Orbital Observatory / The mathematics of shape
Follow one point through four views. Explore all 16 Observatory orbitals—1s, 2p, 3d, and 4f—plus three radial-node examples. See how distance and direction combine to create amplitude, probability, and orbital shapes.
01 / Distance
The vertical marker follows the probe radius. Click either graph to move it.
P(r) dr is probability in a thin spherical shell. Dashed vertical lines mark radial nodes.
02 / Direction
Color shows Y(θ, φ) on a unit sphere, scaled to this orbital’s maximum |Y|. Tap to choose a direction; drag to rotate both 3D views.
03 / Product
This is a slice of the 3D field, not a projection. Click within the circular field to move the probe. Dashed curves mark nodes.
04 / Shape
The equation behind the picture
q = r/a₀ · θ measured from +z · φ measured from +x toward +y
The angular formulas use dimensionless direction cosines: u = x/r = sin θ cos φ, v = y/r = sin θ sin φ, and w = z/r = cos θ. The symbol Y(θ, φ) denotes the full angular function below. At r = 0, direction is undefined.
ψ(r, θ, φ) = R(r)Y(θ, φ) · ρ = |ψ|² · ∫ρ dV = 1 · dV = r² sin θ dr dθ dφ
For the nonrelativistic hydrogen atom with a fixed nucleus, the stationary Schrödinger equation in atomic units is [−½∇² − 1/r]ψ = Eψ. Separating distance from direction produces a radial equation and an angular equation. Finite, normalizable solutions restrict the quantum numbers and energies.
Here En = −1/(2n²) hartree, ℓ = 0, …, n−1, and m = −ℓ, …, +ℓ. The collection contains 1s, three 2p, five 3d, and seven 4f real orbitals, plus 2s, 3s, and 3pᶻ for radial-node comparisons. This is a selected teaching set, not every orbital through n = 4. All states with the same n have the same energy in this hydrogen model, even when their shapes differ.
The complex harmonics Yℓm are eigenfunctions of L² and Lz. Real orbitals such as pₓ and dₓᵧ combine the +m and −m functions at the same n and ℓ. They remain energy and L² eigenstates, but generally do not have a definite Lz. The displayed basis label distinguishes m = 0 from a real ±m combination.
The radial factor has n−ℓ−1 radial nodes. For the separated real angular basis here, the conventional angular-node count is ℓ: polar nodal surfaces plus azimuthal nodal planes. This counts the separated nodal factors or conical sheets, not disconnected pieces of the full nodal set; nodal planes intersect, and conical sheets meet at the origin.
These are stationary states: their full time dependence is ψ exp(−iEt/ℏ), whose overall phase changes while |ψ|² stays constant. This display shows the real spatial factor. The overall sign of an orbital is arbitrary; the relative signs across its nodes are what the two colors reveal.
Blue and coral encode positive and negative wavefunction amplitude. They do not represent positive and negative electric charge. Squaring removes the sign, but nodes remain zeros. A position has a probability density; probability is obtained by integrating that density over a volume. These pictures are spatial probability distributions, not electron paths.
Cartesian orbital names are conventional abbreviations. For example, fz³ has angular polynomial w(5w²−3), not simply w³, and fxz² uses u(5w²−1). The full normalized angular formula for the selected orbital is shown above. Orbitals within the same subshell share their radial factor, but angular shape can change an isodensity boundary's extent.
The angular sphere has a fixed radius and no atomic length scale. The cross-section and orbital surface use physical coordinates in Bohr radii. Each orbital has its own displayed window size, which is labeled; this explorer is for mathematical structure, while the companion size-comparison viewer uses a common window across orbitals.
The full, uncut isodensity region contains approximately 90% probability, summing every lobe and both signs. The clipped display shows only part of it. This is not a hard atomic boundary, and the threshold can omit low-density regions between nodes. The slicing plane and probe are annotations. Cutting a mesh does not add a physical wall or a nodal surface.
The cumulative value in the radial view is the probability inside a sphere of radius r. It is a different region from the 90% isodensity surface. For s orbitals only, P(r) = 4πr²|ψ(r)|². For normalized angular functions generally, P(r) = r²|R(r)|².
These one-electron hydrogen orbitals neglect nuclear motion, relativistic effects, external fields, and electron–electron interactions. They are not quantitative many-electron orbitals. Numerical isosurfaces approximate the analytic functions; normalization, nodal roots, probability thresholds, and mesh checks are documented in the accompanying package.