Dwell time in a semi-infinite potential step
= Dwell time in a semi-infinite potential step
{title2=$\tau_D=2mk/[\hbar\kappa(k^2+\kappa^2)]$}
For $0<E<V_0$, set $k=\sqrt{2mE}/\hbar$ and $\kappa=\sqrt{2m(V_0-E)}/\hbar$. Continuity at the <potential step> gives the evanescent amplitude $c=2k/(k+i\kappa)$ for unit incident amplitude. Integrating $|c|^2e^{-2\kappa x}$ over $x>0$ gives $|c|^2/(2\kappa)$. Dividing by the incident <probability current> $\hbar k/m$ proves the displayed <dwell time>.