Single Electron Atom Wave Functions
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Visualization
There is endless variety in the ways that one can visualize these
wave functions. Here we write a few functions that automate the
process of plotting these functions.
The first function, PlotYlm,
provides a spherical polar plot of the angular part of the wave
function. In addition to the options for the ParametricPlot3D
function, PlotYlm
can take on three other options. The option Orbitals
determines whether
or the linear combination
is plotted. In the latter case the option Parity
specifies the choice in the
sign. The option ShowQuantumNumbers
determines whether or not the graphic includes information on the
angular momentum quantum numbers. The coloring of the
graphic was chosen rather arbitrarily (in the color directive ):
it is fun to experiment with other possibilities.
The function MakeYlmGraphicsArray
plots an array of angular wave functions. An example is shown after
the function definition.
Here is an example of MakeYlmGraphicsArray.
Here is a 3dimensional plot of
in the
plane.
A SurfaceGraphics
object can be turned into an equivalent ContourGraphics
object simply by wrapping it in the ContourGraphics
function.
Here we create plots of the absolute squares of the radial and
angular wave function. The display of these plots is suppressed
by using the option .
First the radial plot.
Now the angular plot.
All three plots can be shown together by using Rectangle
and GraphicsArray:
One could go on to create a function that automatically
creates a graphic of this type.
ExtendGraphics package use
Here is an
odd parity orbital:
We can alter the visual display of this with some of the functions
contained in the ExtendGraphics`
set of packages that were written by Tom WickhamJones and are described
in his book "Mathematica Graphics". The ExtendGraphics`
packages are available free of charge from MathSource.
These functions from ExtendGraphics`
alter the lighting properties.
Now redisplay the graphic. To see the effects of the lighting
modifications we need to set the Lighting
option to True
(it was set to False
within the definition of PlotYlm).
The following uses pattern matching and a replacement rule to
take this graphic and remove the surface mesh.
As we stated earlier, the possibilities for visualization are
endless. The various three dimensional graphics above can also be
animated with Mathematica (by plotting the graphics from
various ViewPoints,
or by using the standard addon package Graphics`Animation`,
or by a variety of other means). The results can be exported as
animated GIFs, and so on.
