The bound-state wavefunctions separate as ψnℓm(r,ϑ,φ) = Rnℓ(r) · Yℓm(ϑ,φ). Choose the quantum numbers — only the physically allowed values are offered: ℓ = 0…n−1 and m = −ℓ…+ℓ. Lengths are in units of the Bohr radius a (a = 1), nuclear charge Z = 1.
The energy depends only on n. The selected level is split into its n² degenerate states — one segment per state, grouped by ℓ (s, p, d, …) with 2ℓ+1 values of m each (colors = ℓ, selected state circled); the other levels are plain lines. Levels get shallower and denser as they converge to the ionization limit E = 0.
Normalized Laguerre function. Radial nodes: n − ℓ − 1.
Rnℓ(r) — amplitude
P(r) = r²|Rnℓ|² — radial probability density
Legendre Θ(ϑ) × particle-on-a-ring Φm(φ) — amplitude and probability density |·|².
Polar plots (z-axis up) — Θℓm(ϑ) = Nℓm Pℓm(cos ϑ)
Θ(ϑ) amplitude — polar
|Θ(ϑ)|² density — polar
Cartesian plots — Θ(ϑ) and the ring Φm(φ) = eimφ/√(2π)
Θ(ϑ) & |Θ(ϑ)|² vs ϑ
Φm(φ): Re & Im vs φ
3D boundary surface of |ψ|² (L̂z eigenstate) — axially symmetric about z · drag to rotate
3D boundary surface of ψ — blue: ψ>0, red: ψ<0 (same enclosed % as above) · drag to rotate