The Pretty Picture Gallery
This image is a picture of a quantum mechanical particle scattering
off a spherical square well in three dimensions. The white areas
are regions of enhanced probability for the particle to be found.
The figure must be rotated about its axis of symmetry to get the full 3D
picture. The square well is acting like a lens - note the focus of
"light" in the lower part of the well. Note also the strong backscattered
wave.
One can think of the well as a material of a high index of refraction
if the scattered particle is thought of as a light wave.
This image is a picture of a particle scattering off of two nearby
square wells in two dimensions. The left hand well has a clear focus,
but the focus of the right hand well is substantially distorted by the
presence of the left well. The incident wave is coming from the upper
left corner.
Here is a wave hitting a shallow well in 2D. The energy of the incident
wave is much greater than the depth of the well - this means that the wavelengths
are almost the same inside and outside the well. In other words,
the index of refraction of the well is close to one. One can see
that we are approaching the classical limit (in the sense of ray paths)
here: one can almost imagine reproducing the wavefronts inside the
well using Snell's Law. The well is trying hard to be a good lens,
but it suffers from a phenomenon called "spherical aberration" - which
the Hubble Space Telescope suffered
from once upon a time. That's why the focus of this lens is so stretched
out. Perhaps in this case I should rather call it "cylindrical aberration,"
because the particle lives in two dimensions and is hitting a disc, not
a sphere!
And here is a picture of a scattering resonance in a system of 100 randomly
placed scatterers in three dimensions. Only the scattered wave is
shown, without the incident wave. It's a bit like an asymmetric Rorschach
test! Actually, this picture is more complicted than I am letting
on. It is actually a slice through a 3D vector wavefunction.
So you are looking at a single component of the scattered wave. The
full wave is really three complex numbers at every point in space.
By the way, did you know that the late Hermann Rorschach looked a lot like the modern day
movie star Brad Pitt? What is the significance of this?
One more: this is a picture of a wave from below hitting 50 scatterers arranged in a semicircle.
The scatterers are hard discs with a radius 1/50th the diameter of the semicircle. There is a nice shadow behind the "mirror."
Can you see where the focus of this mirror is? Looks like about half the
radius, just like the optics books say for a large diameter spherical mirror.
The focus is blurred by cylindrical aberration again. We could do better with a parabolic mirror...
If you look closely, you can also see the "creeping wave," or surface wave, which snakes around the backside of the mirror, and runs tangent to the surface. It is not very bright, but it is there. As it works its way around the mirror, it peels off the surface and enters the classically forbidden shadow region. (a=1,k=.25,R=100).
OK, I can't resist. One more. This is a picture of a plane wave coming from below impacting a
"bumpy mirror." There are 15
point scatterers arranged in a diagonal line. The point scatterers are scattering both s- and p-waves,
using a new method I devised for adding higher partial waves into the point scatterer model. The point
scatterers are mimicking hard discs with radius r = 1 unit, scatterer separation is 5 units,
and kr = 2, where k is the wavenumber of the incident wave.
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