Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 15B (Quantum Mechanics)

The Time-dependent Schrodinger equation is $i\hslash {\partial }_t\psi =\hat{H}\psi$. Differentiate $⟨\hat{A}⟩=⟨\psi |\hat{A}|\psi ⟩$, use the equation and its adjoint, and include explicit time dependence of the operator. This gives Ehrenfest theorem

Substitution of the stated phase intocancels the potential and scalar phase terms, leaving

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

The phase has unit modulus and $X=x+g{t}^2/2$, henceThe free-particle equations imply $⟨X⟩=a+vt$. ThereforeThe wavepacket centre obeys the classical constant-acceleration trajectory, an instance of the correspondence principle.