Hydrogen Atom Wave Function Animation

TODO

-Add a better parsing to set easly m, l and n.
-Setup the animation.

This repository contains a Python script for calculating and animating the wave function of a hydrogen atom in a plot. It utilizes the numpy, matplotlib, and scipy libraries for calculations and visualization.

The wave function of a quantum system describes its state and provides the probability distribution for observable measurements. For a hydrogen atom, the wave function can be expressed as a product of radial and angular parts:

$$[ \Psi(r, \theta, \phi) = R(r) Y(\theta, \phi) ]$$

where $(r)$, $(\theta)$, and $(\phi)$ are spherical coordinates, $(R(r))$ is the radial part, and $(Y(\theta, \phi))$ is the angular part.

Radial and Angular Parts of the Wave Function

The radial part is given by:

$$[ R_{n,l}(r) = \sqrt{ \left( \frac{2}{n a_0} \right)^3 \frac{(n-l-1)!}{2n[(n+l)!]^3} } e^{-\frac{r}{na_0}} \left( \frac{2r}{na_0} \right)^l L_{n-l-1}^{2l+1} \left( \frac{2r}{na_0} \right) ]$$

where $( n )$ is the principal quantum number, $( l )$ is the orbital angular momentum quantum number, $( a_0 )$ is the Bohr radius, and $( L_{n-l-1}^{2l+1} )$ is the associated Laguerre polynomial.

The angular part is given by spherical harmonic functions:

$$[ Y_{l,m}(\theta, \phi) = (-1)^m \sqrt{ \frac{2l+1}{4\pi} \frac{(l-m)!}{(l+m)!} } P_l^m(\cos(\theta)) e^{i m \phi} ]$$

where $( l )$ is the orbital angular momentum quantum number, $( m )$is the magnetic quantum number, $( \theta )$ is the polar angle, $( \phi )$ is the azimuthal angle, and $( P_l^m )$ is the associated Legendre polynomial.

Calculating the Wave Function and Probability Density

The compute_wave_function function calculates the wave function for specified quantum numbers $( n, l, )$ and $( m )$, along with a Bohr radius scale factor. It then computes the probability density using the compute_probability_density function.

Plotting the Wave Function

The plot_wave_function function plots the probability density using matplotlib's imshow function, creating a 2D image of the probability density. The update function animates the plot by updating it at each frame with the wave function and probability density for given quantum numbers and a Bohr radius scale factor.

Running the Script

Execute the main function in a Python environment to run the script. The output is a plot of the wave function and probability density for the specified quantum numbers and Bohr radius scale factor. The plot is animated with a time-dependent wave function.

Dependencies

This script requires the following Python libraries:

• numpy
• matplotlib
• scipy

License

This project is licensed under the MIT License. See the LICENSE file for details.

Acknowledgments

The script includes the Bohr radius value from the physical_constants module of the scipy library. The genlaguerre and lpmv functions from scipy are used to calculate associated Laguerre and Legendre polynomials, respectively. The imshow function from matplotlib is used to create the 2D image of the 3D probability density.