## Activity: 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.
• This activity is used in the following sequences
What students learn
• To calculate quantum probabilities in Dirac and Wavefunction notation
• To identify when probabilities depend on time
• Media

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?

## Time Dependence for a Particle on a Ring: Instructor's Guide

### Introduction

This activity is about calculating probabilities with states that depend on time. It is a good idea to remind students how to time evolve a state by giving students the initial state (that is on the handout) and ask them to write $\vert\psi(t)\rangle$ on small whiteboards.

### Student Conversations

• Degeneracy: Students may experience some difficulty due to the degeneracy of some states, in particular that you are calculating probability of measuring a certain value, you have to include all the states that share that eigenvalue.. $P_{E={m^2\,\hbar^2\over 2I}}=\vert \langle m\vert \psi\rangle\vert^2+\vert \langle -m\vert \psi\rangle\vert^2$
• Probability v. Probability Density: Students struggle with the two different ways of finding probability: for discrete and continuous measurements. Most recognize that they need to do an integral for a continuous quantity, but are not sure when to square (before integration or after).

$\left|\int \phi_n^*(x)\Psi(x,t) dx\right|^2$ vs. $\int\left|\Psi(t) \right|^2 dx$

In particular, many students will forget to do the squaring for the calculation on the left because $\int \phi_n^*(x) \Psi(x,t) dx$ looks a lot like $\int \Psi^*(x,t) \Psi(x,t) dx$.

• Notation: Depending on how much you have done beforehand, some students still struggle with writing the energy eigenstates for the ring in wave function notation. In particular, they may forget to include the normalization constant. $|m\rangle\doteq\frac{1}{\sqrt{2\pi}}e^{im\phi}$
• When calculating the probability of finding the particle in a region, some students will recognize some of the exponential cross terms can be written as a sine function $2i\sin \omega t = e^{i\omega t}-e^{-i\omega t}$. This is a nice detail to point out in the wrap up, but it is more important that the students take the integral over position and see that the position probability changes with time.
• Some will state without showing that the energy and angular momentum probabilities do not change with time. Ask them to calculate this explicitly to make sure that everyone in the group understands why (because the time-dependent phases norm square to 1 and there are no cross terms remaining). Understanding this calculation makes the comparison to position probability much easier - you see the cross terms go away for energy and angular momentum.

### Wrap-up

This is a good activity to have a group present their results. This allows the whole class to see the worked out solution without redoing it for them, but still allows you to point out the important features of the problem.
• Remind students how to deal with degeneracy.
• Reiterate the two ways of finding a probability and how they are connected.
• For discrete measurements:$P_{a_n}=\vert\langle a_n \vert \psi\rangle \vert^2\doteq\left|\int_{0}^{2\pi}\Phi_n^*(x)\psi(x)dx\right|$
• For continuous measurements:$P_{a<x<b}=\sum_{x=a}^b\vert\langle x \vert \psi\rangle \vert^2 \doteq\int_a^b\vert\psi(x) \vert^2 dx$
• Mention that the exponential cross terms can often be written as a sine or cosine (depending on the phase).
• In discussing under what circumstances measurement probabilities change with time, it is a good idea to connect it to earlier activities (from Spins and Waves courses) where they saw that probabilities were only time-dependent if the operator did not commute with the Hamiltonian.

### 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 another activity and/or homework from the Quantum Ring Sequence that allows students to practice this more general method.

• group Energy and Angular Momentum for a Quantum Particle on a Ring

group Small Group Activity

30 min.

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

Quantum Ring Sequence

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.
• group Superposition States for a Particle on a Ring

group Small Group Activity

30 min.

##### Superposition States for a Particle on a Ring

Quantum Ring Sequence

Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.
• group Expectation Values for a Particle on a Ring

group Small Group Activity

30 min.

##### Expectation Values for a Particle on a Ring
Central Forces 2023 (3 years)

Quantum Ring Sequence

Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.
• format_list_numbered Quantum Ring Sequence

format_list_numbered Sequence

##### Quantum Ring Sequence
Students calculate probabilities and expectation values for a quantum mechanical particle confined to a circular ring in bra/ket, matrix, and wave function representations and compare the different calculation methods. Several different graphical representations of the time dependence for both states with special symmetry and arbitrary states are explored in a Mathematica notebook. Compared to the analogous particle-in-a-box, this quantum system has a new feature---degenerate energy eigenstates.
• group Working with Representations on the Ring

group Small Group Activity

30 min.

##### Working with Representations on the Ring
Central Forces 2023 (3 years)
• 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 $r_0=1$ $$\left|{\Phi_a}\right\rangle = i\sqrt{\frac{ 2}{12}}\left|{3}\right\rangle - \sqrt{\frac{ 1}{12}}\left|{1}\right\rangle +\sqrt{\frac{ 3}{12}}e^{i\frac{\pi}{4}}\left|{0}\right\rangle -i\sqrt{\frac{ 2}{ 12}}\left|{-1}\right\rangle +\sqrt{\frac{ 4}{12}}\left|{-3}\right\rangle$$ $$\left| \Phi_b\right\rangle \doteq \left( \begin{matrix} \vdots \\ i\sqrt{\frac{ 2}{12}}\\ 0 \\ -\sqrt{\frac{ 1}{12}} \\ \sqrt{\frac{ 3}{12}}e^{i\frac{\pi}{4}} \\ -i\sqrt{\frac{ 2}{12}}\\ 0 \\ \sqrt{\frac{4}{12} }\\ \vdots \end{matrix}\right) \begin{matrix} \leftarrow m=0 \end{matrix}$$ $$\Phi_c(\phi) \doteq \sqrt{\frac{1}{24 \pi}} \left( i\sqrt{2}e^{i 3 \phi} -e^{i\phi} +\sqrt{3}e^{i\frac{\pi}{4}} -i \sqrt{2} e^{-i\phi} + \sqrt{4}e^{-i 3 \phi} \right)$$

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?

• group Magnetic Field Due to a Spinning Ring of Charge

group Small Group Activity

30 min.

##### Magnetic Field Due to a Spinning Ring of Charge
Static Fields 2022 (6 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use the Biot-Savart law $\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}$ to find an integral expression for the magnetic field, $\vec{B}(\vec{r})$, due to a spinning ring of charge.

In an optional extension, students find a series expansion for $\vec{B}(\vec{r})$ either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• group Electrostatic Potential Due to a Ring of Charge

group Small Group Activity

30 min.

##### Electrostatic Potential Due to a Ring of Charge
Static Fields 2022 (7 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Warm-Up

Students work in groups of three to use the superposition principle $V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}$ to find an integral expression for the electrostatic potential, $V(\vec{r})$, everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for $V(\vec{r})$ either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• group Magnetic Vector Potential Due to a Spinning Charged Ring

group Small Group Activity

30 min.

##### Magnetic Vector Potential Due to a Spinning Charged Ring
Static Fields 2022 (5 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use the superposition principle $\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}$ to find an integral expression for the magnetic vector potential, $\vec{A}(\vec{r})$, due to a spinning ring of charge.

In an optional extension, students find a series expansion for $\vec{A}(\vec{r})$ either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• computer Visualization of Quantum Probabilities for a Particle Confined to a Ring

computer Mathematica Activity

30 min.

##### Visualization of Quantum Probabilities for a Particle Confined to a Ring
Central Forces 2023 (3 years)

Quantum Ring Sequence

Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.

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