artmenlope / double-slit-2d-schrodinger

This repository contains Python 3 scripts for simulating the passage of a 2D Gaussian wave packet through a double slit. For this, the 2D Schrödinger equation is solved using the Crank-Nicolson numerical method.

double-slit-2d-schrodinger

A detailed description of the methods used for these simulations can be found at "Solving the 2D Schrödinger equation using the Crank-Nicolson method".

Summary

This repository contains Python 3 scripts for simulating the passage of a 2D Gaussian wave packet through a double slit. For this, the 2D Schrödinger equation is solved using the Crank-Nicolson numerical method.

The method

To simulate the passage of the 2D Gaussian wavepacket through the double slit, the discretized 2D schrödinger equation is solved at each time step using the Crank-Nicolson numerical method. It is important to note that this method is computationally expensive, but it is more precise and more stable than other low-order time-stepping methods [1]. It calculates the time derivative with a central finite differences approximation [1].

For further information on the Crank-Nicolson method, the mathematical details and the creation of the scripts for the simulations, please remember to check this blog post.

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For this problem we consider the 2D time-dependent Schrödinger equation:

$$ i\dfrac{\partial \psi(x, y, t)}{\partial t} = - \nabla^2 \psi(x, y, t) + V(x, y, t) \psi(x, y, t) $$

which can be expanded as

$$ i\dfrac{\partial \psi(x, y, t)}{\partial t} = - \left( \dfrac{\partial^2 \psi(x, y, t)}{\partial x^2} + \dfrac{\partial^2 \psi(x, y, t)}{\partial y^2}\right) + V(x, y, t) \psi(x, y, t) $$

Here, for simplicity we have considered that $\hbar/2m=1$. Then, to solve the problem, we discretize the simulation space and, consequently, the particle's wave function $\psi$ (our Gaussian wavepacket) and the Scrödinger equation.

The chosen initial (unnormalized) Gaussian wavepacket has the form

$$ \psi(x, y, t=0) = e^{-\frac{1}{2\sigma^2}\left[(x-x_0)^2+(y-y_0)^2\right]}\cdot e^{-ik(x-x_0)} $$

and the double slit is parametrized as the following image shows:

The results

The two main scripts in this repository doubleSlit_HW_CN.py and doubleSlit_FPB_CN.py are used to simulate the system in the case of a double slit with infinite potential barrier walls and finite potential barrier walls respectively. The following two animations show how the results look like:

Hard walls double slit	Potential barrier walls double slit

References

[1]: Landau, R.H., Páez Mejía, M.J. & Bordeianu, C.C., 2015. "Heat Flow via Time Stepping". In: "Computational physics: problem solving with Python", 3rd ed. Weinheim: Wiley-VCH.