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

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

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.