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 S 1 {\displaystyle S^{1}} ) is − ℏ 2 2 m ∇ 2 ψ = E ψ {\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi =E\psi } with boundary conditions ψ ( θ + 2 π ) = ψ ( θ ) {\displaystyle \psi (\theta +2\pi )=\psi (\theta )} expressing the fact that the particle is in a ring.