Activity: Energy and Angular Momentum for a Quantum Particle on a Ring

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
  • Media
    • activity_media/EnergyAngularMomentumParticleRing.tex

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).

Energy and Angular Momentum for a Quantum Particle on a Ring: Instructor's Guide

Energy and Angular Momentum for a Quantum Particle on a Ring Handout

Introduction

This activity flows naturally from a lecture in which the eigenstates for energy and angular momentum on a ring are found. Many of the calculations done here are similar to calculations they have done before, but this activity emphasizes the different representations we use for quantum calculations and highlights when each representation is most useful.

Remind the students that that an arbitrary state \(|\Phi\rangle\) can be written in the \(L_z\) eigenbasis as

\[ \eqalign{\left| \Phi\right\rangle &\doteq \pmatrix{\vdots \cr \langle 2|\Phi\rangle \cr \langle 1|\Phi\rangle \cr \langle 0|\Phi\rangle \cr \langle -1|\Phi\rangle \cr \langle -2|\Phi\rangle \cr \vdots} = \pmatrix{\vdots \cr a_{2} \cr a_{1} \cr a_{0} \cr a_{-1} \cr a_{-2} \cr \vdots}} \]

Including this in the introduction to this activity should help students avoid confusion about the ordering of the elements in the column vectors used in this activity.

Student Conversations

  • Notation:
    • For the bra-ket notation, at this stage, many students can find probabilities by inspection. It is a good idea to encourage them to write out their process at least once for each notation in order to be able to better compare them.
    • The three initial states on the handout are identical, but it may take most of the activity for students to realize this, especially for the wavefunction notation.
      • In particular, students struggle with the fact that the coefficient and the normalization constant are combined in the wave function notation (e.g. "Where does this \(\pi\) come from?")
      • If they do recognize that they are the same state, encourage them to do at least one calculation in each notation in order to show that they get the same answer.
  • Zero: Students may be confused in the case where they measure zero probability or when the observed value/eigenvalue for a quantity is zero.
  • Degeneracy: Students may experience some difficulty due to the degeneracy of some states. Remind them:
    • they have to include all states that have the same eigenvalue
    • probability is then the sum of the squares (of the norm of the coefficients) and not the square of the sum of coefficients. \[P_{E={4\hbar^2\over 2I}}=\vert \langle 2\vert \psi\rangle\vert^2+\vert \langle -2\vert \psi\rangle\vert^2\neq \vert \langle 2\vert \psi\rangle+\langle -2\vert \psi\rangle\vert^2\]
  • Operators vs Measurements: Students commonly attempt to determine the values resulting from a quantum experiment by allowing the operator corresponding to the observable of interest to act on the initial state. Students who do this should be encouraged to consider the nature of this transformation (it's a vector, not a scalar) and to recognize that the transformation does not necessarily yield an eigenvector (the state of the system should be an eigenstate after the measurement).

Wrap-up

There are several important ideas to bring up in the wrap-up discussion:
  • Notation: Empasise the fact that all three states are the same state written in different representations. You can talk about which types of questions might be easier answered with one representation over another and being able translating between notations is important.
  • This activity serves as a great introduction to a mini-lecture reminding students that you can write any state \(|\psi\rangle\) as a linear combination of eigenstates \(|m\rangle\).
  • This is a good opportunity to emphasize to students that there are only a limited number of different types of quantum calculations they can do once they know the eigenstates and eigenvalues for a system. In particular this activity encourages them to tie these measurements directly to the postulates of quantum mechanics. In general this should be relatively straight forward, however, students may struggle to relate the quantum postulates and calculations they know in bra-ket and matrix notation to the calculations they do in the position basis. This activity may lead to a mini-lecture/review of how wavefunctions can be written as states in the continuous, position basis using bra-ket notation. \[|\psi\rangle = \sum_m c_m |m\rangle\]
  • Degeneracy: Reiterate that the probability of a degenerate eigenvalue is the sum of the square of the norm of the coefficients associated with the degenerate eigenstates. \[P_{E={m^2\, \hbar^2\over 2I}}=\vert \langle m\vert \psi\rangle\vert^2+\vert \langle -m\vert \psi\rangle\vert^2\]
  • This activity can be used to help review and solidify students understanding of the formalism used to make quantum calculations. This is especially important if you expect to proceed from this activity to calculations of the rigid rotor and hydrogen atom which use the same formalism with substantially more challenging eigenfunctions.

Extensions and Related Material

Students readily grasp the strategy of finding probability amplitudes “by inspection” when they are given an initial state written as a sum of eigenstates. We find that students then find it extremely difficult to find probability amplitudes of wavefunctions that are not written this way (i.e. using an integral to find the expansion coefficients of a function). This activity should be followed up with other activities in the Quantum Ring Sequence and homework that allow students to practice this more general method.

Associated Homework Problem: QM Ring Compare

  • assignment Working with Representations on the Ring

    assignment Homework

    Working with Representations on the Ring
    Central Forces 2023 (3 years)

    The following are 3 different representations for the \(\textbf{same}\) state on a quantum ring for \(r_0=1\) \begin{equation} \left|{\Phi_a}\right\rangle = i\sqrt{\frac{ 2}{12}}\left|{3}\right\rangle - \sqrt{\frac{ 1}{12}}\left|{1}\right\rangle +\sqrt{\frac{ 3}{12}}e^{i\frac{\pi}{4}}\left|{0}\right\rangle -i\sqrt{\frac{ 2}{ 12}}\left|{-1}\right\rangle +\sqrt{\frac{ 4}{12}}\left|{-3}\right\rangle \end{equation} \begin{equation} \left| \Phi_b\right\rangle \doteq \left( \begin{matrix} \vdots \\ i\sqrt{\frac{ 2}{12}}\\ 0 \\ -\sqrt{\frac{ 1}{12}} \\ \sqrt{\frac{ 3}{12}}e^{i\frac{\pi}{4}} \\ -i\sqrt{\frac{ 2}{12}}\\ 0 \\ \sqrt{\frac{4}{12} }\\ \vdots \end{matrix}\right) \begin{matrix} \leftarrow m=0 \end{matrix} \end{equation} \begin{equation} \Phi_c(\phi) \doteq \sqrt{\frac{1}{24 \pi}} \left( i\sqrt{2}e^{i 3 \phi} -e^{i\phi} +\sqrt{3}e^{i\frac{\pi}{4}} -i \sqrt{2} e^{-i\phi} + \sqrt{4}e^{-i 3 \phi} \right) \end{equation}

    1. With each representation of the state given above, explicitly calculate the probability that \(L_z=-1\hbar\). Then, calculate all other non-zero probabilities for values of \(L_z\) with a method/representation of your choice.
    2. Explain how you could be sure you calculated all of the non-zero probabilities.
    3. If you measured the \(z\)-component of angular momentum to be \(3\hbar\), what would the state of the particle be immediately after the measurement is made?
    4. With each representation of the state given above, explicitly calculate the probability that \(E=\frac{9}{2}\frac{\hbar^2}{I}\). Then, calculate all other non-zero probabilities for values of \(E\) with a method of your choice.
    5. If you measured the energy of the state to be \(\frac{9}{2}\frac{\hbar^2}{I}\), what would the state of the particle be immediately after the measurement is made?

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    1. Find the normalized eigenfunctions for a particle in a 2-D box with sides of length \(L_x\) in the \(x\)-direction and length \(L_y\) in the \(y\)-direction.
    2. Find the Hamiltonian for a 2-D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies
    3. Any sufficiently smooth spatial wave function inside a 2-D box can be expanded in a double sum of the product wave functions, i.e. \begin{equation} \psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y) \end{equation} Using your expressions from part (a) above, write out all the terms in this sum out to \(n=3\), \(m=3\). Arrange the terms, conventionally, in terms of increasing energy.

      You may find it easier to work in bra/ket notation: \begin{align*} \left|{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{n}\right\rangle \left|{m}\right\rangle \\ &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{nm}\right\rangle \end{align*}

    4. Find a formula for the \(c_{nm}\)s in part (b). Find the formula first in bra ket notation and then rewrite it in wave function notation.

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