A scarred solution of Schrödinger's equation in a stadium, rendered in 3D
Channel: Nils Berglund
Date: 2022-05-15

This three-dimensional analog of the videos https://youtu.be/32ry0ZtSOvI and https://youtu.be/ivYNYnjFJ3c shows the evolution of a quantum particle in a stadium shaped domain, that is, a rectangle two sides of which have been extended by semi-circles. In the classical case, though it is globally chaotic, the billiard in a stadium admits trajectories that bounce back and forth between opposite sides of the rectangle. These trajectories are not stable, but they are not strongly unstable either, in the sense that trajectories starting with a slightly wrong angle will diverge proportionally to time from the bouncing orbits, as opposed to exponentially fast. As a consequence, the wave function of the quantum system tends to remain concentrated for some time near the classical bouncing orbits, forming so-called "scars". 
This simulation shows the same solution of Schrödinger's equation with two different color schemes: 
Probability density: 0:00
Phase: 2:02
In the first part, both the z-coordinate and the color hue represent the modulus of the wave function, which gives the probability density of finding the particle at any given point. In the second part, the z-coordinate and the luminosity depend on the modulus of the wave function, while the color hue depends on the phase, or argument, 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 boundary conditions in the simulation are periodic, which causes some interference as the wave packets approach the boundary. Absorbing boundary conditions for the Schrödinger equation are more difficult to implement for the wave equation, as they would involve some non-local operators. The small ripples appearing near the end of the simulation are likely due to a numerical instability. 

Render time: 1 hour 26 minutes
Color scheme: Part 1 -  Inferno by Nathaniel J. Smith and Stefan van der Walt
https://github.com/BIDS/colormap
Part 2 - Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Parzival" by Will Rosati@willrosati353 

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!