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:

\[-\dfrac{\hbar^2}{2I} \dfrac{d^2\Phi(\phi)}{d\phi^2} = E \Phi(\phi)\]

Let’s try to look at this based on operators. Consider $L_z = xp_y - yp_x$ in position space.

  1. Question: Write down the operator in position space using Cartesian coordinates.
  2. Question: Given our problem, we want this operator in spherical coordinates. Determine $\partial/\partial x$ and $\partial /\partial y$ 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 $\theta$ and $\phi$. (WolframAlpha can handle finding these derivatives.)
  3. Question: Find the $L_z$ 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?

Relationships between Cartesian and Spherical Coordinates

\[x = r_0 \sin \theta \cos \phi \qquad y = r_0 \sin \theta \sin \phi \qquad z = r_0 \cos\theta\] \[r_0 = \sqrt{x^2+y^2+z^2} \qquad \theta = \cos^{-1}\left(\dfrac{z}{\sqrt{x^2+y^2+z^2}}\right) \qquad \phi = \tan^{-1}\left(\dfrac{y}{x}\right)\]