A hydrogen atom is prepared in the normalised state | = 0.5 ,|2,0,0 + 0.3 ,|2,1,0 +…
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 hydrogen atom is prepared in the normalised state∣ψ⟩=0.5∣2,0,0⟩+0.3∣2,1,0⟩+0.2∣3,0,0⟩,written in the simultaneous eigenbasis ∣n,l,m⟩ of H^, L^2 and L^z, with each ket normalised and mutually orthogonal.
((a)) The total energy is measured. Using the projector onto the degenerate eigenspace of each energy, find the probability of obtaining E2 (the n=2 level) and the probability of obtaining E3 (the n=3 level).
((b)) Suppose the energy is measured non-selectively: its result is recorded for every run, but no run is discarded. Treating each recorded outcome as a branch with its own conditional post-measurement state, compute the probability that a subsequent measurement of L^2 then yields the eigenvalue for l=0.
((c)) Compute the probability of obtaining l=0 if L^2 is instead measured directly on ∣ψ⟩, with no energy measurement first. Compare this with the answer to part (b), and explain in one or two sentences why the intermediate non-selective ensemble in part (b) cannot be written as a single ket even though the two numbers agree.