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:

$$-{\displaystyle \frac{{\hslash }^2}{2I}}{\displaystyle \frac{d^2\mathrm{\Phi }(\varphi )}{d{\varphi }^2}}=E\mathrm{\Phi }(\varphi )$$

Let’s try to look at this based on operators. Consider $L_z=x{p}_y-y{p}_x$ 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 $\varphi$. (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

$$x=r_0\sin \theta \cos \varphi y=r_0\sin \theta \sin \varphi z=r_0\cos \theta$$ $$r_0=\sqrt{x^2+y^2+z^2}\theta ={\cos }^{-1}\left({\displaystyle \frac{z}{\sqrt{x^2+y^2+z^2}}}\right)\varphi ={\tan }^{-1}\left({\displaystyle \frac{y}{x}}\right)$$