Analytic solutions to the time-dependent Schrödinger equation for cutoff wave initial conditions are used to investigate the time evolution of the transmitted probability density for tunneling. For a broad range of values of the potential barrier opacity $\alpha ,$ we find that the probability density exhibits two evolving structures. One refers to the propagation of a forerunner related to a time domain resonance [Phys. Rev. A 64, 0121907 (2001)], while the other consists of a semiclassical propagating wave front. We find a regime where the forerunners are absent, corresponding to positive time delays, and show that this regime is characterized by opacities $\alpha <{\alpha }_c.$ The critical opacity ${\alpha }_c$ is derived from the analytical expression for the delay time, which reflects a link between transient effects in tunneling and the delay time.

• Received 6 May 2002

DOI:https://doi.org/10.1103/PhysRevA.66.032104