Potential Well Simulator

This programm simulates the movement of a particle in a 1D potential well.

Installation/setup

You need to have Python 3 installed on your computer.

To set up the project, do the following:

git clone https://github.com/gucio321/potential-well && cd potential-well
python3 -m virtualenv venv
. venv/bin/activate
python3 -m pip install -r requirements.txt

Then to run it, do python3 main.py.

Technologies

• Python 3
• Dear PyGui (Dear ImGui warapper for python)
• GitHub (for version control)

USP

• The program uses rejection sampling for simulating theoritical particle movement.
• There is a numerical equation solver for the transcendental equation (which cannot be solved analytically).
• We're using Simpson's rule for numerical integration (needed to apply boundry conditions for Schrödinger equation solutions).
• A whole lot of math behind the scenes used to compute SE solutions' constants.

Math

Schrödinger equation

After solving the Schrödinger equation for a particle in finite potential well, we get the following solutions:

$$\psi (x)=\{\begin{array}{ll}A\sin (kx)+B\cos (kx)& \text{for }0\le x\le L\\ C\exp (\alpha x)& \text{for }x<0\\ D\exp (-\alpha (x-L))& \text{for }x>a\end{array}$$

where

$$k=\sqrt{\frac{2mE}{{\hslash }^2}}$$

$$\alpha =\sqrt{\frac{2m(V-E)}{{\hslash }^2}}$$

After applying boundry conditions, we get the following coefficients conditions:

$$B=C$$

$$A=\frac{\alpha }{k}B$$

$$D=B(\frac{\alpha }{k}\sin (kL)+\cos (kL))$$

$$-\alpha D=B\ast k\ast (\frac{\alpha }{k}\cos (kL)-\sin (kL))$$

As you can see it is impossible to calculate all parameters. The last condition is the normalization condition:

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|\psi (x){|}^{2}dx=1$$

From 2 last conditions we can get transcendental equation for:

$$(\frac{k}{\alpha }-\frac{\alpha }{k})-2\frac{1}{\tan (kL)}=0$$

We solve this numerically.

Here is a graph that shows above equations:

Click to open desmos.

As you can see from the above, the Energy of a particle could not have any random value in range [0, V] (to keep $\psi$ continous) - try yourself on the graph above.

walls height

We need max Psi value (e.g. for wall width):

$$f(x)=A\sin (kx)+B\cos (kx)$$

$$f^{\prime }(x)=k(A\cos (kx)-B\sin (kx))=0$$

$$\tan (kx)=\frac{A}{B}$$

$$x=\frac{\arctan (A/B)}{k}$$

$$x=\frac{\arctan (\alpha /k)}{k}$$

Motivation

This is a project for Python Lab https://sylabusy.agh.edu.pl/pl/document/8cdc4249-ac97-43c9-b0f7-ea2abdeb50a9.html. Coop with @michal1563.

References

• Dear PyGui documentation: https://dearpygui.readthedocs.io/en/latest/reference/dearpygui.html
• Rejection sampling: https://youtu.be/kYWHfgkRc9s