Stern-Gerlach Chain
Shoot a spin-½ particle through a Stern-Gerlach apparatus oriented along axis n̂. The beam splits into two: spin-up (along n̂) and spin-down. Block one beam. Send the survivors through another SG oriented at angle θ from the first. The transmission probability is cos²(θ/2). Stack a sequence of axes and watch the beam survive in quantum, not classical, proportions.
why spin doesn't commute with itself
A spin-½ particle measured along axis n̂ gives outcome +½ or −½. Block the −½ beam; you're left with particles in state |+n̂⟩. Now measure along axis m̂ at angle θ from n̂:
P(+m̂ | +n̂) = cos²(θ/2)
At θ = 0 (same axis): probability = 1 (all transmitted). At θ = π/2 (perpendicular): probability = 1/2 (random). At θ = π (opposite): probability = 0 (all blocked).
The non-classical part: try SG_z → block down → SG_x → block down → SG_z. After the first two, you have a clean "x-up" beam — no information about z. The third SG_z gives 50/50, even though the beam was originally "z-up" after the first magnet! The middle SG destroys the z-information that the first SG established. This is the canonical demonstration of non-commuting observables: σ_x and σ_z don't commute, so measuring one destroys information about the other.
Heisenberg's uncertainty principle for spin: ⟨σ_x²⟩ ⟨σ_z²⟩ ≥ (1/4) ⟨[σ_x, σ_z]⟩² = (1/4) ⟨2σ_y⟩². You cannot simultaneously have definite x and z components of spin — measurement collapses one or the other.
Stern and Gerlach 1922: this experiment, originally on silver atoms passing through an inhomogeneous magnetic field, was the first direct evidence of spin angular momentum quantization. They expected a continuous smear (classical magnets); they got two discrete spots. The textbook image of quantum mechanics emerging from a single experiment is this one.