Скачать или смотреть Quantum Sinai billiard: Wave function of a particle colliding with a circular obstacle

This #short video is a 3D version of the simulations    • A quantum Sinai billiard: probability density   and    • A quantum Sinai billiard, phase evolution   showing a solution of Schrödinger's equation, when the potential is infinite inside a disc. The initial state is a coherent state, describing a particle with a momentum directed towards the disc. This can be seen as an approximation of what happens when a particle, such as an electron, collides with a much heavier particle, such as a heavy atom, which can be approximated by a fixed potential with rotation symmetry. Simulating a real system of interacting quantum particles requires much more computing power.
The simulation shows the same solution of Schrödinger's equation with two different color schemes:
Probability density: 0:00
Phase: 0:26
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.

For more simulations of Schrödinger's equation, see the playlist    • Schrödinger's equation  

Render time: 52 minutes 20 seconds
Color scheme: Part 1 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
https://github.com/BIDS/colormap
Part 2 - Twilight by Bastian Bechtold
https://github.com/bastibe/twilight

Music: Lift Motif by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 licence. https://creativecommons.org/licenses/...
Source: http://incompetech.com/music/royalty-...
Artist: http://incompetech.com/

See also https://images.math.cnrs.fr/Des-ondes... 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...
C code: https://github.com/nilsberglund-orlea...
https://www.idpoisson.fr/berglund/sof...
Many thanks to my colleague Marco Mancini for helping me to accelerate my code!