Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.
1. < < Energy and Angular Momentum for a Quantum Particle on a Ring | Quantum Ring Sequence | Time Evolution Refresher (Mini-Lecture) > >
Consider the following normalized abstract quantum state on a ring: \begin{equation} \Phi(\phi)= \sqrt{\frac{8}{5\pi r_0}}\cos^3{(2\phi)} \end{equation}
- If you measured the \(z\)-component of angular momentum, what is the probability that you would measure \(2\hbar\)? \(-3\hbar\)?
- If you measured the \(z\)-component of angular momentum, what other possible values could you have obtained with non-zero probability?
- If you measured the energy, what possible values could you have obtained with non-zero probability?
- What is the probability that the particle can be found in the region \(0<\phi<\frac{\pi}{2}\)?
If the previous activities (Energy and Angular Momentum for a Quantum Particle on a Ring and Time Dependence for a Quantum Particle on a Ring) have been done, little introduction is needed. It might be helpful to ask a small whiteboard question to help them remember what the eigenfunctions for a particle on a ring are.
In many cases, students will not think to rewrite the function as a linear combination of eigenstates. Even if they do, many students will have forgotten how. Therefore, you might start this activity in class and have students finish the calculations for homework.