A Particle Moving Along a Ring with Variable Potential

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1. Oct 16, 2016

Automata-Theory

1. The problem statement, all variables and given/known data
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 $\Psi \left(x\right)$?

2. Relevant equations
Here's my Schrodinger 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.

3. 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!)

Last edited: Oct 16, 2016
2. Oct 16, 2016

Staff: Mentor

This ain't nothing.