Lecture: Unit Learning Outcomes: Quantum Mechanics on a Ring

Central Forces 2023

In this unit, you will explore the quantum mechanics of a simple system: a particle confined to a one-dimensional ring.

Motivating Questions

  • What are the energy eigenstates, i.e. eigenstates of the Hamiltonian?
  • What physical properties of the energy eigenstates can be measured?
  • What other states are possible and what are their physical properties?
  • How do the states change if this system and their physical properties depend on time?

Key Activities/Problems

Unit Learning Outcomes

At the end of this unit, you should be able to:

  • Describe the energy eigenstates for the ring system algebraically and graphically.
  • List the physical measurables for the system and give expressions for the corresponding operators in bra/ket, matrix, and position representations.
  • Give the possible quantum numbers for the quantum ring system and describe any degeneracies.
  • For a given state, use the inner product in bra/ket, matrix, and position representations, to find the probability of making any physically relevant measurement, including states with degeneracy.
  • Use an expansion in energy eigenstates to find the time dependence of a given state.

Equation Sheet for This Unit

Learning Outcomes