rigid rotor
The rigid rotor is a simple, idealised model of a rotating object — most usefully a small molecule — treated as masses held a fixed distance apart, free to tumble in space. Picture two atoms joined by a stiff, unstretchable bar, spinning end over end. Because the only motion is rotation, all of its energy is rotational, and quantizing that motion is one of the cleanest applications of angular-momentum theory.
Solving the quantum rigid rotor, the allowed energies turn out to be E = ℏ²·ℓ(ℓ+1) / 2I, where I is the moment of inertia and ℓ is the familiar angular-momentum quantum number. The levels are not evenly spaced; they spread further apart as ℓ grows. Each level also comes with 2ℓ+1 orientations, the same m values as before, which matters when a field or collision can tell those orientations apart.
This model is the backbone of rotational spectroscopy. When a molecule jumps between rotational levels it absorbs or emits light at microwave frequencies, producing a ladder of evenly spaced spectral lines that fingerprints the molecule and reveals its bond length. Real molecules stretch and bend a little, so the rigid rotor is an approximation — but a remarkably good first one, and the natural starting point for understanding molecular rotation.
Rotational energy levels widen as ℓ grows, set by the moment of inertia I.
Energy levels grow as ℓ(ℓ+1), so neither the levels nor the gaps between them are evenly spaced — the gaps widen as 2ℓ. Yet the spectral lines themselves are evenly spaced, because each successive transition energy is larger than the last by the same fixed amount, which is why a pure rotational spectrum looks like a comb of equally spaced lines.