This visualization demonstrates how the shapes of atomic orbitals up to g orbitals are related to spherical harmonics, which are solutions to the angular part of the Schrödinger equation.

Explanation:

The 3D plot shows the shape of the spherical harmonic \( Y_{l,ml}(θ,φ) \) for the selected quantum numbers n, l, and ml.

These shapes represent the angular part of the wavefunction \( ψ \), which is a solution to the Schrödinger equation for hydrogen-like atoms.

The radial distance from the origin represents the absolute value of the spherical harmonic \( |Y_{l,ml}(θ,φ)| \).

The shapes correspond to the probability density of finding an electron in different regions around the nucleus.

Notice how different combinations of n, l, and ml produce various shapes, including cloverleaf-like patterns for certain d and f orbitals.

Quantum Numbers:

n (principal quantum number): Determines the energy level and overall size of the orbital. Not visualized here but ranges from 1 to ∞.

l (angular momentum quantum number): Determines the shape of the orbital. Ranges from 0 to n-1.

ml (magnetic quantum number): Determines the orientation of the orbital. Ranges from -l to +l.

ms (spin magnetic quantum number): Represents the spin of the electron. Can be +1/2 or -1/2. Does not affect the spatial shape of the orbital.