Scattering from a DOWN stepping potential

Now I will present a few movies that demonstrate some aspects of scattering. The development of theoretical and computational tools to describe the scattering of particles has been very important for the advancement of many fields of physics. The reason is that our information about small systems often only derives from the analysis of how a particle scatters from the small system. The importance of understanding scattering processes is usually not reflected in standard quantum text books; this is possibly due to the relatively few cases where analytic solutions can be obtained.
Although scattering type situations are neglected in many courses, the qualitative behavior of wave packets scattering from short ranged disturbances can often be understood quite easily. In this movie, I show how an electron scatters from a potential that decreases over a small range in x in the direction the electron is travelling. Classically there would be no scattering in this situation; the electron would only accelerate in the direction it is moving. I've chosen the potential to have the form V(x)=1/(1+exp[x^3]) to be able to obtain numerical results in an efficient manner. [Almost all of the qualitative results can be obtained using the potential V(x)=1 atomic unit for x<0 and V(x)=0 for x>0; for this potential, you can obtain the exact wave function by forcing the wave function and its first derivative to be continuous at x=0.] In this movie the wave function for an electron at t=0 is proportional to exp{-[(x+50)/10]^2}exp(ix/2) which represents a particle localized at -50 atomic units of distance with a velocity of 1/2 atomic unit to positive x. One frame from the movie is given below.


1.2 Mb MPEG movie of an electron moving in a down stepping potential

Before focussing on the new features in this movie, I want to first point out that while the packet moves from x=-50 atomic units to 0 it spreads and decreases in height due to dispersion as in the movie of the free electron hitting the infinite wall. There is another feature similar to one in the free electron movie: there is an interference pattern that makes the probability density oscillate in x when the packet is near 0. You can see it in the GIF image above. But there is a peculiarity: the interference pattern is only at negative x not at positive x (examine the image above). Why?
There are some features in this movie that arise purely from quantum mechanics. Perhaps the most important is that the packet splits in two at x=0 with a small part moving to negative x. Classically the whole packet should move to positive x. For the step potential [V(x)=1 atomic unit for x<0 and V(x)=0 for x>0], you can show that the reflection probability goes to 1 as the electron's initial velocity goes to 0! Although classically nonintuitive, it is a generic property of waves to have a large amount of reflection when there is a large fractional change in wave length when moving from one medium to another. This generic property is independent of whether the wave length increases or decreases when changing media. There are some features in this movie that can be understood from analogy with classical mechanics. The packet at positive x has a larger speed than the packet at negative x because its kinetic energy is greater. When the packet moves from negative x to positive x, its width increases by a factor (roughly) of the ratio of the velocities in the two regions. Why?
There is one important difference between the results from the smooth potential I used in the simulation and the step potential for which you can obtain analytic results. The reflection and transmission probabilities for the step potential only depends on the ratio of momenta in the two regions (the exact transmission probability equals 4 r^2/(1+r)^2 where r is the ratio of the momentum for negative x to the momentum at positive x); thus, the hbar->0 operation does not give the classical result of 100% transmission. However, for the smooth potential you can show that the reflection goes to zero as hbar->0. The reason for the difference is that for the step function potential the potential always changes rapidly compared to the wavelength (h/p) whereas the smooth potential does not change rapidly with wavelength in the h->0 limit. This is only one of the many odd situations that can arise when you use discontinuous potentials; so be cautious when interpreting results from simple, but discontinuous, potentials.