A quantum parabolic resonator
Channel: Nils Berglund
Date: 2021-09-09

This #short simulation shows a solution of Schrödinger's equation in a parabolic resonator similar to the one in the video https://youtu.be/n19XjuK_Dgs The boundary is made of two parabolas that share the same focal point. The initial state is a Gaussian wave packet representing a particle moving to the right. The luminosity indicates the modulus squared of the wave function, representing the probability of finding the particle at different places, while the hue indicates the phase/argument of the wave function. Since mixing all colors of the rainbow in equal proportions produces white, the colors often appear quite faded. 
For classical wave equations, parabolas have the property of transforming a planar wave into a circular wave converging on the focal point, and vice versa. By contrast with the classical linear wave equation, Schrödinger's equation is dispersive, meaning that different wave lengths travel at different speeds. Therefore, one cannot expect such good focusing properties, but still, the animation shows a relatively stable oscillation along the symmetry axis of the cavity. See https://youtu.be/9Hnvd0ZAlPo for comparison in a different cavity. 

Render time: 3 hours 47 minutes
Yes, I know this is ridiculously longer than comparable wave videos. The reason is that I have to use a much smaller time step, since otherwise a weird instability makes the phase completely chaotic after a while. In many other Schrödinger equation simulations, I used half the resolution, which decreases the computation time by a factor 16 (a factor 4 because of the larger spatial discretization, and another factor 4 because one can increase the time step). 

Music:  Heavy Interlude by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 licence. https://creativecommons.org/licenses/by/4.0/
Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100515
Artist: http://incompetech.com/

The simulation solves the Schrödinger equation by discretization. 
The C code is available on 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!

You may be interested in some science outreach articles: https://images.math.cnrs.fr/_Berglund-Nils-1343_.html