MATSLISE: Schrödinger |
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The File-Menu
Closes the current Schrödinger problem and opens a new problem: a new Schrödinger problem, a Sturm-Liouville problem or a distorted Coulomb problem
Open ProblemDisplays a dialog box that enables you to retrieve a problem.
Some (test)problems are predefined in the directory
predefined_problems
.
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 partitionWhen 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
.
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 eigenvaluesy(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 |
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