Wikipedia: Particle in a ring

Particle in a ring
From Wikipedia, the free encyclopedia.

In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. We have to solve the Schrodinger equation

As in the other case, the potential V is zero.

Now as we have 1D ring, we can use polar coordinates, and as the wave function only depends on θ,

the wave function takes the form

Substituting this form into the Schrodinger equation and simplifying we find

and so

The value for E can be found using the periodic boundary conditions.

As the system is peroidic in &theta

i.e.

simplifying we have

now as

rearranging

where

Substituting for E in the wave function we have

n=0

For n=0, the un-normalised wave function ψ = 1 and E=0.

n=1,2,3...

As these states are doubly degenerate, with one state for and one for .

This means that the total number of states is 2n+1.