A particle with orbital angular momentum quantum number is prepared in the state | =…
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 particle with orbital angular momentum quantum number l=1 is prepared in the state
∣ψ⟩=31(∣1,−1⟩+∣1,0⟩+∣1,1⟩),
a superposition of the three L^z eigenstates with eigenvalues −ℏ,0,+ℏ. It interacts with a three-position dial (a macroscopic pointer with three distinguishable resting positions), initially in a "ready" state ∣Dr⟩. The interaction is unitary and linear, and acts on each basis term as
∣1,m⟩∣Dr⟩→∣1,m⟩∣Dm⟩,m=−1,0,1,
where ∣D−1⟩,∣D0⟩,∣D1⟩ are three mutually orthonormal dial states, the three readings.
((a)) Using linearity alone, derive the joint particle-plus-dial state immediately after the interaction.
((b)) Give the Born-rule probability, according to P3, that the particle alone would be found with Lz=+ℏ if measured directly before any interaction with the dial, and compare it with the probability, according to P3 applied to the dial's own eigenbasis after the interaction, of finding the dial pointing to the position labelled m=+1.
((c)) State exactly which two postulates are in conflict for this three-outcome apparatus, and explain why enlarging the number of outcomes from two (as for a spin pointer) to three changes nothing about the structure of the conflict.