Activity: Visualization of Quantum Probabilities for a Particle Confined to a Ring

Central Forces 2021
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.
• This activity is used in the following sequences
What students learn
• Students use Mathematica to visualize eigenfunctions and linear combinations of eigenfunctions in one dimension on a ring.
• Students use Mathematica to observe the time evolution of linear combinations of eigenfunctions
• Students can review the relationship between visual and algebraic representations of wavefunctions in a simple context before working with the more complicated functions in the hydrogen atom solution.
• Media
• face Time Evolution Refresher (Mini-Lecture)

face Lecture

30 min.

Time Evolution Refresher (Mini-Lecture)
Central Forces 2021

Quantum Ring Sequence

The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.
• assignment Circle Trig

assignment Homework

Circle Trig
trigonometry cosine sine math circle Quantum Fundamentals 2021

On the following diagrams, mark both $\theta$ and $\sin\theta$ for $\theta_1=\frac{5\pi}{6}$ and $\theta_2=\frac{7\pi}{6}$. Write one to three sentences about how these two representations are related to each other. (For example, see: https://phet.colorado.edu/sims/html/trig-tour/latest/trig-tour_en.html)  • assignment Sum Shift

assignment Homework

Sum Shift
Central Forces 2021

In each of the following sums, shift the index $n\rightarrow n+2$. Don't forget to shift the limits of the sum as well. Then write out all of the terms in the sum (if the sum has a finite number of terms) or the first five terms in the sum (if the sum has an infinite number of terms) and convince yourself that the two different expressions for each sum are the same:

1. \begin{equation} \sum_{n=0}^3 n \end{equation}
2. \begin{equation} \sum_{n=1}^5 e^{in\phi} \end{equation}
3. \begin{equation} \sum_{n=0}^{\infty} a_n n(n-1)z^{n-2} \end{equation}

• assignment Visualization of Wave Functions on a Ring

assignment Homework

Visualization of Wave Functions on a Ring
Central Forces 2021 Using either this Geogebra applet or this Mathematica notebook, explore the wave functions on a ring. (Note: The Geogebra applet may be a little easier to use and understand and is accessible if you don't have access to Mathematica, but it is more limited in the wave functions that you can represent. Also, the animation is pretty jumpy in some browsers, especially Firefox. Imagine that the motion is smooth.)
1. Look at graphs of the following states \begin{align} \Phi_1(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +\left|{-2}\right\rangle )\\ \Phi_2(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle -\left|{-2}\right\rangle )\\ \Phi_3(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +i\left|{-2}\right\rangle ) \end{align} Write a short description of how these states differ from each other.
2. Find a state for which the probability density does not depend on time. Write the state in both ket and wave function notation. These are called stationary states. Generalize your result to give a characterization of the set of all possible states that are stationary states.
3. Find a state that is right-moving. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are right-moving.
4. Find a state that is a standing wave. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are standing waves.
• 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 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 (5 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 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 (4 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.

• group Time Dependence for a Quantum Particle on a Ring

group Small Group Activity

30 min.

Time Dependence for a Quantum Particle on a Ring
Central Forces 2021

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.

Use this Mathematica notebook or this GeoGebra applet to explore the time dependence of quantum states on a ring.

Instructor's Guide

Introduction

When students first open up Mathematica it is useful to walk them through the worksheet explaining the basic purpose of each step in the worksheet. It is not important to focus on the details at this point, but going through the steps in the worksheet helps keep some groups from getting stuck or just blasting through the worksheet without understanding what Mathematica is doing.

It may be helpful to review with students the meaning of the probability density since this is the primary quantity plotted in this worksheet.

Student Conversations

The most common student mistake is to apply the time dependence to the wavefunction as a whole rather than applying the time dependence to each energy eigenfunction. Because these animations are premade, this activity will not help to address this issue unless you point to where and how the time dependence is added in the code.

After students have some time to play with the worksheet and understand what is being calculated and what value is being plotted, it is useful to ask some of the following questions.

• What is being plotted?
• What causes the time dependence? How is that related to what you've seen in the case of spins?
• What do you expect to happen if you change the amplitudes in front of one of the components of the wave function?
• What do you expect to happen if you make one of the coefficients complex or imaginary?
• What is the difference between those wave functions that appear to be sloshing versus those that appear to be rotating?

Wrap-up

Discussing what patterns students observed is a good start to a wrapup. This conversation can often lead to discussions of which wavefunctions result in observable time dependence and which do not. It is useful to look at the case of two wave functions added together, since this shows the simplest time dependence.

***Update this discussion*** This observation can be connected back to the time evolution of the two state system being sinusoidal with a frequency related to the energy difference between the two states.

$|\Psi\rangle = a_m e^{-i {E_m\over\hbar} t} |m\rangle +a_n e^{-i {E_n\over\hbar}t} |n\rangle$

$\langle\phi|\Psi\rangle = a_m e^{-i {E_m\over\hbar}t} \langle\phi|m\rangle +a_n e^{-i {E_n\over\hbar}t} \langle\phi|n\rangle$

$\langle\phi|\Psi\rangle = e^{-i {E_m\over\hbar}t} (a_m \Phi_m(\phi) +a_n e^{-i {(E_n-E_m)\over\hbar}t} \Phi_n(\phi))$

$|\langle\phi|\Psi\rangle|^2 = e^{i {E_m\over\hbar}t} (a_m^* \Phi_m^*(\phi) +a_n^* e^{i {(E_n-E_m)\over\hbar}t} \Phi_n^*(\phi)) e^{-i {E_m\over\hbar}t} (a_m \Phi_m(\phi) +a_n e^{-i {(E_n-E_m)\over\hbar}t} \Phi_n(\phi))$

$|\langle\phi|\Psi\rangle|^2 = e^{i {E_m\over\hbar}t} e^{-i {E_m\over\hbar}t} (a_m^*a_m \Phi_m^*(\phi)\Phi_m(\phi)) + a_m^*a_n e^{i {(E_n-E_m)\over\hbar}t} \Phi_m^*(\phi)\Phi_n(\phi)) + a_n^*a_m e^{-i {(E_n-E_m)\over\hbar}t} \Phi_n^*(\phi)\Phi_m(\phi) + a_n^*a_n e^{i{(E_n-E_m)\over\hbar}t}e^{-i {(E_n-E_m)\over\hbar}t} \Phi_n^*(\phi)\Phi_n(\phi))$

If $a_m$ and $a_n$ are real then

$|\langle\phi|\Psi\rangle|^2 = a_m^2 |\Phi_m(\phi)|^2 + 2a_m a_n \cos\bigl(\frac{\Delta E_{mn}}{\hbar} t\bigr) \Phi_m(\phi)\Phi_n(\phi) + a_n^2 |\Phi_n(\phi)|^2$

It it is sometimes helpful to have a follow up discussion after students have had some time to play with the worksheet outside of class. This gives each student some time to be in control of the worksheet and gives them time to play without the distractions and pressures of class.

Extensions and Related Materials

This activity can be used as part of a sequence of Mathematica activities that allows one to explore probability densities for a particle first confined to a ring, then to the surface of a sphere, and finally for the entire three-dimensional hydrogen atom.

Related homework question: Particle on a Ring Time Dependence: Mathematica Notebook

Author Information
Corinne Manogue, Kerry Browne, David McIntyre
Keywords
central forces quantum mechanics angular momentum probability density eigenstates time evolution superposition mathematica
Learning Outcomes