Hydrogen Orbitals 3D

This is the hydrogen atom solved exactly by quantum mechanics, shown as a 3D cloud. The electron has no definite position; the brightness at each point is the probability of finding it there, $|\psi|^2$. Three integers set the shape: $n$ (1 to 5) is the energy level and overall size, $\ell$ is how much angular structure the cloud has (0 is a sphere, 1 a dumbbell, 2 a cloverleaf), and $m$ tilts and twists that pattern around the axis. The sliders are clamped to the only allowed combinations, $\ell < n$ and $|m| \leq \ell$, so some settings refuse to move; that restriction is the physics, not a bug, and it is what builds the periodic table. Switch the view to read probability density (viridis), the wavefunction phase (hue wheel), or a lit isosurface that makes the lobes look solid; a colour key in the corner says which scale is active. Drag to orbit, scroll to zoom. The readout shows the energy $E_n = -13.6 \text{ eV} / n^2$ and the mean radius, which grows like $n^2$.

Figure 1. Hydrogen orbitals: the electron probability cloud

|\psi_{n\ell m}|^2

for quantum numbers

(n, \ell, m)

. Brighter regions are where the electron is more likely to be found, and the colour key in the corner shows the active scale (density, phase, or sign). Method: exact hydrogenic wavefunctions (radial Laguerre times spherical harmonic) volume ray-marched on the GPU.

WHAT TO TRY

Change the quantum numbers n, l, m: the electron cloud reshapes into s spheres, p dumbbells, d cloverleaves, the exact solutions of the hydrogen Schrodinger equation. Brightness is the probability density.
Raise n: the cloud grows and gains radial nodes (dark shells), since the orbital energy and size scale with the principal quantum number.
Increase l and vary m: the angular nodes appear, carving the cloud into lobes. The shapes are the spherical harmonics that set chemical bonding geometry.