Particle in a ring

The Schrödinger equation for a free particle which is restricted to a ring (technically, whose configuration space is the circle

(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 The energy eigenvalues

are quantized because of the periodic boundary conditions, and they are required to satisfy The eigenfunction and eigenenergies are Therefore, there are two degenerate quantum states for every value of

The case of a particle in a one-dimensional ring is an instructive example when studying the quantization of angular 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.

The statement that any wavefunction for the particle on a ring can be written as a superposition of energy eigenfunctions is exactly identical to the Fourier theorem about the development of any periodic function in a Fourier series.

This simple model can be used to find approximate energy levels of some ring molecules, such as benzene.

This ring behaves like a circular waveguide, with the valence electrons orbiting in both directions.