A Particle Moving Along a Ring with Variable Potential

[Automata-Theory]

Homework Statement

Alrighty, so here's my problem in a nutshell:

Some particle of mass m is confined to move along a ring of radius R. Since it's on a ring, it has periodic boundary conditions--i.e.:

For the boundary defined as $-\pi R \leq x \leq \pi R$, $x = -\pi R$ and $x = \pi R$ is the same physical point.

Now here's where things get interesting. The circular ring has a varied potential. At the region enclosed by $- \frac \pi 2 R \leq x \leq \frac \pi 2 R$, the potential $V \left( x \right) = 0$. For the remaining half of the ring, however, the potential $V \left(x\right) = V_0 \gt 0$.

The question is, how does a varied potential along a ring change how you solve for the wave function $\Psi \left(x\right)$?

Homework Equations

Here's my Schrödinger equation:

$H \Psi = -\frac {\hbar^2} {2m} \frac {1} {R^2} \frac {\partial^2 \Psi} {\partial \phi^2} + V \left( R \right)\Psi = i \hbar \frac {\partial \Psi} {\partial t}$

I figured that $\Psi$ would be independent of $R$ and $\theta$ because of the periodic boundary conditions, turning this into a one-dimensional problem. I know how to solve for where the potential across the entire ring is zero, but I'm stumped at where to start when it's varied across the ring.

The Attempt at a Solution

This is how I solved for a ring where the potential was zero:

$- \frac {\hbar^2}{2m} \frac {d\psi^2}{dx^2} + V\left(x\right) = E\psi$

Solutions are in the form of $\cos\left(kx\right), \sin\left(kx\right)$, or $e^{\pm ikx}$. There's degeneracy since there are multiple solutions for energy levels. The plane is infinitely degenerate, since the energy depends only on the magnitude and not on direction.

$k^2 = \frac {2mE}{\hbar^2} \gt 0$

$k = \sqrt \frac {2mE}{\hbar^2} \gt 0$

So I could use that value of $k$ to represent the energy levels, due to the form of the solutions depending on $k$. Could I do something similar with my current problem? I already think I can treat it like a one-dimensional problem, and certainly for half of the ring I can apply the above approach, where $V\left(x\right) = 0$. How would I go about solving for the other half though, and how would I formulate a cohesive transcendental equation with both halves? Thank you in advance for any insights.

EDIT: So I just thought, since it has repeating boundary conditions, it essentially "oscillates," so should I approach the second region as a harmonic oscillator problem? I'll try it and see what I get.

EDIT #2: I ran into some problems with the harmonic oscillator approach, and I'm trying out a different method. I remembered that a ring can be related to an infinite square potential well so I'm going with that idea.

EDIT #3: SOLVED IT! Man, I feel so smort right now. THANKS FOR NOTHING, CHUMPS! (JK--Posting my question really helped me brainstorm! Thanks, Physics Forums!)

 

[fresh_42]

THANKS FOR NOTHING, CHUMPS! (JK--Posting my question really helped me brainstorm! Thanks, Physics Forums!)

This ain't nothing.

 

[Trock]

Hi Automata-Theory,

I am also dealing with a similar problem and I am curious as to how you went about solving this. How did you set up a transcendental equation, or did you have any luck relating the ring to an infinite square well problem?

Cheers!