Spin & Measurement

Everything so far has been a wave spread through space, with infinitely many places to be. Now shrink the world until only two answers remain. Electrons carry a kind of internal compass, called spin, and however you orient your measuring device, that compass reports one of exactly two things: aligned, or anti-aligned. Never anything in between. Never zero.

This is the simplest quantum system in existence, and precisely because there is so little of it, the strangeness has nowhere to hide. In this chapter you will watch a measurement erase the result of the one before it – not by clumsiness, not by nudging the atom, but because the two questions cannot both have answers at once.

In 1922, in Frankfurt, Otto Stern and Walther Gerlach fired a beam of silver atoms between the poles of a lopsided magnet. A little bar magnet flying through such a field gets pushed sideways by an amount that depends on its tilt, so a beam of randomly tilted atoms should smear into a continuous band – a classical prediction as safe as they come.

Instead, the beam split cleanly in two. Not a band: two spots, one high, one low, with nothing in between. (The spots were only visible at all because Stern smoked such cheap, sulfurous cigars: the fumes turned the invisible silver film into black silver sulfide as they breathed over the plate.) Whatever the atom’s compass was doing, it refused to point anywhere except fully along the magnet, or fully against it.

Two outcomes, then: call them $|{\uparrow}\rangle$ and $|{\downarrow}\rangle$. But quantum states superpose, so the general state is a blend of both – and the blend takes two numbers to describe: how much of each, and the phase between them. Two numbers means a surface, and that surface is a sphere.

It is the friendliest picture in quantum mechanics. Every possible spin state is an arrow from the centre of a ball. Straight up is “definitely up”, straight down is “definitely down”, and everywhere else is a superposition. Point your measuring device along any axis you like, drop a shadow of the arrow onto it, and that shadow tells you the odds. Drag the figure around; then press Measure and watch the arrow do the one thing it is allowed to do: snap onto the axis, up or down, never in between.

One warning about the picture. Up and down are opposite ends of the ball, 180° apart – yet as states they are as different as two states can be. Angles on this ball are doubled. An arrow tipped 90° from the pole, lying on the equator, is not “half up”: it is a perfect 50/50 gamble.

Now chain three magnets together, as in the second figure. The first sorts atoms up-or-down and we throw the “down” pile away. What survives is a beam of certified, guaranteed up atoms. Send it into a second up-down magnet and, reassuringly, every single atom comes out up. Ask the same question twice, get the same answer twice. Nothing to see.

Now slip a third magnet in between, turned on its side, and throw away its reject pile. Everything reaching the final magnet has now passed two purity checks. And the final magnet splits the beam clean in half. Half of your certified up-atoms come out down.

Resist the obvious explanation. The sideways magnet did not knock them about; it is not a matter of clumsy equipment, and no gentler apparatus would help. Asking “which way along sideways?” gives the atom a definite sideways answer, and an atom with a definite sideways answer has no up-down answer to remember. The question overwrote it.

Collect the rules the figures have been obeying. A quantum system has a state – an arrow. A measuring device picks an axis. The possible answers are the two ends of that axis, and nothing else is ever observed. The odds come from the shadow of the arrow on the axis. And once the answer is in, the arrow is the answer: ask again and you get the same result, forever, until something rotates it away.

That last clause is the one that costs. The old answer, the one from the previous axis, is not stored anywhere. There is only one arrow, and the measurement moved it.

Spin gave us quantum mechanics with the scenery removed: two answers, one arrow, and a measurement that rewrites what it reads. It is also the most practical thing in this course – that arrow, controlled and measured exactly as in the figure above, is a qubit, the unit of every quantum computer being built today, and its precession is what an MRI machine detects inside you.

One thing remains. We began, in Chapter 0, with an atom that should have collapsed in a picosecond and with light emitted at maddeningly specific colours. We now have every tool needed to go back and finish that story properly, in three dimensions: the hydrogen atom, its energy ladder, and the orbitals that give chemistry its shape.