## Quantum Ring Sequence

Students calculate probabilities and expectation values for a quantum mechanical particle confined to a circular ring in bra/ket, matrix, and wave function representations and compare the different calculation methods. Several different graphical representations of the time dependence for both states with special symmetry and arbitrary states are explored in a Mathematica notebook. Compared to the analogous particle-in-a-box, this quantum system has a new feature---degenerate energy eigenstates.
##### 1. Activity: Energy and Angular Momentum for a Quantum Particle on a Ring
Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.
##### 2. Activity: Superposition States for a Particle on a Ring
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.
##### 3. Activity: Time Evolution Refresher (Mini-Lecture)
The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.
##### 4. Activity: Time Dependence for a Quantum Particle on a Ring Part 1
Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.
##### 5. Activity: Time Dependence for a Quantum Particle on a Ring Part 2Unpublished
This activity is a follow up to Part 1, where students practice time evolution, visualize the time dependent wavefunction, and calculation probabilities of $\hat{L}_z$ and $\hat{H}$ which are NOT time dependent. Here we follow up on that work by having them calculate the probability of finding the particle in a region of the ring which WILL depend on time. These two activities work together and highlight how probabilities come to be time dependent, which can be followed up with homework problems to drive these messages home.
##### 6. Activity: Visualization of Quantum Probabilities for a Particle Confined to a Ring
Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.
##### 7. Activity: Expectation Values for a Particle on a Ring
Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.