Stern-Gerlach Experiment

What to observe

1. Start with a single analyser. A classical magnetic moment could be tilted at any angle, so the beam should arrive as one smeared band, and the checkbox draws that prediction. What arrives is two spots. The component of spin along any axis has only two possible values, ±ħ/2, and nothing in between is ever measured.
2. Now z, z: two analysers on the same axis. Every atom that came out of the first one's upper beam goes up again, so the lower spot stays completely empty. A measurement is repeatable: the first one left the atom in the state it reported.
3. Turn the second analyser to x, ninety degrees away. The beam that was purely "spin up along z" splits again, exactly fifty-fifty. Being definite along z means being completely undecided along x.
4. Slide analyser 2 anywhere between and the split followscos²(Δ/2), the fraction shown on the card. At Δ = 0 it is 1, at Δ = 180° it is 0, and the halfway point is not 90° of angle but 90° of half-angle, which is the fingerprint of a spin-½ object.
5. The one that should feel wrong: z, x, z. The first analyser selects spin up along z, the third one asks about z again, and the answer is 50/50. Measuring x destroyed what was known about z. The two questions cannot be answered at once, which is what it means for their operators not to commute.
6. Watch a single atom at a time. Nothing about it is halfway: it goes up or it goes down. The smooth cos²(Δ/2) only appears in the counts, after thousands of atoms have each made a discrete choice.