The Stern-Gerlach Experiment

z → z. One magnet splits the beam in two. Take the up half and send it into a second magnet pointing the same way. Every single atom comes out up. Nothing surprising — the beam was measured, and measuring it again agrees.

z → x. Now turn the second magnet sideways. The beam that was definitely up on z splits evenly. Also not surprising yet: nobody ever measured its x component, so why would it have one?

z → x → z. Take the x-up half and put it back through a z magnet. The atoms in that beam were all measured z-up, and you threw none of them away. Watch the last two counters. Wait for a few thousand atoms.

The third analyser reads 50/50. Half the atoms come out down on an axis where every one of them had already been measured up.

Nothing went wrong with the apparatus and nothing was randomised by hand. The middle magnet did not disturb a value the atom was quietly carrying: after it, there is no z value to disturb. The x measurement replaced the state, and the replacement has no z component to remember. Set the middle analyser back to 0° and the last counter snaps to 100% again — the loss is caused by the angle, not by the extra magnet.

That is the difference between a quantum measurement and reading a label off a box. Spin along z and spin along x are not two properties an atom has at once; committing to one erases the other. The 50/50 you just watched accumulate is the erasure, counted.

Rotate the last analyser and watch the amber marker slide along the curve. If the spin arriving at an analyser points along one direction and the analyser measures along another, Δ apart, the fraction leaving the up port is cos²(Δ/2).

At Δ = 0 that is 1 — perfect agreement. At 60° it is exactly ¾. At 90° it is ½, which is the z → x step. At 180° it is 0: the analyser is pointing the opposite way, and every atom leaves through the other port. Let a few thousand atoms through at each angle and the teal points settle onto the violet curve; they are measurements, not the formula drawn twice.

The half in Δ/2 is not a typo, and it is the tell that this is a spin-½ and not an arrow. Two directions 180° apart on the Bloch sphere describe states that exclude each other completely, while two perpendicular axes — z and x — give no information about each other at all.

Otto Stern and Walther Gerlach fired a beam of silver atoms through a non-uniform magnetic field in Frankfurt and caught them on a glass plate. Classical physics said the atomic magnets would point every which way and the beam would smear into a band. The old quantum theory of Bohr and Sommerfeld, which assigned silver one unit of orbital angular momentum, said the magnets could only point along or against the field: two lines. They got two lines, cleanly separated — which is what the first magnet on this page is doing — and Stern and Gerlach announced it as a triumph for Bohr.

The famous part of the story, as Stern later told it, is that the deposit was too faint to see until he breathed over the plate — sulphur from the cheap cigars he could afford on his salary turned the silver into black silver sulphide, and the two lines appeared. (The recollection is Stern’s own; Friedrich and Herschbach retell it in Physics Today, December 2003.)

The part that matters more: it was the right answer for the wrong reason. A silver atom’s orbital angular momentum is in fact zero, so the theory they thought they had confirmed should have given a single undeflected beam. What split their beam was the spin of the lone 5s electron — a property nobody proposed until Uhlenbeck and Goudsmit did, three years later. The experiment found spin before anyone had the concept, and for a while nobody noticed that it had.

The single magnet is the 1922 experiment. The chain of two and three magnets with beam stops between them is the idealised sequence used to teach spin — Feynman’s filters, Sakurai’s opening chapter — rather than a photograph of a real bench; keeping an atomic beam collimated through three successive magnets is hard, and modern equivalents of these results are usually obtained with light, trapped ions or NMR.

The physics is not idealised. Every branch on this page is the Born rule applied to a spin-½ state, sampled one atom at a time, and the numbers it produces are checked inlib/viz/sternGerlach.test.tsagainst cos²(Δ/2) and against the exact ⅛ yield of the z → x → z chain. If you set the simulation up as the real experiment, it gives the real experiment’s answer.