A particle in a three-dimensional isotropic harmonic oscillator of frequency has…
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Density matrices, open systems, entanglement and interpretation
The Structure of Quantum Mechanics, Ch The postulates, measurement and collapse
Problem
A particle in a three-dimensional isotropic harmonic oscillator of frequency ω has energy levels EN=(N+3/2)ℏω, with simultaneous eigenstates ∣N,l,m⟩ of H^, L^2 and L^z; for N=0 the only state is l=0,m=0, while for N=2 the allowed values are l=0 (m=0) and l=2 (m=−2,…,2). The particle is prepared in the normalised state∣ϕ⟩=0.36∣0,0,0⟩+0.16∣2,0,0⟩+0.48∣2,2,0⟩.
((a)) The total energy is measured. Find the probability of obtaining E0 and the probability of obtaining E2, using the projector onto each degenerate energy eigenspace.
((b)) The energy is measured non-selectively: its result is recorded for every run, but no run is discarded. Compute the probability that a subsequent measurement of L^2 then gives the eigenvalue for l=0, by weighting each branch's conditional l=0 probability by its energy probability from part (a).
((c)) Compute the probability of l=0 if L^2 is instead measured directly on ∣ϕ⟩, with no prior energy measurement, and state briefly why the two answers in (b) and (c) can agree numerically even though the object produced by the non-selective measurement in (b) is not a single ket.