⚛️ Hydrogen Atom Orbitals 3D

The hydrogen atom is the only atom whose Schrödinger equation has an exact solution. Each stationary state factorises into a radial part and an angular part, ψₙₗₘ(r,θ,φ) = Rₙₗ(r) · Yₗₘ(θ,φ). The radial function uses a generalised Laguerre polynomial, Rₙₗ ∝ ρˡ e^(−ρ/2) L^(2l+1)ₙ₋ₗ₋₁(ρ) with ρ = 2r / (n a₀); the angular part uses associated Legendre polynomials. This tool shows the real orbitals chemists draw (pₓ, d_xy, d_z² …), which are sums of the complex ±m states.

Every dot is a random position drawn from the probability density |ψ|². The radius is sampled exactly from the radial distribution P(r) = r²R² (inverse CDF), and the direction by rejection sampling on |Y|², so dense regions really are where the electron is most likely to be found. Colour shows the sign of ψ — the phase that decides whether orbitals overlap constructively when bonds form.

• Energy: Eₙ = −13.598 eV / n² (reduced-mass Rydberg). It depends only on n — all n² orbitals in a shell are degenerate.
• Nodes: n−l−1 radial (spherical shells, best seen in Slice view) and l angular (planes or cones), n−1 in total.
• Mean radius: ⟨r⟩ = a₀ [3n² − l(l+1)] / 2, with a₀ = 52.918 pm.
• Hydrogen-like ions: for a single electron around a nucleus of charge Z, energies scale by Z² and every length by 1/Z — so He⁺ is a quarter the size of H with four times the binding. The cloud’s shape is unchanged; its axis is in units of a₀/Z. Selection rules for a photon: Δl = ±1, Δm = 0, ±1.
• Transitions: 1/λ = R_H (1/nₗ² − 1/nᵤ²), R_H = 1.09678×10⁷ m⁻¹. Hα (3→2) is 656.47 nm in vacuum (656.28 nm in air).

Worked example: 2→1 (Lyman-α) gives 1/λ = 1.09678×10⁷ × 3/4 → λ = 121.57 nm, E = 10.20 eV — deep ultraviolet.