The crack. Planck, 1900

Wavelength and frequency are two sides of one fact. All light travels at the same speed, about 300,000 kilometres a second, so a shorter wave must wiggle faster to keep up. Our eyes see only a narrow band, from about 380 to 750 nanometres. Longer waves are infrared, which you feel as warmth from a radiator. Shorter ones are ultraviolet, which causes sunburn.

Speed equals wavelength times frequency:

Red at λ = 700 nm gives ν ≈ 4.3 × 10¹⁴ Hz. Violet at 400 nm gives ν ≈ 7.5 × 10¹⁴ Hz.

K is kelvin, a temperature scale that starts at the coldest possible point, about minus 273 °C. In the 1890s physicists in Berlin measured exactly how much of each colour a hot oven gives off. The curve was the same for every material: only the temperature mattered. Double the temperature and the brightest wavelength halves, a rule Wilhelm Wien found in 1893.

Wien's law for the peak, and the curve itself, which is Planck's law:

The old rule is called equipartition: in heat, every way of moving gets the same average share. It works beautifully for gases. For light in an oven it fails at the short end. Lord Rayleigh and James Jeans worked out the prediction between 1900 and 1905, and in 1911 Paul Ehrenfest named its failure the ultraviolet catastrophe. Planck did not start from this problem. He was chasing precise Berlin measurements that no formula yet matched.

Each way of wiggling (a mode) gets an average energy kT. The number of modes per unit wavelength grows as 8π/λ⁴, which gives the Rayleigh-Jeans law. It runs to infinity as λ goes to zero:

Planck found his formula in October 1900, and it fitted the Berlin measurements perfectly. In December he showed that it follows if energy comes in lumps. He applied the lumps only to the jiggling matter in the oven walls, not to light itself. The "pocket money" is the average share of heat the old rule promised every wave. With lumps, a wave either gets a whole coin or nothing, and big coins are rarely affordable.

One lump is E = hν. The pocket money is kT, Boltzmann's constant times the temperature. The chance that a wave holds at least one lump, and its average energy, become:

When hν is much smaller than kT, ⟨E⟩ goes back to kT. When it is much bigger, ⟨E⟩ falls towards zero.

h is called Planck's constant. Since 2019 it has had a fixed value by definition, and the kilogram itself is defined from it. Halving h would make a hot oven give off eight times as much energy.

Total power per area is σT⁴ with σ = 2π⁵k⁴ / (15c²h³), so it scales as 1/h³. As h → 0, Planck's law turns into the Rayleigh-Jeans law.

A swing rocks about once every two seconds and carries tens of joules of motion, so it would take some 10³⁵ steps to get it going. A nitrogen molecule in the air wiggles about 70 trillion times a second. Its steps are some ten times bigger than the heat it gets at room temperature, so it nearly always sits on the bottom step. That is the same rule that keeps violet light out of the oven.

A swing at 0.5 Hz has steps hν ≈ 3.3 × 10⁻³⁴ J. N₂ at ν ≈ 7.0 × 10¹³ Hz has hν ≈ 0.29 eV, against kT ≈ 0.025 eV at room temperature.

Looking back in a letter of 1931 to the American physicist Robert Williams Wood, Planck called his guess an act of desperation. He was cautious and hoped to save the old physics. Making light itself lumpy was Einstein's bolder step, in 1905.