= Half revival in an infinite square well
{title2=$t_*=2ma^2/(\pi\hbar),\quad U(t_*)=-R_a$}
For the <infinite square well> of length $a$, the <energy eigenvalues> are $E_n=n^2E_1$, with $E_1=\pi^2\hbar^2/(2ma^2)$. At $t_*=\pi\hbar/E_1$, the <dynamical phase> is $e^{-iE_nt_*/\hbar}=(-1)^n$. Since the <midpoint parity operator> has eigenvalues $(-1)^{n+1}$ in the same complete <orthonormal basis>, the <time-evolution operator> is exactly $U(t_*)=-R_a$ on every state. At $2t_*$ every phase is one, giving a full revival.
Thus a state becomes its reflected copy, up to a <global phase>, at the half revival. Its position <probability density> is reflected, while its mean momentum changes sign because $PR_a=-R_aP$. For an initially reflection-symmetric <probability density>, the initial and half-revival position distributions agree even when their <probability currents> point in opposite directions.
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