This must be solved under the circularity condition. Let the ring's radius be r, then in one dimension:
where α is the position angle on the ring. We get
The solution of this is
The circularity condition is now, that
eli
Now the wave function becomes
Quantum states found:
:
Application
In organic 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). These molecules are exceptionally stable.
The above explaines why: the ring behaves like a circular wave guide. The excess (valency) electrons spin around in both directions.
Every energy level is filled by electrons, as electrons have additionally two possible orientations of their spins.
The rule that excess electrons in the ring produces an exceptionally stable ("aromatic") compound, is know as the Hückel rule.