By the year 1900 physics looked all but finished. Newton's mechanics ran the planets, Maxwell's equations ran light and electricity, and thermodynamics ran heat. A famous (probably apocryphal) remark of the age was that all that remained was to measure the next decimal place. And yet, tucked into the corners of the laboratory, sat three small, stubborn facts that the whole magnificent edifice could not explain. Not "not yet" — could not, in principle. Each of them, pushed hard enough, forced the same shocking conclusion: energy at the smallest scales does not flow smoothly. It comes in lumps.

This page tells the story of those three cracks — the glow of a hot furnace, electrons knocked out of metal by light, and the sharp bright lines in the spectrum of hydrogen — and how each one demanded the same strange new ingredient: quantisation. Classical physics assumed energy was a continuous quantity, like water; the experiments insisted it was granular, like sand. The name of the grain is the quantum, and the story of how physicists were dragged, protesting, to accept it is the birth of the whole of quantum mechanics.

Crack one: the ultraviolet catastrophe

Heat any object and it glows: a poker goes red, then orange, then white. A perfect absorber and emitter of radiation is called a black body, and the way its brightness is spread across the colours (the spectral radiance against frequency) depends only on its temperature. Measuring that curve was a triumph of nineteenth-century experiment. Explaining it was a disaster.

Classical physics — counting up the electromagnetic waves that can rattle around inside a hot cavity and giving each an equal share of energy k_B T — gives the Rayleigh–Jeans law:

It fits beautifully at low frequency. But look what it does as the frequency climbs: the \nu^2 grows without limit, so the predicted energy shoots off to infinity in the ultraviolet. Integrated over all frequencies, a warm teacup should blast out infinite energy and instantly freeze the universe. This absurdity was christened the ultraviolet catastrophe. The graph below shows the classical curve running away from the (finite, humped) reality.

In late 1900 Max Planck found a formula that fit the real curve perfectly. To derive it he had to make an assumption he himself called an "act of desperation": the little oscillators in the cavity walls cannot emit or absorb energy in arbitrary amounts, but only in whole multiples of a basic packet proportional to the frequency,

with h \approx 6.63\times 10^{-34}\ \text{J·s}, now called Planck's constant. High-frequency modes need one enormous packet h\nu just to get started, and at a given temperature there is rarely enough thermal energy lying around to pay for it — so those modes stay quiet, and the catastrophe is cured. Planck thought his packets were a mathematical trick. They were the first quantum.

Crack two: the photoelectric effect

Shine light on a clean metal surface and it can knock electrons loose. Classically, light is a wave, and a wave's energy depends on its brightness (its amplitude), not its colour. So dim light should still eject electrons if you wait long enough for energy to build up, and brighter light should eject faster electrons. Both predictions are flatly wrong.

What experiment shows is this: below a certain threshold frequency no electrons come out at all, however blindingly bright the light. Above it, electrons come out instantly, even for the feeblest beam, and their maximum kinetic energy depends only on the light's frequency — not its brightness. Turning up the brightness gives you more electrons, each with the same energy.

In 1905 Einstein explained it in one stroke by taking Planck's packets literally: light itself arrives as discrete quanta, later called photons, each carrying energy

One photon knocks out one electron. It must first pay the work function \phi — the energy binding the electron to the metal — and whatever is left becomes kinetic energy:

This is the equation that won Einstein his Nobel Prize — not relativity. It made the quantum undeniable: light, the very archetype of a smooth wave, is delivered in grains.

Crack three: the hydrogen spectrum and the Bohr atom

Pass an electric current through hydrogen gas and it glows; split that glow through a prism and you do not get a rainbow but a handful of razor-sharp bright lines at very particular colours. Every element has its own such fingerprint. Why should an atom emit only these frequencies and no others?

Classically the situation was even worse: an electron orbiting a nucleus is an accelerating charge, so it should radiate energy continuously, spiral inward, and crash into the nucleus in about 10^{-11} seconds. Matter should not exist. In 1913 Niels Bohr fixed both problems at once with two audacious postulates: the electron may only occupy certain stationary orbits, in which — decree — it does not radiate; and it jumps between them by emitting or absorbing a single photon whose energy is exactly the gap:

Quantising the angular momentum in units of \hbar = h/2\pi gives the allowed energies of hydrogen a strikingly simple form:

The lowest rung, n=1, sits at -13.6\ \text{eV}: the ground state, the most tightly bound the electron can be. The rungs crowd closer and closer together as they climb toward E = 0, where the electron is finally free and the atom is ionised. Reveal the ladder step by step:

Every sharp line in the spectrum is one particular jump. The gaps between fixed rungs can only take fixed values, so the emitted photons can only take fixed frequencies — sharp lines, exactly as seen. Bohr's model gets the hydrogen spectrum numerically right, which is why, however crude it looks today, it convinced the world that the atom itself is quantised.