A particle's orbital state is | = 12|0,0 + 12|1,-1 + 12|1,0 + 12|1,1
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Symmetry, angular momentum and spin
The Structure of Quantum Mechanics, Ch Compatible observables and uncertainty as a theorem
Problem
A particle's orbital state is ∣ψ⟩=21∣0,0⟩+21∣1,−1⟩+21∣1,0⟩+21∣1,1⟩, written in the usual ∣l,ml⟩ basis of simultaneous eigenstates of L^2 and L^z, with L^2∣l,ml⟩=l(l+1)ℏ2∣l,ml⟩ and L^z∣l,ml⟩=mlℏ∣l,ml⟩. Two sharp, ideal measurements are to be made on this particle, one of L^2 and one of L^z, each followed immediately by the projective collapse of the reduction postulate.
((a)) Using the degenerate-eigenspace projectors for L^2 and for L^z, compute the joint probability of obtaining L^2=2ℏ2 and then, on the collapsed state, L^z=0, if L^2 is measured first.
((b)) Compute the joint probability of the same pair of outcomes, L^z=0 then L^2=2ℏ2, if L^z is measured first instead.
((c)) State the property of L^2 and L^z that guarantees these two numbers had to agree, without redoing the calculation.