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

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.
• This activity is used in the following sequences
What students learn
• How to find expansion coefficients of wavefunctions using inner products (integration)
• How to write a compact wavefunction as a superposition (sum) of eigenstates
• Media
Consider the following normalized abstract quantum state on a ring: $$\Phi(\phi)= \sqrt{\frac{8}{5\pi r_0}}\cos^3{(2\phi)}$$
1. If you measured the $z$-component of angular momentum, what is the probability that you would measure $2\hbar$? $-3\hbar$?
2. If you measured the $z$-component of angular momentum, what other possible values could you have obtained with non-zero probability?
3. If you measured the energy, what possible values could you have obtained with non-zero probability?
4. What is the probability that the particle can be found in the region $0<\phi<\frac{\pi}{2}$?

## Superposition States for a Particle on a Ring: Instructor's Guide

This activity is the same as homework Superposition States for a Particle on a Ring

### Introduction

If the previous activities (Energy and Angular Momentum for a Quantum Particle on a Ring and Time Dependence for a Quantum Particle on a Ring Part 1) have been done, little introduction is needed. It might be helpful to ask a small whiteboard question to help them remember what the eigenfunctions for a particle on a ring are.

In many cases, students will not think to rewrite the function as a linear combination of eigenstates. Even if they do, many students will have forgotten how. Therefore, you might start this activity in class and have students finish the calculations for homework.

### Student Conversations

• Finding Coefficients:
• 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).
• If the students write the cosine in terms of exponentials, they can find the coefficients without using an integral. Group that recognize this will often finish early, so having them go back and do the integrals. Remind them that, while writing things in terms of exponentials is a great strategy, knowing how to perform the integrals is the method that will work in any instance.
• Remind the students that the sum of the square of the norm of the coefficients they find should add to one for a normalized quantum state.
• Degeneracy: Students may experience some difficulty calculating probabilities due to the degeneracy of some states, in particular, that you have to include all the states that share that particular eigenvalue. $P_{E={m^2\,\hbar^2\over 2I}}=\vert \langle m\vert \psi\rangle\vert^2+\vert \langle -m\vert \psi\rangle\vert^2$

### Wrap-up

Use their work to demonstrate how finding all of the probabilities allows you to rewrite the wavefunction as a linear combination of eigenstates. $P_{L_z=m\hbar}=\vert\langle m\vert\Psi\rangle\vert^2=\left|\int_{-\infty}^{\infty} \Phi_m^*(\phi)\Psi(\phi)\,d\phi\right|^2=\vert c_m \vert^2$ $\vert\Psi\rangle=\sum_m c_m \vert m\rangle \doteq \sum_m c_m \left( \frac{1}{\sqrt{2\pi r_0}}e^{im\phi}\right)$

### Extensions and Related Materials

This is a part of Quantum Ring Sequence of activities.
Associated Homework Problem: QM Ring Function
• 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 Time Dependence for a Quantum Particle on a Ring Part 1

group Small Group Activity

30 min.

##### Time Dependence for a Quantum Particle on a Ring Part 1
Theoretical Mechanics (6 years)

Quantum Ring Sequence

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.
• 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.
• 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.
• face Unit Learning Outcomes: Quantum Mechanics on a Ring

face Lecture

5 min.

##### Unit Learning Outcomes: Quantum Mechanics on a Ring
Central Forces 2023
• assignment Ring Function

assignment Homework

##### Ring Function
Central Forces 2023 (3 years) Consider the normalized wavefunction $\Phi\left(\phi\right)$ for a quantum mechanical particle of mass $\mu$ constrained to move on a circle of radius $r_0$, given by: $$\Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)}$$ where $N$ is the normalization constant.
1. Find $N$.

2. Plot this wave function.
3. Plot the probability density.
4. Find the probability that if you measured $L_z$ you would get $3\hbar$.
5. What is the expectation value of $L_z$ in this state?
• 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 2023 (8 years)

Power Series Sequence (E&M)

Warm-Up

Ring Cycle Sequence

Students work in small groups 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 2023 (6 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in small groups 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.

• group Wavefunctions on a Quantum Ring

group Small Group Activity

30 min.

##### Wavefunctions on a Quantum Ring
Central Forces 2023 (2 years)
• group Going from Spin States to Wavefunctions

group Small Group Activity

60 min.

##### Going from Spin States to Wavefunctions
Quantum Fundamentals 2023 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Completeness Relations

Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.

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