Dwell time (quantum mechanics, ${\tau }_D={\int }_{\Omega }|\psi (x){|}^{2}dx/j_{\mathrm{in}}$)

For a stationary one-dimensional scattering state with incident probability current $j_{\mathrm{in}}>0$, the dwell time in a region $\Omega$ is its integrated probability density divided by incident flux. It measures occupancy per incoming rate, not a classical trajectory or a uniquely defined tunnelling traversal time.

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.