Solution
= Solution
With $X=x+gt^2/2$ and $T=t$, the chain rule gives $\Phi_t=\Theta_T+gt\Theta_X$ and $\Phi_x=\Theta_X$. The drift terms cancel, so
$$
i\hbar\Theta_T=-\frac{\hbar^2}{2m}\Theta_{XX}.
$$
Applying Ehrenfest's theorem with $[X,P]=i\hbar$ and $[P,P^2]=0$ gives
$$
\boxed{\frac d{dT}\langle X\rangle=\frac1m\langle P\rangle,
\qquad \frac d{dT}\langle P\rangle=0}.
$$