Student handout: Time Dependence for a Quantum Particle on a Ring

Theoretical Mechanics (6 years)
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.
What students learn
  • To calculate quantum probabilities in Dirac and Wavefunction notation
  • To identify when probabilities depend on time

In this activity, your group will carry out calculation on the following quantum state on a ring: \begin{equation*} \left|{\Phi}\right\rangle =\sqrt{\frac{2}{3}}\left|{-3}\right\rangle +\frac{1}{\sqrt{6}}\left|{-1}\right\rangle +\frac{i}{\sqrt{6}}\left|{3}\right\rangle \end{equation*}

  1. Imagine you carry out a measurement to determine the \(z\)-component of the angular momentum of the particle at time, \(t\). Calculate the probability that you measure the \(z\)-component of the angular momentum to be \(3\hbar\). What representation/basis did you use to do this calculation and why did you use this representation?
  2. Imagine you carry out a measurement to determine the energy of the particle at time, \(t\). Calculate the probability that you measure the energy to be \(\frac{9\hbar^2}{2I}\). What representation/basis did you use to do this calculation and why did you use this representation?
  3. Calculate the probability that the particle can be found in the region \(0<\phi<\frac{\pi}{3}\) at some time, \(t\). What representation/basis did you use to do this calculation and why did you use this representation?
  4. Under what circumstances do measurement probabilities change with time?


Author Information
Corinne Manogue, Kerry Browne, Elizabeth Gire, Mary Bridget Kustusch, David McIntyre
Keywords
central forces quantum mechanics eigenstates eigenvalues angular momentum time dependence hermitian operators probability degeneracy quantum measurements
Learning Outcomes