Orbital Observatory / A sense of scale
Same ruler. Different orbitals.
Explore how far hydrogen-like orbitals extend. Every view uses the same physical scale and a surface enclosing approximately 90% electron probability.
Hydrogen · Z = 1 / One scale across all views
The radial picture
Where is the electron likely to be?
The curves include the spherical-shell volume factor.
Their peaks are not the edges of the surfaces above.
What “to scale” means here
These are nonrelativistic, one-electron Coulomb orbitals with a fixed nucleus and the same nuclear charge Z. Coordinates are measured in Bohr radii (1 a₀ ≈ 0.529177 Å). At fixed Z, the 1s, 2p, 3d and 4f surfaces retain their relative physical dimensions. Increasing Z contracts every orbital by 1/Z. They are alternative states of a one-electron system, not a picture of one atom occupied by 16 electrons.
Each surface bounds the region where |ψ|² exceeds an orbital-specific threshold, chosen to contain approximately 90% of that orbital’s integrated probability. The threshold is not the same density for every orbital. The remaining probability lies outside the shown regions, including near nodes; the wavefunction has no hard outer edge. Blue and coral indicate wavefunction sign, not electric charge.
The 16 real orbitals match the original collection: one 1s, three 2p, five 3d and seven 4f. They have no radial nodes. This is not every orbital through n = 4, and these are not quantitatively realistic orbital sizes for many-electron atoms. The f labels are conventional shorthand; the full Cartesian polynomial is shown beneath each selected f orbital, where r² = x² + y² + z².
All panels use an orthographic camera, identical view widths, and synchronized rotation and zoom. Nucleus markers and coordinate guides are annotations. The average radius ⟨r⟩ is an expectation value, not a surface radius. Different real angular forms in the same subshell share ⟨r⟩ but can have different outermost surface points.
For a fixed n and ℓ, ⟨r⟩ = [3n² − ℓ(ℓ + 1)]a₀/(2Z). Here ℓ = n − 1, so the radial-probability peak is at n²a₀/Z. The three-dimensional density ρ = |ψ|² obeys ρZ(r) = Z³ρ1(Zr), where r is the position vector. The radial shell density instead obeys PZ(r) = ZP1(Zr). Both remain normalized as the region contracts. The radial chart keeps its distance axis fixed when Z changes; its probability-density axis rescales and remains labeled.
In many-electron atoms, electron repulsion and shielding modify this simple relation. Adding a proton while also adding electrons is a different comparison. A single effective nuclear charge can be a useful approximation, but is not a general exact orbital-size law.
Mathematical references: Oregon State: hydrogenic radial functions and harmonics; University of Texas: hydrogen atom; NIST: Bohr radius. Geometry is a finite-resolution numerical approximation; numerical validation and rebuild provenance accompany the downloadable package.