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## Quantum Calculations for a Particle Confined to a Ring

This sequence of activities gives students a chance to practice a variety of calculations for a specific system: a quantum particle confined to a ring. In particular, these activities help students to see that all of the calculations that they have done previously in the Spins and Waves courses can be applied to these new systems. In addition to the main goal(s) for each activity, they all provide an opportunity for students to deal with degeneracy and to move between the different representations/notations that they have used (bra-ket notation, matrix notation, and wave function notation).

Typically, the first few activities are done in class and the rest are incorporated into the homework using these or similar problems (wiki version, cfqmringhomeworkcombined.pdf).

### Activities

**Energy and Angular Momentum for a Particle on a Ring:**This small group activity allows upper-division students to identify eigenstates and associated eigenvalues of a particle confined to a ring and then calculate probabilities for energy and angular momentum for a particle confined to a ring in Dirac “bra-ket” notation, matrix notation, and wavefunction notation. One of the main purposes of this activity is to help students see the parallel between similar calculations in these three representations.

**Time Dependence for a Particle on a Ring:**Students work in small groups to calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of eigenstates for a particle confined to a ring. This activity is intended to build student understanding of when they can expect a measurement probability to be time dependent. Notation fluency is required as students are initially asked to work in Dirac notation but will need to use wavefunction notation for some of the calculations; students are asked to actively reflect on their chosen notation for each part.

**Superposition States for a Particle on a Ring:**In this small group activity, students are asked to calculate probability for energy, angular momentum and position on a wavefunction that is a specific superposition state for a particle on a ring. The state students work with is not easily separated into eigenstates which prevents students from using the strategy of finding probability amplitudes “by inspection” as is done when given an initial state written as a sum of eigenstates.

**Expectation Values for a Particle on a Ring:**Students calculate the expectation values for energy and angular momentum as a function of time.

### Quantum Calculations for a Hydrogen Atom

The ring activities can also be sequenced with activities that ask the students to do similar calculations for a particle confined to sphere or for the hydrogen atom. For example:

**Quantum Calculations on the Hydrogen Atom:**Students calculate probabilities and expectation values for energy and angular momentum for a hydrogen in a linear combination of $\vert n\ell m\rangle$ states.

**Probability of Finding an Electron Inside the Bohr Radius:**Students work in small groups to calculate the probability that an electron in a hydrogen atom will be within one Bohr radius of the center.