MATSLISE: Schrödinger    

The File-Menu

New Problem

Closes the current Schrödinger problem and opens a new problem: a new Schrödinger problem, a Sturm-Liouville problem or a distorted Coulomb problem

Open Problem

Displays a dialog box that enables you to retrieve a problem.

Some (test)problems are predefined in the directory predefined_problems.

Save Problem

Allows you to save the problem you defined. This saved problem can then be re-opened later. All problems should be saved in a .mat-file.

Please note that information about the problem can be entered in the text-field at the bottom of the input window. This information is also saved with the rest of the problem.


The Options-Menu

Show partition

When this option is checked, an additional plot is produced by the construct-button. On this plot the partition is displayed. An example: The Mathieu potential with tol = 1e-8.

The dotted line represents the potential function and the full blue line the corresponding reference potential. This reference potential function is piecewise constant and has a jump in every meshpoint. The x-values of the blue points are the meshpoints of the partition. The number of meshpoints in the partition is remarkably small.

Half-range reduction

Half-range reduction is useful for symmetric Schrödinger or Sturm-Liouville problems. A Schrödinger problem is symmetric when the problem is posed on the interval -b to b, where b may be inf, and the potential function is even (V(x)=V(-x)) and the boundary conditions are similarly symmetric, which means that

A0 = A1,     B0 = -B1.

In this case the eigenfunctions belonging to eigenvalue E_k (k=0,1,...) are even or odd functions according as k is even or odd. Hence, the eigenvalues can be obtained by solving the given equation, but on the interval [0,b], with the given boundary condition at b and with

y'(0) = 0 to get the even eigenvalues
y(0) = 0 to get the odd eigenvalues.

The normalized eigenfunctions of the full-range problem are reconstructed from those of the half-range problem by extending in the appropriate way and dividing by sqrt(2).

For symmetric double well problems, this reduction may make the difference between a highly ill-conditioned problem and a perfectly straightforward one (See e.g. The close-eigenvalues problem). Such problems often have their lower eigenvalues occurring in close pairs or large clusters: solving on the half-range makes the problem more tractable.

When the half-range reduction option is checked, half-range reduction is automatically applied.

Reference: J. D. Pryce, Numerical Solution of Sturm-Liouville Problems (Oxford Univ. Press, Oxford, 1993), §2.5.2.

   The construct-button  Sturm-Liouville