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. 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.
See also: Quantum mechanics: one-dimensional periodic case.
|
