Gregor Schiwietz
Time-dependent Schrödinger equation:
Numerical solution of the one-dimensional time-dependent Schrödinger equation on a grid. The initial state of the electron is a Gauss wave packet that moves to the right and interacts with a potential barrier. The barrier height is 0.2 a.u. (blue box) and the mean initial momentum of the electron is 0.5 a.u. The main part of the wave packet is reflected at the barrier and only a minor part is tunneling through the barrier. The last picture frame of this animation shows the tunneled fraction on the right-hand side of the barrier in comparison with an undisturbed wavepacket (dotted curve).
It is seen that the position of the pulse maximum of the tunneled fraction appears to move faster than the maximum of an undisturbed wavepacket. This should not be misunderstood as an increased signal-transport velocity. In fact, the barrier acts as a filter, selecting high velocity components of the rising edge for the tunneled fraction of the wave packet. These components consist even to a large extend of electron momenta above the classical threshold. This can directly be seen from the Fourier transform of the resulting wavefunction (kcut is the classical threshold in the figure below).
The blue curve shows the momentum distribution of the backscattered wavepacket as well as a small fraction inside the barrier at k = 0. The small contribution at positive k-values (red curve) corresponds to the transmitted (tunneled) fraction.
The black dotted curve at positive k-values corresponds to the initial and also to the undisturbed Gauss distribution. In coordinate space as well as in momentum space the transmitted wavepattern is enveloped by the undisturbed pulse and the modification by the barrier is consistent with the action of a simple high-pass filter.
