A beam of silver atoms is prepared in the spin state
PHYSICS · Subjective · 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 beam of silver atoms is prepared in the spin state ∣ψ0⟩=∣↑x⟩=21(∣↑⟩+∣↓⟩), where ∣↑⟩,∣↓⟩ are the eigenstates of S^z with eigenvalues ±ℏ/2. The beam passes through three Stern-Gerlach magnets in succession, each used selectively (only the stated outcome's beam is kept and sent on to the next magnet). (a) Magnet 1 measures S^z; keep the +ℏ/2 ("up") beam. Magnet 2 measures S^n, the spin component along a direction n^ in the xz-plane at 60∘ to z, whose +ℏ/2 eigenstate is ∣↑n⟩=cos30∘∣↑⟩+sin30∘∣↓⟩; keep the +ℏ/2 beam. Magnet 3 measures S^z again; keep the +ℏ/2 beam. State the normalised state of an atom immediately after each magnet, using the projection postulate at each selective step. (b) Find the fraction of the original beam that emerges from magnet 3 in the "up" beam, i.e. the joint probability of the outcome sequence up, +, up. (c) Suppose magnet 2 is removed, so magnets 1 and 3 (unchanged, both measuring S^z, both selecting "up") are adjacent. Find the fraction of the beam surviving magnet 3 in this case, and identify which magnet's projection is responsible for the difference between this fraction and the one found in part (b).