## 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.

1.2
Mb MPEG movie of electron hitting infinitely hard wall

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.