Scattering at a repulsive smooth potential barrier

This movie shows a wave packet with negative energy moving according to the Dirac equation. Its initial condition is roughly a Gaussian function. This wave packet hits a potential barrier that is repulsive for electrons. Interestingly, the negative energy wave packet gets attracted by the barrier. Therefore, virtually everything is transmitted, there is no visible reflected part. You can even see that the wave packet is accelerated inside the barrier. (The local wave length is shorter inside the barrier).

A wave packet with negative energy behaves like an antiparticle. The antiparticle (a positron) has the opposite charge, and hence the potential barrier acts as an attractive potential well.

The motion of the negative-energy wave packet shows another interesting fact: The momentum is opposite to the velocity. (The momentum is determined by the sequence of colors in the wave packet. The momentum always points in the direction from red to yellow to green to blue, that is, in the positive direction of the color circle).