Colliding 1D fermions with different speeds, rendered in 3D
Channel: Nils Berglund
Date: 10/08/2022

This simulation is the fermion equivalent of the simulation https://youtu.be/w6FzHuziJvg , using similar initial states, except for the symmetry. 
In quantum mechanics, the state of two 1D particles is described by a wave function Psi(x,y), whose modulus squared gives the probability of finding one particle at x, and the other one at y. Fermions are particles for which the wave function satisfies Psi(y,x) = -Psi(x,y), which implies in particular that Psi(x,x) is always zero. This implies Pauli's exclusion principle, which states that two fermions cannot be in the same quantum state.
The boundary conditions in this simulation are periodic, meaning that each of the two particles moves on a circle (a line segment with extremities glued together). There is an interaction potential between the particles that behaves like a regularized Coulomb potential when the particles are close to each other. This potential is of the form c/sqrt(r^2 + r0^2), where r0 is a small constant. The initial state is an antisymmetrized version of a Gaussian wave packet, describing one particle at rest, and the other particle moving towards the first one. Antisymmetrizing means that one starts with a wave packet of equation Psi0(x,y), and then sets Psi(x,y) = [Psi0(x,y) - Psi0(y,x)]/2, so that one indeed has  Psi(y,x) = -Psi(x,y). One can indeed see in the first part of the video, showing the real part of Psi, that peaks (in yellow) and troughs (in purple) are reversed.It is harder to see that this is still the case near the diagonal x = y, but I made sure in the code that the wave function does remain antisymmetric.  
The simulation shows the same solution of Schrödinger's equation with two different color schemes: 
Real part: 0:00
Modulus squared: 0:50
In the first part, the color hue represents the real part of the wave function, and the z-coordinate shows the sum of the real part and the interaction potential. In the second part, the the luminosity represents the modulus squared of the wave function, which gives the probability density of finding the particles at any given point, and the z-coordinate is the sum of the modulus squared and the potential. 

Render time: 2 hours 17 minutes
Color schemes: Part 1 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
https://github.com/BIDS/colormap
Part 2 -  Magma by Nathaniel J. Smith and Stefan van der Walt
https://github.com/BIDS/colormap

Music: "Beach House" by Text Me Records@socialxwork

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!