Delay Time and the Hartman Effect in Quantum Tunneling

A general relation between the group delay and the dwell time is derived for quantum tunneling. It is shown that the group delay is equal to the dwell time plus a self-interference delay. The Hartman effect in quantum tunneling is explained on the basis of saturation of the integrated probability density (or number of particles) under the barrier.

Figure 1

Schematic of the barrier tunneling problem.

Figure 2

Transmission of a rectangular potential barrier of height $V_0$ versus energy $E/V_0$. Here $\sqrt{2m{V}_0}L/\hslash \equiv \gamma L=3\pi$.

Figure 3

Dwell time (dashed line), self-interference delay (dotted line), and phase time (solid line) versus normalized energy $E/V_0$ for a particle in a rectangular potential barrier. Here $\sqrt{2m{V}_0}L/\hslash \equiv \gamma L=3\pi$. The times are normalized by ${\tau }_0=L/v_0$, where $v_0=\hslash \gamma /m$.