A particle free to move on a circle of fixed radius. The eigenfunctions are the particle-on-a-ring states Φm(φ) = eimφ/√(2π) — the azimuthal building block of the sphere and the hydrogen atom. Energies depend on m²: Em = ℏ²m²/2I, so +m and −m (opposite circulation) are degenerate. Choose the integer m.
Energy grows as m². Because E depends only on m², the states +m and −m (clockwise vs counter-clockwise circulation) share each level: every level except m=0 is doubly degenerate. The selected level is split into its states; the selected one is circled.
Φm(φ) and |Φm|² vs φ — cartesian
Wave on the ring — height = Φ (real part), color = phase/sign · drag to rotate