The Wavefunction

Chapter 1 kept using a symbol without apologising for it: $\psi$ (“psi”), the electron’s wave. But it dodged the obvious question. When the sea waves, water goes up and down. When sound waves, air squeezes and relaxes. When $\psi$ waves… what goes up and down?

The honest answer shocked the physicists who found it: nothing you can point to. At every point of space there is not a height or a pressure but a little arrow, and the “waving” is the arrows turning like clock hands. Everything quantum mechanics can say about a particle is stored in those arrows: how long they are, and which way they point. By the end of this chapter you will have seen the wavefunction whole – it looks like a corkscrew – and seen why the arrows can’t be flattened into an ordinary wave.

First, meet the arrows. A complex number is just an arrow drawn on a dial: it has a length and a direction. You can add two arrows (walk along one, then the other), and you can multiply them – and multiplication is the beautiful part: multiplying by an arrow rotates you by its angle. There is a special arrow called $i$, a quarter-turn: multiply by $i$ twice and you have turned half way round, facing backwards – which is why mathematicians write $i \times i = -1$.

Centuries ago that looked like nonsense – what number squares to a negative? – and Descartes mocked such numbers as “imaginary,” a sneer that stuck as the official name. But they are no more imaginary than negative numbers (what does “minus three sheep” look like?). They are simply the arithmetic of things that turn. And quantum mechanics, it turns out, is about things that turn.

Now place one of these clock arrows at every point of space. That field of arrows is the wavefunction. For a particle moving along a line with a definite speed, the arrows arrange themselves in the simplest possible pattern: each one leans a bit further round than its neighbour, like fans doing a stadium wave, and all of them turn together as time passes.

Two things to hunt for in the figure. The lean travels: the moving pattern is the wave. And the faster the particle, the tighter the winding – that is de Broglie’s rule from Chapter 1, drawn instead of stated. Then switch on the wave’s strength: every arrow has the same length, so the strength is utterly flat. All the waving is in the arrows’ directions – none of it is in their size.

A flat drawing can show an arrow’s height, but each arrow has two directions to point in – up-down and sideways. To see the wavefunction whole you need three dimensions: distance along the line, plus both directions of the dial. Do that, and the moving particle’s wave is revealed as a corkscrew.

Drag the figure around. Looked at from the side, the corkscrew’s shadow is the familiar wiggling wave – that’s all a flat graph ever showed you. But the full object holds more: the direction of its twist is the direction the particle moves. Slide the momentum negative and watch the spiral become its own mirror image. And if you strip the twist away (“show |ψ| only”), almost nothing remains – a featureless tube. The twist isn’t decoration. The twist is the physics.

Fair question: maybe the sideways part of the arrows is just mathematical scaffolding? Perhaps an honest up-down wave – like a guitar string – could do the same job. The figure below lets you run that experiment. Two lumps of wave fly at each other, overlap (making stripes, as waves must), and pass through unharmed. Now tick “pretend ψ were real” and watch the impostor fail twice.

First failure: a lone travelling lump flickers. Its “strength” pulses on and off everywhere even though the particle is just cruising along – nothing should be happening. The true $\psi$ stays smooth, because turning arrows keep a steady length. Second failure: freeze a snapshot of the real wave, and a leftward lump and a rightward lump look identical – the snapshot has amnesia about direction. The corkscrew’s twist stores that memory. Nature, it seems, insists on the arrows.

So: what is waving? An arrow at every point of space. The arrows’ twist stores how the particle moves, and their length stores where it is likely to be found. But notice that word – likely. We’ve been sampling dots from the wave’s strength since the double slit, as if that were obviously the thing to do. It is time to face what that actually means: at the bottom of physics sits a rule about chance, and it is unlike any law science had ever written down. That is Chapter 3.