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Python implementation of the Crank-Nicolson method for solving the one dimensional time-dependent Schrödinger equation

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vguillon/time-dependent-schrodinger-equation

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Examples

The initial wave packet is a (non-normalized) Gaussian function. It evolves over a period T along the x-axis. A potential is placed on its path.

Evolution of the wave packet for a Gaussian potential (more examples available in movies)

movie gaussian potential

Snapshot of the wave packet for a step potential (more examples available in snapshots)

snapshot step potential at t=0.3T

Installation

The program needs numpy and matplotlib to run. For movie generation, see the matplotlib.animation documentation (ffmpeg recommended).

You can clone the project in your current repository with:

$ git clone https://github.com/vguillon/time-dependent-schrodinger-equation.git

Parameters of the potentials

You can choose the parameters of the potentials directly in the potential_parameters.txt file. The parameters are:

  • Potential: select potential. Available potentials are 'step', 'barrier', 'well' and 'gaussian' ;
  • V0: potential height. Only used for 'step', 'barrier' and 'well' ;
  • width: potential width. Only used for 'barrier' and 'well'. Hints: for 'barrier', width = 0.03 and for 'well', width = 0.1 give good results ;
  • x0: center of the gaussian potential ;
  • sigma: width of the gaussian potential.

Running the program

The program allows several command line options. Each of these options possess a default value. Hence, after the potential parameters are configured, you can just run

$ python3 schrodinger.py

to obtain a snapshot similare to the one above.

To show the available command line options, type:

$ python3 schrodinger.py -h

For instance, if you want a .png snapshot of the wave function after half a period for a wave packet starting at x=0.1 at t=0, you can write:

$ python3 schrodinger.py -sf png -t 0.5 -x 0.1

If you want to create a .mp4 movie over a period, type:

$ python3 schrodinger.py -o movie

If you want to create a .gif over a period with 3000 points along the x-axis for a wave packet starting at x=0.2 at t=0, write:

$ python3 schrodinger.py -o movie -mf gif -N 3000 -x 0.2

The generation of the datas can take few minutes.

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Python implementation of the Crank-Nicolson method for solving the one dimensional time-dependent Schrödinger equation

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