Many moving probes sample the hydrogen-like orbital density. The cloud is built from |psi| squared, not classical electron tracks.
psi(n,l,m) = R(n,l)(r) × Y(l,m)(theta,phi). Ball positions are Monte Carlo samples from |psi| squared.
This simulation visualizes hydrogen-like atomic orbitals by sampling the probability density function |ψ(n,l,m)|² using Monte Carlo methods. Thousands of particles orbit the nucleus, forming a 3D cloud that reveals the shape and structure of each quantum state.
Explore all orbitals from 1s to 5g — s, p, d, f, and g shapes with their characteristic lobe structures and nodal patterns.
Particle positions are sampled from |ψ|² using the radial wavefunction R(n,l)(r) and real spherical harmonics Y(l,m)(θ,φ).
Toggle isosurface rendering to reveal the orbital’s 3D boundary surface. Low-probability regions (nodes) are hidden, showing only the shape where electrons are likely found.
Adjust particle count, cloud scale, quantum motion speed, and glow intensity. Drag to orbit, scroll to zoom, or regenerate with a new random seed.
Shortcuts: Press S to save PNG, R to regenerate, Space to toggle pause.
All s orbitals (1s, 2s, 3s, etc.) are spherically symmetric. The probability density depends only on distance from the nucleus. Higher n means more radial nodes — shells of zero probability.
p orbitals (l=1) have two lobes along a specific axis (x, y, or z). The three p orbitals for each n are perpendicular to each other, forming the basis for chemical bonding.
The number of angular nodes equals l (the azimuthal quantum number). s=0 nodes, p=1 planar node, d=2 planar nodes, f=3, g=4. These nodes create the characteristic shapes.
Toggle Isosurface ON to see only particles above a 3% probability threshold. This cuts through the cloud to reveal the orbital’s true geometric shape, matching textbook diagrams.
With 500 particles the orbital is barely recognizable. At 30,000 particles the cloud becomes dense and smooth, clearly showing lobe shapes, nodal planes, and radial structure.
Each particle represents a Monte Carlo sample from the probability density function |ψ(n,l,m)|² of a hydrogen-like atom. The cloud of particles shows where an electron is most likely to be found.
The wavefunction for a hydrogen-like atom is:
where R is the radial wavefunction (associated Laguerre polynomials) and Y are the real spherical harmonics. The probability density is |ψ|².
n (principal quantum number) determines energy and radial extent. l (azimuthal quantum number) determines angular momentum and shape: l=0 (s), l=1 (p), l=2 (d), l=3 (f), l=4 (g). m (magnetic quantum number) determines the orbital’s orientation in space.
The particles oscillate along their probability distribution paths to create a dynamic, living visualization. The motion follows the quantum probability flow and helps reveal the 3D structure of each orbital.
When Guides are ON, red = x-axis, green = y-axis, blue = z-axis. The semi-transparent grid spheres show distance from the nucleus in arbitrary units.
Yes. On smaller screens the HUD panel becomes collapsible and the controls adapt. The simulation uses p5.js WEBGL which works on all modern mobile browsers.
This simulation is built with p5.js using WEBGL 3D rendering. The radial wavefunctions use approximate analytic forms for hydrogen-like orbitals, and the angular part uses real spherical harmonics. Each orbital’s probability density is sampled via Metropolis-style Monte Carlo to distribute particles in 3D space.
For questions, suggestions, or bug reports, please email all4nmr@gmail.com.
Information Collection: This simulation runs entirely in your browser. No data is collected, stored, or transmitted to any server. All computation and rendering happens locally on your device.
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