What it does
The electron in a hydrogen atom does not orbit like a planet. Quantum mechanics gives it a wavefunction ψ(r, θ, φ), and |ψ|² is the probability of finding it at each point. Each allowed state is labelled by three numbers: the shell n (1, 2, 3 …), the subshell l (0 to n − 1, called s, p, d, f …) and m (−l to l). This tool draws any of them up to n = 7, for hydrogen or any one-electron ion (He⁺, Li²⁺ …) with nuclear charge Z.
It shows three pictures:
- A slice through the nucleus (the xz, xy or yz plane), coloured by the sign of ψ: one colour where ψ is positive, another where it is negative, brighter where |ψ| is larger. The dark lines and circles between lobes are the nodes, where the electron is never found.
- A probability cloud of up to 10,000 dots, each a random position drawn from |ψ|² in three dimensions. Turn and tilt it with the sliders. The dots are the same every time you open the same orbital.
- Radial plots of R(r) and the radial probability r²R²(r), which says how likely the electron is to be at distance r whatever the direction.
How to use it
- Pick n, then l and m. The m list shows the chemistry names for real orbitals (2p x, 3d xy, 4f xyz …).
- Choose real or complex orbitals. Chemists use real ones because they point along axes; physicists often use complex ones with a definite m, whose |ψ|² is the same all the way round the z axis.
- Set Z for an ion, the slice plane, the picture size (blank picks one holding 97 % of the electron) and the number of dots.
- Download the pictures and facts as an image, PDF or Word file, or the radial functions as CSV.
Worked example
The page opens on 3d z² of hydrogen. Its energy is E = −1.51174 eV, a ninth of the ground state’s −13.6057 eV. The mean radius is 10.5 a₀ (556 pm), the most probable radius 9 a₀, and 90 % of the electron lies within 15.8 a₀. It has no radial nodes (n − l − 1 = 0) and two angular nodes, the cones where 3cos²θ = 1. In the xz slice the two lobes along z have one colour and the ring round the middle the other. Change n to 4 and l to 0 for 4s: three radial nodes appear as rings in the slice, at r = 1.87, 6.61 and 15.5 a₀.
Tips
- An orbital with ψ = 0 on the whole slice plane (such as 2p z in the xy plane) shows an empty square; the legend says so. Pick another plane.
- For Z = 2 every length shrinks by half and every energy grows four times.
- The radial probability peaks at n² a₀ / Z for the largest l (1s at 1 a₀, 2p at 4 a₀, 3d at 9 a₀): Bohr’s orbits come back as the most probable radii.
FAQ
Why do the 2s and 2p orbitals have the same energy? In a pure Coulomb field the energy depends only on n. Fine structure, the Lamb shift and the other electrons of a bigger atom split them, but these are not included here.
Are the dots real electron positions? They are samples of where a measurement could find the electron, which is what |ψ|² means. It is for study and teaching, not a substitute for professional judgement.
Does anything get uploaded? No. Every picture is computed in your browser.
Where the numbers come from
- Hydrogen-like wavefunctions (associated Laguerre radial functions and spherical harmonics with the Condon–Shortley phase), energies and the closed forms for ⟨r⟩, ⟨r²⟩ and ⟨1/r⟩: standard textbook results, for example Griffiths, D. J., Introduction to Quantum Mechanics, and Bransden, B. H. and Joachain, C. J., Physics of Atoms and Molecules.
- Rydberg energy and Bohr radius: CODATA 2018 values, rounded.
- Checked against SciPy (scipy.special.genlaguerre, sph_harm_y, quad and brentq) written for the tests.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.