Helmholtzificator

Explore eigenfunctions of the 2D Laplacian, subject to Dirichlet boundary conditions. In addition to the Helmholtz equation, Hemholtzificator also supports similar eigenvalue problems of the form

$$ \nabla^2 f(\vec{x}) - V(\vec{x})f(\vec{x}) = \lambda f(\vec{x}) $$

This is the form of the time-independent Schrödinger equation with potential $\propto V(\vec{x})$.
^{(Want to play with something time-dependent instead? Try https://github.com/benburrill/toad_seer)}

Helmholtzificator provides an interactive matplotlib user interface with a slider to select eigenvalues.

Examples

By drawing the silhouette of a cat face in an image editor and passing it as input to helmholtzificator, we can create some immortal, time-independent Schrödinger's cats:

Here's an eigenfunction for a quartic double-well potential:

Usage

To start the interactive eigenfunction explorer, run:

$ python3 -m helmholtzificator FILE

Where FILE can be either a .png file (transparent pixels specify boundary conditions of 0, no way to specify a potential) or a .npz file (with up to two numpy arrays, boolean array "mask" is False in the boundary region and float array "potential" is $V(\vec{x})$ as described above).

In addition to the mask given by the input, the boundary condition of 0 is also enforced outside the bounds of the image/array.

All images are considered to have a pixel-centered width of 1, ie the top left pixel has coordinates (0, 0) and the top right pixel has coordinates (0, 1).

Uncertainties are estimated rather crudely, so take them with a grain of salt. They give some indication of how well the solution has converged, but I typically see more variance in eigenvalues by pressing refresh a bunch of times than can be explained by the reported uncertainties.

The resolution of your input images/arrays should be low! I recommend below 100x100 px.