This experiment provides a
beautiful demonstration of electron's wave-nature and absolutely requires the Schrodinger
equation for its explanation. In short, we find that for certain energies -- energies
where the electron's wavelength is equal to the size of the atom -- electrons will pass
right through a class of symmetrical atoms. The effect may only be explained
using quantum mechanics and is a good "mental model" to have as you proceed to
advanced courses.We set up a target area, filled with xenon atoms, and shoot low speed
(1-12 eV) electrons through it at a collecting plate. Some of the electrons go
straight through to the plate; that is our "plate current." Some of the
electrons collide with the xenon atoms and are scattered into a shield which surrounds the
whole device; these scattered electrons form the "shield current."
By comparing the shield current and the plate current, we would like to figure out the scattering
cross section: how solid a xenon atom appears to the free electron. For
instance, it is easier for an energetic electron to "punch through" an atom than
for a pokey one. Indeed, classically this is the only variation in the cross
section, and a graph of cross section versus energy appears as in the figure at right. (Under
contruction -- ed.) Unfortunately, we cannot separate the
geometry of the device -- the shield and the plate have different shapes and sizes -- from
the variation in cross section.
However, we can remove all of the xenon atoms from our target field, by freezing them
out. Then, we can compare the currents with xenon present to the current with xenon
absent, and notice a startling, amazing fact: when the electron is the same size as
the xenon atom, it goes straight through the xenon!
In fact, we can predict this effect with the Schrodinger equation (and only with the
Schrodinger equation). This formula gives the propagation of a particle -- in this
case our free electron -- through space in terms of the particle's energy and the
potential energy of each point in space. It does so by describing the evolution of
the probability wave function of the electron. This formula is very deep,
very fundamental, and you will spend your next year's worth of quantum mechanics exploring
it.
Now, the electron is at a lower energy when it is near the xenon atom (because of the
attraction of the nucleus), and so the xenon atom is an area of lower potential energy.
A (very) crude model of the xenon's effect on the electron is that of a square well, at
right. (Under contruction -- ed.) Of course,
the xenon is fuzzy (not discontinuous), has a positive nucleus, and lives in three
dimensions (among many other details) but we are just going to say it is a one-dimensional
square hole in the ground. Using advanced methods, one may show that our answer is
basically right anyway.
When an electron wave hits this well, part will be reflected and part transmitted.
Recall the similar example, from mechanics, of a wave on a string. When a
travelling wave hits a portion of the string with a heavier section tied in, it will be
partially reflected at both the initial and final boundaries. In our case, when the
electron wave hits the well, the first transition will give a reflected wave out of phase,
and the second transistion will give a reflected wave in phase with the incident wave (the
figure at right is clearer). Of course, again we are making many approximations: the
electron will have a lower wavelength within the well, and the total intensity of the
transmitted and reflected waves must equal that of the incoming wave. We need the
Schrodinger equation to describe these effects, and you may derive them if you are daring.
The important point, however, is that when the well is equal to one half the electron
wavelength, the two reflected waves add out of phase. The reflected wave has
amplitude zero: there is no probability of reflection: there is no reflected wave.
Where does the reflected wave go? With the correct solution, we find that the transmitted
wave has amplitude one; it is certain to go straight through. That is good: our
probabilities balance out to one. ("Something" is certain to happen!)
