Schrödinger's equation in a pentagon (rotating 3D view)
Channel: Nils Berglund
Date: 28/06/2022

This simulation is a 3D rendering of a solution to Schrödinger's equation in a regular pentagon. Unlike the equation in a regular hexagon, recently featured in the video https://youtu.be/dEFHatRpcYc , there is no simple expression for the resonant modes/eigenfunctions of a pentagonal domain. Actually, not much seems to be known on resonant frequencies and modes of regular polygons other than the equilateral triangle, the square and the regular hexagon. In the video with the hexagon, the question arose whether the resonant modes of the pentagon would show some features related to quasicrystals. I would not claim this simulation is conclusive in this respect, but I would not be surprised if the golden ratio plays a role in its resonant modes.
The simulation shows the same evolution with two different representations:
Real part: 0:00
Phase: 1:06
In the first part, both the z-coordinate and the color hue represent the real part of the wave function. In the second part, the z-coordinate and the luminosity represent the modulus squared of the wave function, which gives the probability density of finding the particle at any given point. The color hue shows the phase of the wave function. In other words, if psi(t,x) denotes the complex wave function, the z-coordinate and luminosity depend in |psi(t,x)|^2, while the hue depends on the angle phi such that psi(t,x) = |psi(t,x)|*exp(i*phi(t,x)).  The initial state is a Gaussian wave packet directed at one corner of the hexagon. It is interesting to compare the quantum evolution near the corners to the classical one, shown for instance in the videos https://youtu.be/q1Rmjhp0UcQ and https://youtu.be/GkCYmRX8hCI

Update 30 December, 2022: Bob Jones has an impressive list of eigenpairs of the Laplacian in a regular pentagon, see https://www.hbelabs.com/sweep/index.html and in particular https://www.hbelabs.com/helmholtz/pentagon.html for plots of many eigenfunctions.

Render time: 40 minutes 17 seconds
Color scheme: Part 1 -  Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
https://github.com/BIDS/colormap
Part 2 - Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "You're No Help" by Silent Partner

See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html for more explanations (in French) on  a few previous simulations of wave equations. 

The simulation solves the Schrödinger equation by discretization. The algorithm is adapted from the paper https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf 
C code: https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Many thanks to my colleague Marco Mancini for helping me to accelerate my code!