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

${\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi =E\psi }$