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 semicircle (technically, whose configuration space is the circle) is

Wave function

Using cylindrical coordinates on the 1-dimensional semicircle, the wave function depends only on the angular coordinate, and so

Substituting the Laplacian in cylindrical coordinates, the wave function is therefore expressed as

The moment of inertia, which is easily solvable.

, the general solution to this differential equation is

are continuous and the wave function is normalizable:

. Basically

so we must apply this boundary condition. Discarding the trivial solution where B only when m where m is any integer. The condition m everywhere, meaning that the particle is not in the potential at all. Negative integers are also ruled out since they can easily be absorbed in the normalization condition.

. The normalized wave function is

The ground state are both 0, which means that the probability of finding the particle at these nodes are 0.

Analysis

respectively.

respectively, where these observables play a role similar to position and momentum for the particle in a box. The commutation and uncertainty relations for angular position and angular momentum are given as follows:

Boundary conditions

, the particle is subject only to a periodic boundary condition (see particle in a ring, the issue of even and odd parity becomes important.

The wave equation for such a potential is given as: are for odd and even m respectively.

replace the linear operators x and p.

See also

• Particle in a ring

• Particle in a box

• Finite potential well

• Delta function potential

• Gas in a box

• Particle in a spherically symmetric potential