Visualizing the Impossible

Software for generating animations of the time dependent schrödinger equation using the finite difference method in python, as seen in Visualizing the Impossible.

Article

The accompanying article to this repo can be found here.

About

This repo contains software for generating animations of the TDSE using the finite difference method in python. This is done by recursively evalutating the following expression:

$$ \begin{aligned} \psi(x, t + \Delta t) &= \left(\frac{i}{2m} \right) \left(\frac{\Delta t}{{\Delta x}^2} \right) \left( \psi(x + \Delta x, t) - 2 \psi(x, t) + \psi(x - \Delta x, t) \right) \\ &- \left(\frac{i}{\hbar}\right)(\Delta t)V(x)\psi(x,t) \\ &+ \psi(x,t) \end{aligned} $$

Derivation of this expression can be found in this repo's accompanying article, Visualizing the Impossible.

This repo and the accompanying article were create for the Summer of Math Exposition 3 (SoME3), read more about the competition at: https://some.3b1b.co/.

Render Your Own Animations

First, clone the repo,

git clone https://github.com/liam-ilan/time-dependent-schrodinger-equation.git

The software used to generate these renders was built with Python, Scipy, Numpy, and Matplotlib. To install all necessary packages through pip,

pip install -r requirements.txt

From there, run main.py to render all animations and thumbnails for the accompanying article, as well as the demo gif at the top of the readme,

python main.py

Render Method

renderer.py provides a render() method that creates a thumbnail and animation of the TDSE.

render(path, thumb_path, config)

• path: the path to save the animation to
• thumb_path: the path to save the thumbnail to
• config: a dictionary containing all nescacarry configuration
  • k: the wave number of the initial wave packet
  • std_dev: the standard deviation of the initial wave packet
  • x_0: initial position of wave packet
  • m: mass of particle
  • bounds: an array of length 2, containing the low (1st item) and high (2nd item) spatial bounds
  • sample_number: number of discrete elements to split space into
  • detla_t: discrete timestep
  • real_time: total simulated time
  • fps: fps of animation
  • anim_duration: length of animation
  • pot_func: a function, that gets a single position, and returns the potential at that position

Example config:

config_free = {
  'k': 6e9,
  'std_dev': 9e-10,
  'x_0': -1.5e-8,
  'm': constants.m_e,
  'bounds': [-2e-8, 2e-8],
  'sample_number': 800,
  'delta_t': 5e-54,
  'real_time': 3e-48,
  'fps': 30,
  'anim_duration': 15,
  'pot_func': lambda x: 0,
}

All units are SI units.

Credit

• Built by Liam Ilan
• Check the accompanying article for a list of references