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:

22Id2Φ(ϕ)dϕ2=EΦ(ϕ)

Let’s try to look at this based on operators. Consider Lz=xpyypx 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 /x and /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 θ and ϕ. (WolframAlpha can handle finding these derivatives.)
  3. Question: Find the Lz 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=r0sinθcosϕy=r0sinθsinϕz=r0cosθ r0=x2+y2+z2θ=cos1(zx2+y2+z2)ϕ=tan1(yx)