In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box.
The particle follows the path of a semicircle from
Instead there is total reflection, meaning the particle bounces back and forth between
The Schrödinger equation for a free particle which is restricted to a semicircle (technically, whose configuration space is the circle
) is 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 for a semicircle, best expressed in cylindrical coordinates, is
Solving the integral, one finds that the moment of inertia of a semicircle is
Since the particle cannot escape the region from
, the general solution to this differential equation is Defining
We then apply the boundary conditions, where
are continuous and the wave function is normalizable: Like the infinite square well, the first boundary condition demands that the wave function equals 0 at both
Basically Since the wave function
so we must apply this boundary condition.
Discarding the trivial solution where B=0, the wave function
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.
We then normalize the wave function, yielding a result where
The normalized wave function is The ground state energy of the system is
Like the particle in a box, there exists nodes in the excited states of the system where both
are both 0, which means that the probability of finding the particle at these nodes are 0.
Since the wave function is only dependent on the azimuthal angle
, the measurable quantities of the system are the angular position and angular momentum, expressed with the operators
Using cylindrical coordinates, the operators
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: As with all quantum mechanics problems, if the boundary conditions are changed so does the wave function.
If a particle is confined to the motion of an entire ring ranging from 0 to
If a particle is confined to the motion of
, the issue of even and odd parity becomes important.
The wave equation for such a potential is given as: where
Similarly, if the semicircular potential well is a finite well, the solution will resemble that of the finite potential well where the angular operators
replace the linear operators x and p.