MATSLISE: Calculating the eigenfunctions |
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The Eigenfunctions window
This window is generated when the Eigenfunction-button in the Eigenvalues-window is clicked. All the selected eigenfunctions are plotted. For a Sturm-Liouville problem first the eigenfunctions of the associated Schrödinger problem are computed. These Schrödinger eigenfunctions are then transformed into the Sturm-Liouville eigenfunctions and displayed. This (Liouville) transformation is the reason why the computation of the eigenfunctions of a Sturm-Liouville problem needs in general more time than the computation of Schrödinger eigenfunctions.
The Select All-button
This selects all the entries in the eigenfunctions-list. An entry can also be selected using the SHIFT- and CTRL-keys.The Test orthonormality-button
After the plotting it is possible to test the orthonormality of a selection of the eigenfunctions. The results of this test are then listed in the Results-box. e.g.:
<y_0,y_0> = 1
means that the first eigenfunction has
been well normalized.<y_0,y_1> = 9.23e-13
means that the integral of the
product of the first and second eigenfunction is very
small. The two eigenfunctions may be expected to be
orthogonal.Note1: For regular Sturm-Liouville problems <y_i,y_j>
is the
integral of y_i * y_j * w
. (with w = w(x)
see Sturm-Liouville problem )
Note2: The calculations are interrupted by a click on the stop-button.
Show with potential - button
With a click on the Show with potential-button the eigenfunctions-potential plot window is opened.
Show on same plot - button
With a click on this button, the selected eigenfunctions are displayed on one and the same plot.
Focus on one function - button
A new window is opened: Eigenfunction-plot window, where the selected eigenfunction is plotted.
Note: When one of the calculated eigenfunctions is not correct, i.e. when the eigenfunction corresponding to E_k
has more or less than k
zeros, an error message is displayed.
![]() | The Eigenvalues window | Eigenfunction plot window |
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