This applet was generated from our ancient research programs (written in
fortran - of course) which we use to study electron transport properties
in small geometries.
This applet shows you a computer simulation of the time development
of the wavefunction of a particle moving in an arbitrary potential.
When the applet starts, you should be seeing the potential
corresponding to a barrier at the bottom of the page. Above that,
you will be seeing a Gaussian wave-packet (real part in green, imaginary in
blue, and amplitude in red) that is moving in this potential.
At the very top, probability current towards the right is plotted.
(The brown line indicates the zero level.)
You can modify the potential by moving the cursor to the area
where the potential has been plotted at the bottom, and "drawing" in
your modifications with the left mouse button down. (The reset
button will recover the original potential.)
You also have the choice of changing the type of the wave to one that is
being injected from the left, by clicking
on the choices below.
Things to watch:
The wave-packet:
At early times, it is easy to see that the speed with which the
envelope moves (group velocity) is different from the speed
with which the wave moves (the phase velocity).
Note the large oscillations between the barriers as the wave
passes through. The wave is near resonance, so quite a bit of
it passes through.
The part of the wave that does pass through is relatively
"monochromatic", i.e. it has a constant wavelength, due to
the filtering effect of the resonant structure. The reflected
part has a complicated distribution of wavelengths.
The part that passes through moves to the right, and encounters
little interference. The reflected wave however, does have
considerable interference.
At larger times, the wave tends to "hang", with a very slow
variation. That is because the shorter wavelength (high energy)
components have left the picture, and only the slower components
remain.
The injected wave:
Note again the large oscillations between the barriers as the wave
passes through.
Once the wave has "stabilised", note the interference on the
left due to the incoming and the reflected waves. The wave on the
right in contrast experiences no such interference. (Compare envelopes.)
Note also how the probability current eventually becomes uniform, indicating a steady
state. It is this limit that is analysed using the time-independent
Schrödinger Equation.
Modify the potential. Plot in your favorite potential.
Some things you can try are :
Put a barrier in front of the wave. Notice how the wavelength
decreases whenever the wave "slows down" at larger values of
the potential.
Put in quantum wells and try to construct resonant structures.
(You should adjust the length of the features to "fit" a multiple
of the wavelength of the wave.)
