Particle in a ring

In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The Schrödinger equation for a free particle which is restricted to a ring (technically, whose configuration space is the circle ${\displaystyle S^{1}}$) is

with boundary conditions

expressing the fact that the particle is in a ring.

Using polar coordinates on the 1-dimensional ring of radius R, the wave function depends only on the angular coordinate, and so

Requiring that the wave function be periodic in ${\displaystyle \ \theta }$ with a period ${\displaystyle 2\pi }$ (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 ${\displaystyle E}$ are quantized because of the periodic boundary conditions, and they are required to satisfy