## Electron hitting infinitely hard wall

One of the simplest time dependent quantum systems is the motion of a free particle wave packet. This is covered in many (most?) quantum mechanics text books. I will assume that you have enough gumption to visualize the motion of this packet on your own. The next most complicated case is the motion of a free particle except with an infinite wall at x=0. A time dependent wave function for this case can be obtained from the wave function for the free particle. Suppose y(x,t) is a free particle wave packet (the exact solution of Schrodinger's equation with zero potential). Then psi(x,t)=y(x,t)-y(-x,t) is the free particle wave packet when there is an infinite wall at x=0. Hints for deriving this result are given at the bottom of this page.
In this movie the wave function for an electron at t=0 is proportional to exp{-[(x-100)/10]^2}exp(-ix) which represents a particle localized at 100 a.u. of distance with a velocity of 1 a.u. to negative x. One frame from the movie is given below. As you might be able to guess from the labels, this is a movie of the electron probability density as a function of position. There are a couple of features to notice about how the packet moves. The electron density is a perfectly smooth Gaussian until there is nonzero probability for reaching the wall. Once probability reaches the wall and starts reflecting, an oscillatory structure emerges; this structure can be thought of as the interference between the part of the wave function still travelling to negative x and the part of the wave function that has reflected from the wall and is now travelling to positive x. You should also notice in the movie that the packet immediately starts getting wider and shorter: this effect is from dispersion of the packet because the higher energy components of the packet move faster than the lower energy components. A subtle effect that is difficult to see in this movie is that the oscillitory structure has a shorter wavelength when the packet is just starting to hit the wall than when the packet is almost finished colliding with the wall; this is due to the same dispersion mechanism: the higher energy components (short wave length) reach the wall first and the lower energy components (longer wave length) reach the wall later. Finally, notice that after the collision with the wall is completed the packet is again a perfectly smooth Gaussian. Why? To show that psi(x,t)=y(x,t)-y(-x,t)  is a solution of the Schrodinger equation with zero potential but an infinite wall at x=0 if y(x,t) is a solution with zero potential: first show that y(-x,t) is a solution of the zero potential Schrodinger equation if y(x,t) is a solution, then show that the combination y(x,t)-y(-x,t) is zero at x=0 for all times, and the rest is left to you.