Potential landscape view of the 2D quantum hydrogen atom
Channel: Nils Berglund
Date: 22/06/2022

In this simulation of the quantum hydrogen atom, the z-coordinate shows the sum of the Coulomb potential and the modulus squared of the wave function, thereby providing a better idea of what it means that the electron "falls into the potential well created by the proton". 
To avoid numerical instabilities that occurred on previous simulations of the hydrogen atom, the Coulomb potential has been regularized at the origin: instead of being of the form 1/r, the potential is given by 1/sqrt(r^2 + r0^2), where r0 is a small constant that can be interpreted as giving the nucleus a nonzero size. The initial state is a Gaussian wave packet aimed somewhat to the left of the proton. 
The simulation shows the same solution of Schrödinger's equation with two different color schemes: 
Probability density: 0:00
Phase: 0:52
In the first part, the color hue represents the modulus of the wave function, which gives the probability density of finding the particle at any given point. In the second part, the luminosity depends 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, but this has little influence on the simulation, as the wave functions does not stray far from the nucleus. 

Render time: 1 hours 45 minutes
Color scheme: Part 1 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
https://github.com/BIDS/colormap
Part 2 - HSL/Jet

Music: "Classique" by Francis Preve@francispreve

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!