Scattering from a finite potential barrier

For this page, I have two movies where a packet scatters from a short range potential barrier. In both movies the potential is V(x)=exp[-x^4]. As in the movie for the downstepping potential there is a discontinuous potential (V(x)=1 atomic unit for |x|<1 atomic unit and zero otherwise) for which you can obtain analytic wave functions and get qualitatively similar results to those seen below. The first movie is for the case where the classical electron does not have enough energy to go over the barrier whereas the electron does have enough energy to go over the barrier in the second movie. Thus, the classical expectation would be complete reflection for the first movie and complete transmission for the second movie.

In this movie the wave function for an electron at t=0 is proportional to exp{-[(x+50)/10]^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 a number of interesting features that have been seen in the previous movies. The wave packet initially spreads and its height decreases due to dispersion. When the packet hits the potential at x of roughly 0, an interference pattern develops (but why does the pattern only develop for negative x). The important new feature is that the packet splits into two, with one part reflected back to negative x and a part transmitted through the barrier to positive x. You do not see 100% reflection of the wave packet due to quantum mechanical tunneling through the barrier. If the barrier were to extend to +infinity by changing the barrier to V(x)=exp(-x^4) for x<0 and V(x)=1 for x>0, then you would see 100% reflection. One of the features that I don't show is that the probability for tunneling through the barrier decreases as the initial velocity decreases; for a short range barrier (defined to be a barrier that goes to zero at large |x| faster than 1/x^2) the tunneling probability goes to a nonzero constant as E->0; the tunneling probability goes to 0 as E->0 for a potential barrier that goes to zero at large |x| more slowly than 1/x^2. For example, the tunneling probability for the potential 1/(x^2 + b^2) [with b a real constant] goes to zero as E->0.
There is an anology to this quantum mechanics case from simple optics. If you have light going from glass into air you can get total internal reflection if the angle is large enough. If you have two flat pieces of glass separated by a small gap of air, you can get the light to "tunnel" from one piece of glass into the other even though it should be totally internal reflected. This is because contrary to the geometrical optics formulation of light there is some light intensity that extends from the glass into the air even when the light is internally reflected (I believe this is called an evanescent wave). This evanescent wave can then reach into the second piece of glass after which it travels in the original direction, thus giving a small amount of transmitted light. As in the quantum mechanical tunneling, the transmission probability decreases exponentially with the distance the light needs to tunnel.

In this movie the wave function for an electron at t=0 is proportional to exp{-[(x+50)/10]^2}exp(ix3/2) which represents an electron localized at -50 atomic units of distance with a velocity of 3/2 atomic unit to positive x. One frame from the movie is given below.

The only difference between this movie and the one above is that the electron now has enough energy to travel over the top of the barrier. It is fascinating that almost half of the probability distribution is reflected. These two movies show that quantum mechanical particles can tunnel through barriers they do not have enough energy to surmount and also they can be reflected from barriers they should be able to go over. If you analytically solve for the wave function when there is a rectangular barrier, you will find similar behavior. As in the previous movie on scattering from a down stepping potential, the rectangular barrier does have some pathologies not observed for the smooth potential.
You might wonder how the tunneling probability in the case above depends on hbar as hbar->0 and on how the reflection probability for this case depends on hbar in the same limit. The tunneling probability when there is insufficient energy to go over the barrier tends to be proportional to exp(-S/h) where S depends on the potential and on the energy. The reflection probability when there is sufficient energy to go over the barrier roughly depends on the Fourier transform of the potential as [V(p/h)}^2. For infinitely differentiable potentials, V(p/h)->0 as h/p->0 faster than any power of h/p.

Any critical reader of the above discussion probably noted that there was a simplification that affects the discussion somewhat. I treated the energy as if it was v_0^2/2 and did not take into account that there is a velocity distribution due to the finite extent of the initial packet. The normalized distribution is sqrt(50/pi)exp[-(v-v_0)^2 50]. The velocity that gives an electron right at the top of the barrier is sqrt(2). Using this you find that in the first case, only 0.0017% of the probability is for energies above the barrier whereas in the second case 19.5% of the probability is for energies below the barrier. Thus the classical expectation is 0.0017% transmission for the first case and 19.5% reflection for the second case.