QiLing Tools

⚛️Hydrogen Atom Orbital Viewer - Wavefunctions, 3D Electron Cloud, Radial Distribution

Pick the quantum numbers n, l and m and the tool builds the exact hydrogen wavefunction from associated Laguerre polynomials and spherical harmonics, then shows it as a phase-coloured cross-section, a rotatable 3D electron cloud and the radial probability r²R². It also covers energy levels, hydrogen-like ions (He⁺, Li²⁺ …) and superpositions of two states, whose electron cloud sloshes at ΔE/h, with the transition dipole and spontaneous emission rate.

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How to use Hydrogen Atom Orbitals

  1. 1Choose the nuclear charge Z, real or complex orbitals, then click n, l and m (or click a level on the energy diagram)
  2. 2In Cross-section, change the plane and offset to see the sign of ψ, radial node circles and angular nodal surfaces
  3. 3Switch to 3D electron cloud and drag to rotate, or cut away the front half; read the most probable radius and ⟨r⟩ under Radial distribution
  4. 4Under Superposition, pick a second state and press Play to watch the cloud oscillate with its dipole arrow and transition wavelength

FAQ

What is the difference between real and complex orbitals?
A complex orbital ψnlm has a definite magnetic quantum number m (L_z = mħ); its |ψ|² is symmetric about the z axis and its phase winds as e^{imφ}. The real pₓ, p_y, d_xy … orbitals in chemistry books are combinations of +m and −m, which gives the familiar dumbbells and clover leaves but no definite L_z. Both sets have the same energies.
How do I count radial and angular nodes?
There are n − l − 1 radial nodes (spheres where ψ = 0) and l angular nodes (planes or cones), n − 1 in total. 3s has two radial nodes; 3p has one radial node and one nodal plane; 3d has no radial node but two angular nodes (two cones for d_z²). The white lines in the cross-section are the nodes.
Why is the most probable radius different from the mean radius?
The radial probability P(r) = r²R² is skewed with a long tail, so the mean ⟨r⟩ = a₀(3n² − l(l+1))/(2Z) lies beyond the peak. For 1s the peak sits exactly at the Bohr radius a₀ = 52.9 pm while ⟨r⟩ = 1.5 a₀.
Why does a superposition oscillate?
When two stationary states with different energies are combined, the cross term carries e^{−iΔEt/ħ}, so the density sloshes with period h/ΔE. If Δl = ±1 this creates an oscillating electric dipole that radiates at wavelength hc/ΔE — the semiclassical picture of atomic emission. A 1s + 2s mixture only breathes and does not radiate as a dipole, and states with the same n are degenerate, so their mixture does not move at all.
How does this relate to the spectroscopy and electron configuration tools?
This tool is about the shape and properties of one-electron wavefunctions. For full spectral series, the Rydberg formula and the photoelectric effect use the spectroscopy tool; for configurations of many-electron atoms use the electron configuration tool; for chemical bonding see the molecular orbital diagram.

Features

  • Exact ψnlm for n ≤ 7, as real orbitals (pₓ, d_xy …) or complex orbitals with definite m
  • Cross-section heat map with phase colouring, automatic node lines and a |ψ|² readout under the pointer
  • 3D electron cloud sampled point by point from the true distribution, rotatable with a cut-away view
  • Radial distribution with most probable radius, ⟨r⟩, a₀ and nodes; compare every l for the same n
  • Superposition animation with transition dipole, selection rules and spontaneous rate (6.27×10⁸ s⁻¹ for 2p→1s)
  • Energy, degeneracy, node counts, ⟨r⟩, √⟨r²⟩, |L|, L_z and parity for every state