A graduate student with a ruler is assigned to each bottle, and at a signal, they measure the positions of their respective particles. We then construct a histogram of the results, which should match |Ψ|², and compute the average, which should agree with ⟨x⟩. (Of course, since we're only using a finite sample, we can't expect perfect agreement, but the more bottles we use, the closer we ought to come.) In short, the expectation value is the average of repeated measurements on an ensemble of identically prepared systems, not the average of repeated measurements on one and the same system. Now, as time goes on, ⟨x⟩ will change (because of the time dependence of Ψ) and we might be interested in knowing how fast it moves. Referring to Equations 1.25 and 1.28, we see that d⟨x⟩/dt = ∫ x ∂/∂t |Ψ|² dx = iħ/2m ∫ x ∂/∂x (Ψ* ∂Ψ/∂x - ∂Ψ*/∂x Ψ) dx. To keep things from getting too cluttered, I'll suppress the limits of integration.