Day 19 In-Class Activity
Particle on a Ring
Last week, we worked on the particle on a ring problem before truly diving into the details. Let’s back up and try to understand the eigenvalue problem a bit more. The problem we set up was:
Let’s try to look at this based on operators. Consider
- Question: Write down the operator in position space using Cartesian coordinates.
- Question: Given our problem, we want this operator in spherical coordinates. Determine
and in spherical coordinates. There’s some relationships at the end that will be helpful. Start by using the chain rule to see how these partial derivatives depend on and . (WolframAlpha can handle finding these derivatives.) - Question: Find the
operator given your results for these partial derivatives.- Discussion: How does this operator feature in the differential equation above? If we found eigenstates associated with that operator, what would be the expected eigenvalues?