Student handout: Energy and Angular Momentum for a Quantum Particle on a Ring

Theoretical Mechanics 2024
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.
What students learn
  • to determine possible measurement values and probabilities for a superposition state of a particle confined to a ring
  • to relate information and calculations among Dirac bra-ket, matrix, and wavefunction notation.
  • to deal with degeneracy when calculating probabilities

In this activity, your group will carry out calculations on each of the following normalized abstract quantum states on a ring: \begin{equation} \left|{\Phi_a}\right\rangle = \sqrt\frac{ 2}{12}\left|{3}\right\rangle + \sqrt\frac{ 1}{12}\left|{2}\right\rangle +\sqrt\frac{ 3}{12}\left|{0}\right\rangle +\sqrt\frac{ 2}{ 12}\left|{-1}\right\rangle +\sqrt\frac{ 1}{12}\left|{-3}\right\rangle +\sqrt\frac{ 3}{12}\left|{-4}\right\rangle \end{equation} \begin{equation} \left| \Phi_b\right\rangle \doteq \left( \begin{matrix} 0 \\ \sqrt\frac{ 2}{12}\\ \sqrt\frac{ 1}{12} \\ 0 \\ \sqrt\frac{ 3}{12} \\ \sqrt\frac{ 2}{12}\\ 0 \\ \sqrt\frac{1}{12} \\ \sqrt\frac{3}{12} \\ \end{matrix}\right) \end{equation} \begin{equation} \Phi_c(\phi) \doteq \sqrt {\frac{1}{24 \pi r_0}} \left( \sqrt{2}e^{i 3 \phi} +e^{i 2\phi} +\sqrt{3} + \sqrt{2} e^{-i 1 \phi} + e^{-i 3 \phi}+\sqrt{3}e^{-i 4 \phi} \right) \end{equation}

For each question state the postulate of quantum mechanics you use to complete the calculation and show explicitly how you use the postulates to answer the question.

  1. For each state above, what is the probability that you would measure the \(z\)-component of angular momentum to be \(-4\hbar\)? \(0\hbar\)? \(-2\hbar\)? \(3\hbar\)?
  2. What other possible values for the \(z\)-component of angular momentum could you have obtained with non-zero probability?
  3. For each state, what is the probability that you would measure the energy to be \(\displaystyle \frac{16\hbar^2}{2 I}\)? \(0\)? \(\displaystyle\frac{4 \hbar^2}{2 I}\)? \(\displaystyle \frac{9 \hbar^2}{2 I}\)?
  4. If you measured the energy, what other possible values could you obtain with non-zero probability?
  5. How are the calculations you made for the different state representations similar and different from each other? Be prepared to compare and contrast the calculations you made for each of the different representations (ket, matrix, eigenfunction).

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