## Scattering from potential with sine modulation

For this page, I have two movies where a packet scatters from a weak short range potential. I am doing the case where the energy is always much greater than the potential energy. In this case, you might expect that the reflection from the potential will always be small. This is not always true and one example is given here. In this case the potential will be 0.04 sin(2 x) exp[-x^2/100]. And the two energies chosen for the wave packets will be roughly 0.32 atomic units and 0.50 atomic units. In this movie the wave function for an electron at t=0 is proportional to exp{-[(x+50)/20]^2}exp(ix) which represents an electron localized at -50 atomic units of distance with a velocity of 1 atomic unit to positive x. One frame from the movie is given below. There are the (by now) familiar features of the scattering wave function: spreading and decrease in height due to the dispersion of the packet and the interference when the packet starts to collide with the potential. There are two new features. The first is that the interference pattern now extends over almost the whole range of the potential. This is different from the packets that scattered from the simple potentials of the previous examples. What is the reason for the difference? The second new feature is not visible in the movie but it is apparent when Fourier transforming the wave function: most of the scattered wave is from velocity components near v_0=1.
This can be seen by comparing this movie to a second movie where the initial velocity is dectreased to 0.8 atomic units. In this movie the wave function for an electron at t=0 is proportional to exp{-[(x+50)/20]^2}exp(ix4/5) which represents an electron localized at -50 atomic units of distance with a velocity of 4/5 atomic unit to positive x. One frame from the movie is given below. The only change from the movie above is that the energy has been lowered. You might expect that by lowering the energy you are going to have more scattering from the top of the barrier as in the movie from the previous page. This does not happen. There is almost no scattering in this case whereas there was a very noticeable amount of scattering at the higher energy. The reason was alluded to in the discussion above: it is only the velocity components near v=1 atomic unit that are being scattered. The wave packet in this movie has very little v=1 velocity components whereas the packet above is centered at v=1. Why is it that the v=1 components are the only ones being scattered?

This is a little model calculation to demonstrate Bragg scattering. When the potential energy is always small compared to the kinetic energy, the amplitude for the particle to change momentum is proportional to the Fourier component of the potential at the wave number equal to this change in momentum divided by hbar. The above potential has two sharp peaks in its Fourier transform at k=2 and -2 atomic units. The change in the electron's momentum is 2 m v when it is reflected. Using m=1 atomic unit for an electron, you should find that the electron's velocity should be roughly 1 to be strongly reflected.
One implication is that the amount of scattering can depend sensitively on the spatial width of the initial packet. As the spatial width of the packet increases, the width of the momentum distribution decreases and you can make a larger fraction of the packet have momenta that are strongly scattered.