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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 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.
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!)
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