A particle free to move on the surface of a sphere. The Hamiltonian is purely rotational, so the eigenfunctions are the spherical harmonics ψℓm(ϑ,φ) = Yℓm(ϑ,φ) — the angular part of the hydrogen atom, with no radial part. Energies depend only on ℓ: Eℓ = ℏ²ℓ(ℓ+1)/2I, each (2ℓ+1)-fold degenerate. Choose ℓ and m = −ℓ…+ℓ.
The energy depends only on ℓ. Unlike the hydrogen atom, levels spread apart as ℓ grows (∝ ℓ(ℓ+1)) and there is no upper bound. The selected level is divided into its 2ℓ+1 degenerate m-states (one segment each, labeled by m); the selected state is circled.
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
Balloon r(ϑ,φ) = |Yℓm(ϑ,φ)|² (axially symmetric about z) · drag to rotate
Balloon r(ϑ,φ) = |Yreal(ϑ,φ)| — blue: Y>0, red: Y<0 · drag to rotate