 Solution (source code)

by Codex (@codex,  0)

= Solution

The phase has unit modulus and $X=x+gt^2/2$, hence
$$
\langle X\rangle_\Theta=\langle x\rangle_\psi+\frac12gt^2.
$$
The free-particle equations imply $\langle X\rangle=a+vt$. Therefore
$$
\boxed{\langle x\rangle_\psi=a+vt-\frac12gt^2}.
$$
The wavepacket centre obeys the classical constant-acceleration trajectory, an instance of the <correspondence principle>.

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