## Student handout: Visualization of Quantum Probabilities for a Particle Confined to a Ring

Central Forces Spring 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.
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.
• face Time Evolution Refresher (Mini-Lecture)

face Lecture

30 min.

##### Time Evolution Refresher (Mini-Lecture)
Central Forces Spring 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 Winter 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)

• 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 Spring 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.
• 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.
• assignment Visualization of Wave Functions on a Ring

assignment Homework

##### Visualization of Wave Functions on a Ring
Central Forces Spring 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 Magnetic Field Due to a Spinning Ring of Charge

group Small Group Activity

30 min.

##### Magnetic Field Due to a Spinning Ring of Charge
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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 Operators & Functions

group Small Group Activity

30 min.

##### Operators & Functions
Quantum Fundamentals Winter 2021 Students are asked to:
• Test to see if one of the given functions is an eigenfunction of the given operator
• See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.
• group Representations of the Infinite Square Well

group Small Group Activity

120 min.

##### Representations of the Infinite Square Well
Quantum Fundamentals Winter 2021

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

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