PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics
The Structure of Quantum Mechanics, Ch The postulates, measurement and collapse
Problem
A silver atom is prepared in the state ∣↑⟩, the +ℏ/2 eigenstate of S^z. It passes three stages in succession. Stage 1 measures S^x non-selectively: a detector on each output beam records which way each atom went, and neither beam is blocked. Stage 2 measures S^z selectively; only the +ℏ/2 ("up") beam is kept. Stage 3 measures S^n selectively, where n^ lies in the xz-plane at 60∘ to z with +ℏ/2 eigenstate ∣↑n⟩=cos30∘∣↑⟩+sin30∘∣↓⟩; only the +ℏ/2 beam is kept.
((a)) Using the non-selective measurement rule (weight the probability of each later outcome by the Born probability of each recorded stage-1 outcome, applying the projection postulate for that outcome along the way), find the probability that stage 2 gives "up".
((b)) Given that stage 2 gives "up", state the normalised post-measurement state entering stage 3, and find the probability that stage 3 gives its +ℏ/2 outcome.
((c)) Find the overall probability of the recorded sequence "up at stage 2, +ℏ/2 at stage 3". Then explain, referring to the projection postulate, why the probability found in part (a) differs from the probability of getting "up" at stage 2 if stage 1 were omitted entirely.