State: (3, 2, 1)
Radial nodes: 0. Angular nodes: 2.
Hydrogen wave functions
Explore orbital probability density in 3D.
Choose quantum numbers, rotate the probability cloud, inspect nodal surfaces, and compare the density plots against a reference-style hydrogen orbital gallery.
3D simulation
Hydrogen orbital viewer with quantum-number controls.
The main viewer is a live WebGL simulation generated from \(\psi_{nlm}(r,\theta,\phi)=R_{nl}(r)Y_l^m(\theta,\phi)\). Drag to rotate, use the wheel to zoom, and Shift-drag to pan the 3D probability cloud.
Controlled probability-density slice
The slice shows \(|\psi|^2\) through a chosen plane. Use slice depth to move the cut through the orbital. Dark bands are nodes, where the probability density goes to zero because the amplitude changes sign or a radial factor vanishes.
How to read the quantum numbers
\[\begin{gathered} n=1,2,3,\dots \\ l=0,\dots,n-1 \\ m=-l,\dots,l \end{gathered}\]
n sets the energy level and overall size. Larger \(n\) generally spreads the electron probability farther from the nucleus.
l sets orbital shape and angular nodes. \(l=0\) is s-like, \(l=1\) is p-like, \(l=2\) is d-like, and \(l=3\) is f-like.
m sets orientation around the chosen axis. The simulator uses real combinations of spherical harmonics so the lobes are visible in 3D.
\[\rho(r,\theta,\phi)=|\psi_{nlm}(r,\theta,\phi)|^2\]
Reference-style plots
Hydrogen wave function probability density gallery.
Click any tile to load that state into the WebGL 3D viewer. These are live-generated density plots for reference; the learning activity happens in the interactive 3D simulator above.
negative amplitude
positive amplitude
Connect concepts
From wave functions to atoms and measurements.
Use this page with the interactive lab and atom simulations to connect amplitudes, orbitals, nodes, and measurement probability.