Dwell time in a semi-infinite potential step (${\tau }_D=2mk/[\hslash \kappa (k^2+{\kappa }^2)]$)

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