Interactive three-dimensional simulations & visualizations

Visualizing the beauty in physics and mathematics

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Hydrogen orbitals ⚛︎

What are you looking at?

This is a 3D visualization of atomic orbitals: mathematical shapes arising from the solutions of the Schrödinger equation for a hydrogen-like atom.

Each surface represents a region of constant probability density. Colors show the sign (phase) of the wavefunction, and transparency reveals its relative magnitude.

Use the controls to explore how orbital shape and symmetry change.

What can be seen

You are looking at a three-dimensional visualization of atomic orbitals — the quantum-mechanical wavefunctions that describe where an electron is likely to be found around an atomic nucleus.

Each shape represents an orbital defined by the quantum numbers n and ℓ. Rather than showing electrons as tiny particles moving along fixed paths, quantum mechanics describes them as waves. The surfaces you see here are constructed from the angular part of those wavefunctions.

Color and phase

The colors indicate the sign (phase) of the wavefunction:

• Blue and red represent opposite phases of the electron’s wave.
• Where the color changes, the wavefunction passes through zero — these are nodal surfaces,
• regions where the probability of finding the electron is exactly zero.

Transparency and magnitude

The opacity of the surface reflects the magnitude of the wavefunction:

• More opaque regions correspond to larger values of $|\psi|$.
• More transparent regions indicate smaller amplitudes.

This makes both the shape and the structure of each orbital visible at the same time.

What this is — and what it isn’t

• These shapes are not solid objects.
• They are not electron trajectories.
• They are visual representations of mathematical functions that encode probability and symmetry.

What you are really seeing is the geometry of quantum mechanics itself.

The mathematics behind the shapes

Atomic orbitals are solutions of the time-independent Schrödinger equation for a hydrogen-like atom:

\[\left( -\frac{\hbar^2}{2\mu}\nabla^2 -\frac{e^2}{4\pi\varepsilon_0 r} \right)\psi(\mathbf{r}) = E\,\psi(\mathbf{r})\]

Because the potential depends only on the distance (r), the solutions separate in spherical coordinates:

\[\psi_{n\ell m}(r,\theta,\phi) = R_{n\ell}(r)\,Y_{\ell}^{m}(\theta,\phi)\]

• $R_{n\ell}(r)$ is the radial function, controlling the size and radial nodes
• $Y_{\ell}^{m}(\theta,\phi)$ are the spherical harmonics, determining the orbital shape and symmetry

What you see here — and what you don’t

• The surfaces shown are isosurfaces of the wavefunction amplitude
• Color indicates the sign (phase) of $\psi$
• Transparency reflects the relative magnitude

Not shown explicitly:

• The time dependence $e^{-iEt/\hbar}$
• Electron trajectories (electrons do not orbit like planets)
• Exact probability density $|\psi|^2$ — this is a geometric representation

These shapes visualize the structure of quantum states, not literal electron paths.

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