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Feb
4
comment How to numerically solve a 1-d time-independent Schrodinger equation?
You must have made a mistake. I tested it and found excellent agreement. I'll add the comparison soon.
Feb
4
comment Color a single contour from a ListContourPlot3D using values from a list
If I understand it correctly, it's a duplicate of Smooth 4D (3D + color) plot from discrete points
Feb
4
comment Rotating an Interpolating Function
There's nothing wrong with your approach (with is essentially from my answer here), as you can see by calling R[funs[[1]]]. You just have to watch out whether you have an object with Head of InterpolationFunction or not. What your solver returns is an interpolating function, evaluated at a symbolic argument (not the actual function!). In rotating it, you have to replace the symbolic arguments by rotated versions. For what you're after, it may be better to change NDEigensystem by replacing u[x,y] with u in argument 2.
Feb
3
comment Phase Portrait to Differential Equation
@MichaelSeifert You're right, that doesn't seem to make sense globally. All it shows is that the solution isn't unique. You can get a differential equation, but not necessarily the one you expected...
Feb
3
comment How to numerically solve a 1-d time-independent Schrodinger equation?
The higher the excitation, the more nodes the wavefunction has. These are now squeezed into a fixed interval, so you need increased spatial resolution to reduce the error for high-lying states. Ignore the "brain" remark, it was a joke about something I don't use very often (but biologists claim everybody has one - it's just a theory, though).
Feb
2
comment How to numerically solve a 1-d time-independent Schrodinger equation?
Parallelization already happens internally, but as you can see in my updated answer, a huge speedup is possible just using the parallel processor also known as "brain"... I checked this transformation of variables for the pure Coulomb problem, too, and it works there as well.
Feb
2
comment Incorrect Left and Right Eigenvectors in Mathematica
Another way to prove that the eigenvectors have the (unconjugated) orthogonality property is to look at eq. (18) on this MathWorld page and take the determinant, showing that the diagonal of ${\mathbf X}_L{\mathbf X}_R$ is nonzero if there are no zero eigenvalues.
Feb
1
comment Incorrect Left and Right Eigenvectors in Mathematica
@DanielLichtblau But that's the norm which includes complex conjugation, so it doesn't necessarily give the correct scale to measure smallness in this case (I think)...
Feb
1
comment Incorrect Left and Right Eigenvectors in Mathematica
@DanielLichtblau I haven't looked at the example in more detail yet, but are the eigenvectors perhaps small in the sense of the maximum norm already?
Feb
1
comment How to numerically solve a 1-d time-independent Schrodinger equation?
@user21 Standing on the shoulders of giants...
Feb
1
comment Labeling solutions of an Eigenvalue equation involving Bessel functions
@Aegon Looks like the library add-on package also has trouble with duplicate roots. Maybe you can try my new approach and see if it works better now.
Feb
1
comment How to numerically solve a 1-d time-independent Schrodinger equation?
The closer to the "ionization threshold" you get, the larger the true wave functions are. I don't think any simple boundary condition can fix the resulting errors, unless the domain is made larger (which eventually makes NDEigensystem run out of time or memory). Arnoldi is (I believe) the default method, I just added it explicitly so that I can also try out different sub-methods (such as `"Shift"). With or without shift, you should of course get the same eigenvalues, but sorted differently. But different sorting (default is absolute value) means the ground state comes much later in the list.
Feb
1
comment How to numerically solve a 1-d time-independent Schrodinger equation?
I was initially confused about your use of Neumann boundary conditions. I think you must mean something else, because by comparing with your desired eigenvalues I concluded that you're really solving the radial equation for the reduced wave function, where no Neumann boundary condition is needed. The biggest problem is how to deal with the conditions of vanishing amplitude at infinity.
Feb
1
comment Incorrect Left and Right Eigenvectors in Mathematica
Contrary to what you're claiming, the eigenvalues you're comparing are not the same, so there is no contradiction.
Jan
31
comment How to numerically solve a 1-d time-independent Schrodinger equation?
Your code doesn't run because Alpha is undefined. Where did you get the values in the last table?
Jan
31
comment How to numerically solve a 1-d time-independent Schrodinger equation?
You might find this useful: Solving the Schroedinger equation for bound states with Mathematica 3.0 (arXiv link).
Jan
31
comment Non Standard Eigenfunction Plots of the Laplacian Over the Unit Square
I added another paragraph on accidental degeneracies.
Jan
31
comment Incorrect Left and Right Eigenvectors in Mathematica
Possibly related: Orthonormalization of non-hermitian matrix eigenvectors
Jan
30
comment Non Standard Eigenfunction Plots of the Laplacian Over the Unit Square
The projector in the case funs[[5]] returns numerically zero, indicating that these states don't belong to the 2D irreducible representation onto which I project. But there are other (1D) representations, each with their own projection operators. You would have to construct them using the rules of group theory and apply them to each eigenstate to isolate the symmetrized components. You'll need a character table for the group and repeat what I did above - it can be automated, but I'll have to leave that to you or someone else for now.
Jan
30
comment Non Standard Eigenfunction Plots of the Laplacian Over the Unit Square
You can do a lot by applying group theory if there are symmetries. I give an example for the square in my answer.