A silver atom is prepared in the spin state with

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 spin state ∣ψ⟩=cos⁡θ ∣↑⟩+sin⁡θ ∣↓⟩|\psi\rangle=\cos\theta\,|\uparrow\rangle+\sin\theta\,|\downarrow\rangle with θ=30∘\theta=30^\circ, where ∣↑⟩,∣↓⟩|\uparrow\rangle,|\downarrow\rangle are the S^z\hat S_z eigenstates. It enters a Stern-Gerlach apparatus whose pointer is itself treated quantum mechanically, starting in a "ready" state ∣A0⟩|A_0\rangle. Unitary evolution (P5) during the interaction is linear and acts as ∣↑⟩∣A0⟩→∣↑⟩∣A↑⟩,∣↓⟩∣A0⟩→∣↓⟩∣A↓⟩,|\uparrow\rangle|A_0\rangle \to |\uparrow\rangle|A_\uparrow\rangle, \qquad |\downarrow\rangle|A_0\rangle \to |\downarrow\rangle|A_\downarrow\rangle, where ∣A↑⟩|A_\uparrow\rangle and ∣A↓⟩|A_\downarrow\rangle are two distinguishable, orthonormal pointer states ("needle deflected up" and "needle deflected down"). ((a)) Using only the linearity of the evolution and the given action on the two basis terms, derive the joint atom-plus-pointer state immediately after the interaction, starting from the initial joint state ∣ψ⟩∣A0⟩|\psi\rangle|A_0\rangle. ((b)) The projection postulate (P4) asserts that if the pointer position were measured at this instant, one of the two outcomes ("up" or "down") would be obtained, with the joint state reducing accordingly. State the two postulates in tension here, say precisely what each predicts for the joint system immediately after the interaction, and explain why they cannot both be exact descriptions of the same physical process, given that the interaction is itself just a physical process. ((c)) Give the Born-rule probabilities, according to P3 applied to the pointer's own eigenbasis, of finding the pointer showing "up" and "down".

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