This Demonstration studies the pure rotational spectrum of the quantum rigid rotor problem (neglecting centrifugal distortion), described by the Hamiltonian

, where

is the angular momentum operator and

is the moment of inertia. The energy levels are given by

, where

is the rotational constant and the transition frequency between two adjacent levels is

(selection rules only allow transitions that satisfy

). The left graphic shows the spectrum consisting of a series of equally spaced lines, where the relative intensity of each line depends on the probability to occupy the initial state in thermal equilibrium,

, where

is the Boltzmann constant and

is the temperature. The left and right graphics show the spectrum for a given ratio

and the energy level diagram, and the red highlighting shows the correspondence between a pair of energy levels and a specific transition frequency.