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