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 plot is generated from \(\psi_{nlm}(r,\theta,\phi)=R_{nl}(r)Y_l^m(\theta,\phi)\). Bright regions mark high \(|\psi|^2\); cyan/orange and purple show opposite signs of the real wave amplitude.
Animated probability-density slice
The slice shows \(|\psi|^2\) through a chosen plane. 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
\[n=1,2,3,\dots \qquad l=0,\dots,n-1 \qquad m=-l,\dots,l\]
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 3D viewer. The gallery mirrors the attached reference image with the same labeled orbital families from \((2,0,0)\) through \((4,3,3)\).
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.