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https://github.com/hitthecodelabs/quantummechanicsvisualized
Functions to calculate and visualize the probability density of hydrogen atom orbitals
https://github.com/hitthecodelabs/quantummechanicsvisualized
hydrogen-atom physics quantum-mechanics
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Functions to calculate and visualize the probability density of hydrogen atom orbitals
- Host: GitHub
- URL: https://github.com/hitthecodelabs/quantummechanicsvisualized
- Owner: hitthecodelabs
- License: mit
- Created: 2023-11-30T03:04:05.000Z (about 1 year ago)
- Default Branch: main
- Last Pushed: 2023-12-02T01:44:00.000Z (about 1 year ago)
- Last Synced: 2024-01-28T08:11:56.525Z (about 1 year ago)
- Topics: hydrogen-atom, physics, quantum-mechanics
- Language: Jupyter Notebook
- Homepage:
- Size: 4.73 MB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# QuantumMechanicsVisualized
This repository contains functions to calculate and visualize the probability density of hydrogen atom orbitals in both 2D and 3D.
## Theoretical Background
The probability density $|\Psi|^2$ of the hydrogen atom's electron is calculated using the wave function $\Psi$, which is a product of the radial and angular parts. The radial part depends on the principal and azimuthal quantum numbers `n` and `l`, while the angular part depends on the azimuthal and magnetic quantum numbers `l` and `m`.
### Radial Part
The radial part of the wave function is given by:
$$ R_{nl}(r) = \sqrt{\left(\frac{2}{n a_0}\right)^3 \frac{(n-l-1)!}{2n[(n+l)!]^3}} e^{- \frac{r}{n a_0}} \left(\frac{2r}{n a_0}\right)^l L_{n-l-1}^{2l+1}\left(\frac{2r}{n a_0}\right) $$
where `L` are the generalized Laguerre polynomials and $a_0$ is the Bohr radius.
### Angular Part
The angular part of the wave function is given by the spherical harmonics:
$$ Y_{lm}(\theta, \phi) = (-1)^{\frac{m + |m|}{2}} \sqrt{\frac{(2l + 1)}{4\pi} \frac{(l - |m|)!}{(l + |m|)!}} P_l^{|m|}(\cos \theta) e^{im\phi} $$
where $P_l^m$ are the associated Legendre polynomials.
### Probability Density
The probability density $|\Psi|^2$ is the product of the square of the radial part and the square of the absolute value of the angular part:
$$ |\Psi|^2 = |R_{nl}(r) Y_{lm}(\theta, \phi)|^2 $$
## Sample Visualizations
Below are some sample visualizations generated by the functions in this repository.
### 2D Probability Density Plots
*Examples of a 2D probability density plot for hydrogen atom orbital.*
### 3D Probability Density Plots
*Examples of a 3D probability density plot for hydrogen atom orbital.*
## Functions
### `calculate_hydrogen_density_2d`
Calculates the 2D probability density of a hydrogen atom orbital.
#### Parameters
- **n (int)**: Principal quantum number, defines the energy level.
- **l (int)**: Azimuthal quantum number, defines the shape of the orbital.
- **m (int)**: Magnetic quantum number, defines the orientation of the orbital.
- **$a_0$ (float)**: Bohr radius, default is 1.0.
- **grid_size (int)**: Resolution of the grid for calculations.
#### Returns
- **X, Y, density**: 2D arrays representing the X, Y coordinates and the corresponding probability density.
#### Usage
```python
X, Y, density = calculate_hydrogen_density_2d(n, l, m, a0, grid_size)
```
### `calculate_hydrogen_density_3d`
Calculates the 3D probability density of a hydrogen atom orbital.
#### Parameters
- **n (int)**: Principal quantum number, defines the energy level.
- **l (int)**: Azimuthal quantum number, defines the shape of the orbital.
- **m (int)**: Magnetic quantum number, defines the orientation of the orbital.
- **$a_0$ (float)**: Bohr radius, default is 1.0.
- **grid_size (int)**: Resolution of the grid for calculations.
#### Returns
- **X, Y, Z, density**: 3D arrays representing the X, Y, Z coordinates and the corresponding probability density.
#### Usage
```python
X, Y, Z, density = calculate_hydrogen_density_3d(n, l, m, a0, grid_size)
```
## Dependencies
- Numpy
- Scipy (for special functions like `sph_harm`)
## License
This project is open-source and available under the [MIT License](LICENSE.md).