Two preparations of a beam of spin-1/2 particles are compared
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
Two preparations of a beam of spin-1/2 particles are compared, both giving identical statistics (50%, 50%) for a Stern-Gerlach magnet along z: preparation A is the pure superposition ∣ψ⟩=21(∣↑⟩+∣↓⟩); preparation B is a statistical mixture, half the particles in ∣↑⟩ and half in ∣↓⟩. A magnet is now set at angle θ from the z axis, in the plane containing z and x, with eigenstate for the +ℏ/2 outcome ∣+n^⟩=cos(θ/2)∣↑⟩+sin(θ/2)∣↓⟩.
((a)) Predict P(+ℏ/2) for preparation A at θ=60∘.
((b)) Predict P(+ℏ/2) for preparation B at the same angle, using the fact that a mixture's probability is the weighted average of the probabilities for its two components.
((c)) State whether this magnet orientation can distinguish A from B, and if so which preparation gives the larger fraction of +ℏ/2 outcomes.