The Mathematica Journal
Volume 9, Issue 3

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T R O T T ' S C O R N E R
Michael Trott

Visualization of the Eigenfunctions

For the visualization of the eigenfunctions, we define a function eigenDVibration3DPlot.

Here we show the first nine eigenfunctions. The two lowest states appear in the two crossings. Such crossings can hold eigenstates by themselves [2].

The higher eigenfunctions oscillate more and are better visualized using a contour plot, implemented as the function eigenVibrationContourPlot. Here we use the contour values that give homogeneous contour spacing.

The next graphic shows the sum of the squares of the first 24 states. All of the lowest states are localized in the "larger crossing."

Next, we calculate some of the higher eigenfunctions near the eigenvalue . This will be a good test of the method because the function is obviously an eigenfunction for the drum and fulfills the boundary conditions trivially.

Here we show the corresponding eigenfunctions. The fifth of the selected eigenfunctions is the global product state. Overall we see that the eigenfunctions at this eigenvalue are structured along the drum spine with approximately one maxima per square and only a little structure in the local perpendicular directions.

Next, we calculate some eigenvalues and eigenfunctions at still higher eigenvalues.

This time we show the nodal curves of the eigenfunctions in addition to the absolute values of the eigenfunctions. The typical state is localized in a small part of the drum. Only the global product state extends over the whole drum.

It is interesting to look at the functions . The function grad1DValue again uses a finite difference approximation to calculate an approximate value of .

We now define a function eigenVibrationGrad3DPlot to visualize the gradients as 3D plots similar to the function eigenDVibration3DPlot.

Here are the norms values of the gradients of the lowest eigenfunctions. As is well known, the gradients of harmonic functions diverge as interior corners [3].



     
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