An atomic orbital is a mathematical function that describes the wave-like behavior of an electron in an atom. The square of the wavefunction, $|\psi(r, \theta, \phi)|^2$, gives the probability density of finding an electron at a given point in 3D space.
Visualization Modes:
- Solid 3D Mesh (Isosurface): Renders a continuous boundary surface where the electron probability density $|\psi|^2$ equals a specified threshold iso-level. Positive ($\psi > 0$) lobes are rendered in Cyan, and negative ($\psi < 0$) lobes in Orange.
- Point Cloud: Uses Monte Carlo rejection sampling to generate 60,000+ points proportional to electron probability density.
Quantum Numbers:
- Principal Quantum Number ($n \ge 1$): Determines energy shell level and size.
- Azimuthal Quantum Number ($l = 0 \dots n-1$): Defines orbital shape ($s=0, p=1, d=2, f=3$).
- Magnetic Quantum Number ($m = -l \dots +l$): Specifies spatial orientation.
Mathematical Formulation:
\psi_{n,l,m}(r,\theta,\phi) = R_{n,l}(r) \cdot Y_l^m(\theta,\phi)