Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 15B / b / ii / Solution

With $X=x+g{t}^2/2$ and $T=t$, the chain rule gives ${\Phi }_t={\Theta }_T+gt{\Theta }_X$ and ${\Phi }_x={\Theta }_X$. The drift terms cancel, soApplying Ehrenfest's theorem with $[X,P]=i\hslash$ and $[P,P^2]=0$ gives

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