Requiring that the wave function be periodicin with a period (from the demand that the wave functions be single-valued functions on the circle), and that they be normalized leads to the conditions
,
and
Under these conditions, the solution to the Schrödinger equation is given by
Therefore, there are two degenerate quantum states for every value of (corresponding to ). Therefore, there are 2n+1 states with energies up to an energy indexed by the number n.
The case of a particle in a one-dimensional ring is an instructive example when studying the quantizationofangular momentum for, say, an electron orbiting the nucleus. The azimuthal wave functions in that case are identical to the energy eigenfunctions of the particle on a ring.
Inorganic chemistry, aromatic compounds contain atomic rings, such as benzene rings (the Kekulé structure) consisting of five or six, usually carbon, atoms. So does the surface of "buckyballs" (buckminsterfullerene). This ring behaves like a circular waveguide, with the valence electrons orbiting in both directions. To fill all energy levels up to n requires electrons, as electrons have additionally two possible orientations of their spins. This gives exceptional stability ("aromatic"), and is known as the Hückel's rule.
Further in rotational spectroscopy this model may be used as an approximation of rotational energy levels.