Student handout: Time Dependence for a Quantum Particle on a Ring Part 1

Central Forces 2021
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
  • Practice time evolution
  • To calculate quantum probabilities in Dirac and Wavefunction notation
  • To visualize how wave wavefunctions depend on time after time evolution
  • To notice that not all probabilities of operators will depend on time

Consider a quantum particle on a ring. At \(t=0\), the particle is in state: \begin{equation*} \left|{\Phi(t=0)}\right\rangle =\frac{7i}{10}\left|{-2}\right\rangle -\frac{1}{2}\left|{-1}\right\rangle +\frac{1}{2}\left|{0}\right\rangle -\frac{1}{10}\left|{2}\right\rangle \end{equation*}

  1. Find \(\left|{\Phi(t)}\right\rangle \)
  2. Go to the Ring States GeoGebra applet on the course schedule. Explore changing values of initial coefficents in the applet and see how \(Re(\psi)\), \(Im(\psi)\), and \(|\psi|^2\) change with time.
    Then, create the state given above. How do both pieces of the wavefunction and the probability density change with time?
  3. Calculate the probability that you measure the \(z\)-component of the angular momentum to be \(-2\hbar\) at time \(t\). Is it time dependent?
  4. Calculate the probability that you measure the energy to be \(\frac{2\hbar^2}{I}\) at time \(t\). Is it time dependent?

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