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.

One question runs the whole way through, and you have already taken a first side on it. 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 wall between the two came down from both sides at once. Watch your own answer move as we go.

Start with the question that started everything.

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 Aristotle's verdicts ruled. With no way to test the idea, atomism survived as a minority thought. The Roman poet Lucretius kept it alive. Almost everyone else dismissed it.

In 1658 Pierre Gassendi revived it. But reviving a philosophy is not the same as proving it.

The proof came later, and from chemistry. It came from 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 is where the next century begins. Does the atom itself have an inside?

In 1897, J. J. Thomson was studying cathode rays — glowing beams inside vacuum tubes. He found something impossible: a particle far, far lighter than the lightest atom, chipped out of atoms themselves.

The atom had an inside after all. He had found the electron.

(Strictly, he measured its charge-to-mass ratio, not its bare mass. The number itself came later.)

Thomson pictured the atom as a positive pudding studded with electron currants. His student Ernest Rutherford blew that up.

In the 1909 gold-foil experiment, a beam of particles was fired at thin metal. Most sailed through. A few bounced almost straight back.

Rutherford's reaction is famous. It was as if you had 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 sits in a minuscule, dense core — a nucleus. The light electrons define a vast, mostly empty volume around it.

So matter is particles, all the way down. That looked like the whole story.

But nineteenth-century physics already knew particles were not enough. 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, reaching out to push and pull.

Maxwell turned Faraday's picture into equations. Those equations predicted waves in the field. In 1887 Hertz detected them. Light itself turned out to be 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 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 radiate into the field. What they wanted was the exact recipe: how much light at each frequency?

They calculated it with the trusted physics of the day. The answer was impossible.

In December 1900, Max Planck found the escape. He 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 that the lumpiness is invisible in daily life. A beach looks smooth until you grab a handful of sand.

Planck 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.

But the crack had opened. Light was the field, spread everywhere, and it had just started behaving like a hail of particles.

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 idle symmetry until you compute it. Try it:

Now both walls were down. Light, the field, acted like particles. Matter, the particles, acted like waves.

Something had to describe both at once. In 1926 Erwin Schrödinger wrote it: the wave function, and an equation for how it evolves in time, with its rate of change tied to the system's energy.

There is a subtlety, and the next essay lives inside it. The wave function is not 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.

And there is the inversion.

Democritus's smallest, uncuttable bit has become a wave. It is spread over the system's possible configurations rather than through ordinary space. It carries only the probability of a location.

That is the exact opposite of where we began.