Transient effects and delay time in the dynamics of resonant tunneling

We consider an analytic solution of the time-dependent Schrödinger equation with the initial condition ψ(x,0)=exp(ikx) along -∞<x<0 to investigate the time evolution for x>0 of the wave function in a double-barrier resonant structure at resonance. For typical parameters of the structure we find that the single-resonance approximation is valid from a few tenths of the corresponding lifetime onward. Very short times require the contribution of many far away resonances. The buildup time along the internal region takes a few lifetimes. At birth the transmitted wave front is blurred; however, for long times it becomes well defined and moves with classical velocity yielding a delay time τ≈2ℏ/Γ as in the stationary-phase treatment.