From Atoms to the Wave Function

Twenty-four centuries later, physics arrived at an answer Democritus would not have recognised. The smallest bits were not hard little pellets sitting in one place. They were described by a wave — a mathematical object that carries the odds of a position rather than the position itself.

The whole essay is about one question. You have already picked a side: what is the world actually made of — particles, each sitting somewhere, or something spread out through all of space? For most of the story the answer was both. Then, in about thirty years, the distinction broke down on both sides. Watch your own answer move as we go.

Start with the original question. Take anything — a coin, a drop of water — and divide it. Then divide a piece of that. Keep going. Is there a bottom? Or does it never end?

Aristotle's verdict mattered because his authority settled such questions for centuries. With no way to test the idea, atomism survived as a minority thought — kept alive by the Roman poet Lucretius, dismissed by almost everyone else — until the Scientific Revolution. In 1658 Pierre Gassendi revived it. But reviving a philosophy is not the same as proving it. The proof came later, and from chemistry: the careful weighing of what combined with what.

Note what Dalton's atom was. It was chemically indivisible — the smallest unit that still behaved like tin, or oxygen. It was a featureless sphere. He had no idea whether it had parts. That question — does the atom itself have an inside? — took another century to settle.

In 1897, J. J. Thomson was studying cathode rays — glowing beams inside vacuum tubes — when he found something that contradicted the idea of the atom itself: a particle far, far lighter than the lightest atom, chipped out of atoms themselves. The atom had an inside after all. He'd found the electron. Strictly, what he measured was its charge-to-mass ratio; later measurements put the electron's mass at about 1/1836 that of a hydrogen atom.

Thomson pictured the atom as a positive pudding studded with electron currants. His student Ernest Rutherford ruled it out. In the 1909 gold-foil experiment, a beam of particles was fired at thin metal — most sailed through, but a few bounced almost straight back. Rutherford's reaction is famous: as if you'd fired a shell at tissue paper and it had rebounded into your face. Accounting for that scattering forced a new model of the atom: nearly all its mass and positive charge packed into a minuscule, dense core — a nucleus — with the light electrons defining a vast, mostly empty volume.

So matter is particles, all the way down. That seemed to settle it. But nineteenth-century physics already knew particles weren't enough — because it had discovered a second kind of thing entirely.

Hold a magnet near a compass and the needle swings — with nothing touching it. Michael Faraday's radical idea was that the space around a magnet or a charge is not empty: it is filled with a field, something with a value at every single point, that pushes and pulls. Maxwell turned Faraday's picture into equations; in 1887 Heinrich Hertz detected the waves those equations predicted, and light itself was one of them — a ripple travelling through the electromagnetic field.

That left physics, by 1900, with two different kinds of reality. Feel the difference for yourself:

Heat anything enough and it glows — dull red, then orange, then, hotter, white and finally bluish-white. Blacksmiths read a forge's temperature by colour for centuries; astronomers still do, which is how we know a blue star is hotter than our yellow Sun without going near one. By 1900 physicists could explain the glow — jiggling charges radiating into the field — and wanted the exact recipe: how much light at each frequency? They calculated it with the accepted physics of the day, but the answer made no physical sense.

In December 1900, Max Planck found the fix — and treated it, at first, as a mathematical device rather than a new picture of reality. He could match the real curve exactly, but only by assuming energy leaves in discrete lumps, whole packets he called quanta, each sized by its frequency times a tiny new constant, h (about 6.626×10⁻³⁴ joule-seconds). The number is so small the lumpiness is invisible in daily life. He hoped classical physics would eventually explain the trick away.

Einstein, in 1905, took the packets literally. Light, he said, is grains of energy — photons, each carrying $E = hf$. He used it to explain the photoelectric effect, and the numbers held. Most physicists, Planck included, resisted for years; giving up the wave theory of light was not done lightly. The resistance did not change the result. Light was the field — spread everywhere — and it had just started behaving like particles. That raised the opposite question.

If a wave can act like a particle, can a particle act like a wave? In 1924 Louis de Broglie said yes: every particle has a wavelength, set by $\lambda = h/p$ — the more momentum, the shorter the wave. It sounds like empty symmetry until you compute it. Try it:

Now the distinction had failed in both directions. Light, the field, acted like particles. Matter, the particles, acted like waves. Something had to describe both at once — and in 1926 Erwin Schrödinger wrote it: the wave function, and an equation for how it evolves in time, its rate of change tied to the system's energy. The subtlety — the one the next essay is about — is that it isn't one little wave attached to each particle. A whole system of particles is described by a single wave function, defined over all the ways that system could be arranged.

The reversal is complete. Democritus's smallest, uncuttable bit has ended up as a wave — spread over the system's possible configurations rather than through ordinary space, and carrying only the probability of a location. That is the opposite of the picture we started with.