PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Density matrices, open systems, entanglement and interpretation
The Structure of Quantum Mechanics, Ch The wave function is not the state
Problem
A beam of spin-1/2 particles is prepared, atom by atom, in ∣ψ⟩=21(∣↑⟩+eiφ∣↓⟩) with an unknown relative phase φ. Use ∣+x⟩=21(∣↑⟩+∣↓⟩) and ∣+y⟩=21(∣↑⟩+i∣↓⟩).
((a)) Find P(Sx=+ℏ/2) and P(Sy=+ℏ/2) as functions of φ.
((b)) Long runs give the fraction 0.250 of +ℏ/2 outcomes along x and 0.933 along y. Find φ in [0,2π), and explain why the x run alone would not fix it.
((c)) Show that no value of φ gives 21 along both x and y, and hence that one of these two magnets always tells this beam from a fifty-fifty mixture of ∣↑⟩ and ∣↓⟩.