengths that prediction fits the measured curves well. The trouble is that there is no limit to how many short-wavelength standing waves fit in the box: halve the wavelength and many more patterns fit. Giving each one the same energy means the predicted intensity climbs without limit toward the ultraviolet, and the total energy in the box comes out infinite. Real blackbody curves do the opposite: they turn over and fall to zero at short wavelengths. This dramatic failure was later nicknamed the ultraviolet catastrophe.

In 1900 Max Planck found a formula that matched the measured curves perfectly, then searched for a reason it worked. The only way he could derive it was to assume that the vibrating charges in the walls could not have just any energy. An oscillator of frequency $f$ could hold energy only in whole-number steps of $hf$, where $h$ is a new constant of nature, now called Planck's constant. Here is why that fixes the catastrophe. At low frequencies the steps are tiny and the energy behaves almost continuously, so the classical prediction survives. At high frequencies, each step $hf$ is large compared with the thermal energy available, so those oscillators are rarely excited at all and the short-wavelength intensity falls away instead of exploding. Planck himself treated the steps as a mathematical device; it was Einstein who, in 1905, took them literally.