= Quantum reflection at a downward step
{title2=$E\to0^+:\quad r\to-1,\ t\to0,\ R\to1,\ T\to0$}
For $V_0<0$, $k=\sqrt{2mE}/\hbar$ tends to zero as $E\to0^+$ while $q=\sqrt{2m(E+|V_0|)}/\hbar$ stays finite. Matching gives <probability amplitudes> $r=(k-q)/(k+q)$ and $t=2k/(k+q)$, so <quantum reflection probability> $R=r^2\to1$ and <transmission probability> $T=(q/k)|t|^2\sim4k/q\to0$. The final region is classically allowed; reflection results from the abrupt spatial change in the <wavefunction>'s <wavenumber>. These are limits of positive-energy incoming states, not a flux ratio evaluated at exactly zero incident current.
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