The physical problem
Why can a measurement alter the outcome of a later measurement?
Imagine selecting the upward beam from a first analyzer oriented along z. A second analyzer oriented along x selects its upward beam; a third measures z again. What fraction of the original atoms survives all three selections? The central issue is the state prepared by each accepted outcome. We can separate that issue from both detector statistics and the fate of discarded beams.
Exact prediction and sampled outcomes
Follow the surviving beam
Exact surviving fraction after each analyzer; readout includes the original input as 100%. A bypass or nonselective measurement transmits all atoms, but only the latter changes the density operator.
State geometry
The Bloch-vector picture
Projection into the x–z plane. The y component is reported numerically; a nonselective projective measurement generally shortens the Bloch vector, indicating a mixed state.
Mathematical bridge
Projectors, probabilities, state updates
Represent a spin-½ state by ρ = (I + r·σ)/2, with |r| ≤ 1. The projector on the selected spin component along unit vector n is P± = (I ± n·σ)/2.
After a selected outcome, ρ′ = P±ρP±/p±. For this rank-one measurement, the resulting Bloch vector is ±n whenever p± > 0.
Derive the conditional and joint probabilities
For |z+〉 → |x+〉, |〈x+|z+〉|² = 1/2. After selecting x+, the state is |x+〉, so the probability of z+ is again 1/2. Multiply probabilities along the accepted branch: P(x+,z+|z+) = (1/2)(1/2) = 1/4. If the initial z+ selection itself has non-unit success probability, include its factor.
What does “ignore the measurement result” mean?
The two conditional branches are added as density operators, not as state amplitudes: ρ′ = P+ρP+ + P−ρP−. In Bloch language, r′ = (r·n)n. Total probability is preserved. This experiment differs physically from removing the intermediate device.
How is the finite sample generated?
Each incident atom is assigned outcomes by the Born probabilities conditional on the previous outcomes. At a selected analyzer, atoms in the unwanted branch are removed. Each run uses a reproducible pseudorandom seed. An ideal survival probability p gives an incident-count standard deviation √[Np(1−p)]; the simulated count is a sample, not an alternative quantum prediction.
Predict → observe → explain
Three experiments worth doing
- Erase the middle analyzer. Compare selecting x+ with bypassing x. Predict whether the last z+ analyzer transmits 50% or 100% of the beam that reaches it.
- Discard the record, retain the measurement. Change the middle station to “Ignore result”. Explain why a z+ population can fall even when no atoms are discarded there.
- Rotate the angle continuously. Start in z+ and vary the next analyzer angle θ. Verify P(+)=cos²(θ/2), including 0°, 90°, and 180°.
Check: For incident |z+〉 with x+ and final z+ selected (first z+ measurement certain), what survives?
Further exploration: replace the intermediate projective measurement by an instrument with weak information gain. That extension requires specifying the Kraus operators, beyond the projective model implemented here.
Scope, conventions and further reading
The magnetic-gradient apparatus is represented as an ideal spin-½ projective measurement. Beam trajectories, imperfect separation and decoherence dynamics inside an actual device are outside this model. Analyzer directions are restricted to the x–z plane, while the prepared spin may point anywhere on the Bloch sphere. All probabilities are dimensionless.
Read the lecture notes: Chapter 1 · Stern–Gerlach, Chapter 1 · Measurement, and Chapter 3 · Composite systems and instruments.