Activity: Wavefunctions on a Quantum Ring

Central Forces 2023 (2 years)

Consider the following normalized quantum state on a unit ring (\(r_0 = 1\)): \begin{equation} \Phi(\phi)=\sqrt\frac{8}{3 \pi } \sin^{2}\left( 3\,\phi \right)\cos \left( \phi \right) \end{equation}

  1. Translate this state into eigenfunction, bra/ket, and matrix representations. Remember that you can use any of these representations in the following calculations.

  2. What is the probability that the particle can be found in the region \(0<\phi< \frac{\pi}{4}\)? In the region \(\frac{\pi}{4}<\phi< \frac{3 \pi}{4}\)?
  3. What is the expectation value of \(L_z\) in this state?
  4. The wave function and it's probability density are plotted below. What features of these graphs (if any) tell you the expectation value of \(L_z\) in this state?

Learning Outcomes