face Lecture

30 min.

##### A coarse-grained model for transportation
Contemporary Challenges 2022 (4 years)

A short lecture introducing the idea that most of the energy loss when driving is going into the kinetic energy of the air.

group Small Group Activity

30 min.

##### A glass of water
Energy and Entropy 2021 (2 years)

Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.

accessibility_new Kinesthetic

10 min.

##### Acting Out Charge Densities
Static Fields 2022 (5 years)

Ring Cycle Sequence

Integration Sequence

Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear $\lambda$, surface $\sigma$, and volume $\rho$ charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and the we demonstrate the answers with coins under a doc cam.

accessibility_new Kinesthetic

10 min.

##### Acting Out Current Density
Static Fields 2022 (5 years)

Ring Cycle Sequence

Integration Sequence

Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of steady current. Students act out linear $\vec{I}$, surface $\vec{K}$, and volume $\vec{J}$ current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.

group Small Group Activity

5 min.

##### Acting Out Flux
Static Fields 2022 (3 years)

Students hold rulers and meter sticks to represent a vector field. The instructor holds a hula hoop to represent a small area element. Students are asked to describe the flux of the vector field through the area element.

accessibility_new Kinesthetic

10 min.

Static Fields 2022 (5 years)

Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".

groups Whole Class Activity

10 min.

##### Air Hockey
Central Forces 2023 (3 years)

Students observe the motion of a puck tethered to the center of the airtable. Then they plot the potential energy for the puck on their small whiteboards. A class discussion follows based on what students have written on their whiteboards.

face Lecture

10 min.

##### Angular Momentum Commutation Relations: Lecture
Central Forces 2023 (3 years)

group Small Group Activity

30 min.

##### Applying the equipartition theorem
Contemporary Challenges 2022 (4 years)

Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature $T$.

computer Computer Simulation

30 min.

##### Approximating Functions with Power Series
Theoretical Mechanics (12 years)

Power Series Sequence (E&M)

Students use prepared Sage code or a prepared Mathematica notebook to plot $\sin\theta$ simultaneously with several terms of a power series expansion to judge how well the approximation fits. Students can alter the worksheet to change the number of terms in the expansion and even to change the function that is being considered. Students should have already calculated the coefficients for the power series expansion in a previous activity, Calculating Coefficients for a Power Series.

face Lecture

10 min.

##### Basics of heat engines
Contemporary Challenges 2022 (4 years) This brief lecture covers the basics of heat engines.

group Small Group Activity

30 min.

##### Black space capsule
Contemporary Challenges 2022 (3 years)

In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.

computer Computer Simulation

30 min.

##### Blackbody PhET
Contemporary Challenges 2022 (4 years)

Students use a PhET to explore properties of the Planck distribution.

face Lecture

120 min.

##### Boltzmann probabilities and Helmholtz
Thermal and Statistical Physics 2020

These notes, from the third week of Thermal and Statistical Physics cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.

group Small Group Activity

120 min.

##### Box Sliding Down Frictionless Wedge
Theoretical Mechanics (4 years)

Students solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance $d$ down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.

group Small Group Activity

30 min.

##### Calculating Coefficients for a Power Series
Theoretical Mechanics (7 years)

Power Series Sequence (E&M)

This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the coefficients of a power series for a particular function:

$c_n={1\over n!}\, f^{(n)}(z_0)$

After a brief lecture deriving the formula for the coefficients of a power series, students compute the power series coefficients for a $\sin\theta$ (around both the origin and $\frac{\pi}{6}$). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up.

face Lecture

5 min.

##### Central Forces Introduction: Lecture Notes
Central Forces 2023 (2 years)

group Small Group Activity

30 min.

##### Changes in Internal Energy (Remote)

Warm-Up

Students consider the change in internal energy during three different processes involving a container of water vapor on a stove. Using the 1st Law of Thermodynamics, students reason about how the internal energy would change and then compare this prediction with data from NIST presented as a contour plot.

group Small Group Activity

10 min.

##### Changing Spin Bases with a Completeness Relation
Quantum Fundamentals 2022 (3 years)

Students work in small groups to use completeness relations to change the basis of quantum states.

group Small Group Activity

30 min.

##### Charged Sphere

Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.

face Lecture

120 min.

##### Chemical potential and Gibbs distribution
Thermal and Statistical Physics 2020

These notes from the fifth week of Thermal and Statistical Physics cover the grand canonical ensemble. They include several small group activities.

group Small Group Activity

30 min.

##### Covariation in Thermal Systems

Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.

accessibility_new Kinesthetic

10 min.

##### Curvilinear Basis Vectors
Static Fields 2022 (8 years)

Curvilinear Coordinate Sequence

Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).

assignment_ind Small White Board Question

10 min.

##### Curvilinear Coordinates Introduction
Static Fields 2022 (9 years)

Curvilinear Coordinate Sequence

First, students are shown diagrams of cylindrical and spherical coordinates. Common notation systems are discussed, especially that physicists and mathematicians use opposite conventions for the angles $\theta$ and $\phi$. Then students are asked to check their understanding by sketching several coordinate equals constant surfaces on their small whiteboards.

group Small Group Activity

30 min.

##### de Broglie wavelength after freefall
Contemporary Challenges 2022 (4 years)

In this activity students combine energy conservation with the relationship between the de Broglie wavelength and momentum to find the wavelength of atoms that have been dropped a given distance.

group Small Group Activity

30 min.

##### DELETE Navigating a Hill
Static Fields 2022 (4 years) In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.

group Small Group Activity

30 min.

##### Directional Derivatives
Vector Calculus I 2022

This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.

assignment_ind Small White Board Question

10 min.

##### Dot Product Review
Static Fields 2022 (6 years)

This small whiteboard question (SWBQ) serves as a quick review of the dot product. It is also an opportunity to help students see the advantages of knowing many different representations of and facts about a physical concept.

group Small Group Activity

30 min.

##### Earthquake waves
Contemporary Challenges 2022 (4 years)

In this activity students use the known speed of earthquake waves to estimate the Young's modulus of the Earth's crust.

computer Mathematica Activity

30 min.

##### Effective Potentials
Central Forces 2023 (3 years) Students use a pre-written Mathematica notebook or a Geogebra applet to explore how the shape of the effective potential function changes as the various parameters (angular momentum, force constant, reduced mass) are varied.

group Small Group Activity

30 min.

##### Electric Field Due to a Ring of Charge
Static Fields 2022 (7 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use Coulomb's Law $\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}$ to find an integral expression for the electric field, $\vec{E}(\vec{r})$, everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for $\vec{E}(\vec{r})$ 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.

keyboard Computational Activity

120 min.

##### Electric field for a waffle cone of charge
Computational Physics Lab II 2022

Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient.

group Small Group Activity

30 min.

##### Electric Field of Two Charged Plates
• Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity Electric Potential of Two Charged Plates before this activity.
• Students should know that
1. objects with like charge repel and opposite charge attract,
2. object tend to move toward lower energy configurations
3. The potential energy of a charged particle is related to its charge: $U=qV$
4. The force on a charged particle is related to its charge: $\vec{F}=q\vec{E}$

group Small Group Activity

30 min.

##### Electric Potential of Two Charged Plates
Students examine a plastic "surface" graph of the electric potential due to two charged plates (near the center of the plates) and explore the properties of the electric potential.

group Small Group Activity

60 min.

##### Electrostatic Potential Due to a Pair of Charges (with Series)
Static Fields 2022 (5 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in small groups to use the superposition principle $V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}$ to find the electrostatic potential $V$ everywhere in space due to a pair of charges (either identical charges or a dipole). Different groups are assigned different arrangements of charges and different regions of space to consider: either on the axis of the charges or in the plane equidistant from the two charges, for either small or large values of the relevant geometric variable. Each group is asked to find a power series expansion for the electrostatic potential, valid in their group's assigned region of space. The whole class wrap-up discussion then compares and contrasts the results and discuss the symmetries of the two cases.

group Small Group Activity

30 min.

##### Electrostatic Potential Due to a Pair of Charges (without Series)
Static Fields 2022 (3 years) Students work in small groups to use the superposition principle $V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}$ to find the electrostatic potential $V$ everywhere in space due to a pair of charges (either identical charges or a dipole). This activity can be paired with activity 29 to find the limiting cases of the potential on the axes of symmetry.

assignment_ind Small White Board Question

10 min.

##### Electrostatic Potential Due to a Point Charge
Static Fields 2022

Warm-Up

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.

keyboard Computational Activity

120 min.

##### Electrostatic potential of a square of charge
Computational Physics Lab II 2022

Students write python programs to compute the potential due to a square of surface charge, and then to visualize the result. This activity can be used to introduce students to the process of integrating numerically.

keyboard Computational Activity

120 min.

##### Electrostatic potential of four point charges
Computational Physics Lab II 2022

Students write python programs to compute and visualize the potential due to four point charges. For students with minimal programming ability and no python experience, this activity can be a good introduction to writing code in python using numpy and matplotlib.

keyboard Computational Activity

120 min.

##### Electrostatic potential of spherical shell
Computational Physics Lab II 2022

Students solve numerically for the potential due to a spherical shell of charge. Although this potential is straightforward to compute using Gauss's Law, it serves as a nice example for numerically integrating in spherical coordinates because the correct answer is easy to recognize.

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.

face Lecture

5 min.

##### Energy and Entropy review
Thermal and Statistical Physics 2020 (3 years)

This very quick lecture reviews the content taught in Energy and Entropy, and is the first content in Thermal and Statistical Physics.

face Lecture

30 min.

##### Energy and heat and entropy
Energy and Entropy 2021 (2 years)

This short lecture introduces the ideas required for Ice Calorimetry Lab or Microwave oven Ice Calorimetry Lab.

group Small Group Activity

30 min.

##### Energy radiated from one oscillator
Contemporary Challenges 2022 (4 years)

This lecture is one step in motivating the form of the Planck distribution.

face Lecture

120 min.

##### Entropy and Temperature
Thermal and Statistical Physics 2020

These lecture notes for the second week of Thermal and Statistical Physics involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.

face Lecture

30 min.

##### Equipartition theorem
Contemporary Challenges 2022 (4 years)

This lecture introduces the equipartition theorem.

group Small Group Activity

120 min.

##### Equipotential Surfaces

Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.

group Small Group Activity

60 min.

##### Establish Classroom Norms
Theoretical Mechanics (4 years)

In this hour-long activity, students establish classroom norms for being respectful when working in small groups. This is particularly helpful in the first course a cohort of students encounters.

group Small Group Activity

5 min.

##### Events on Spacetime Diagrams
Theoretical Mechanics 2021

Students practice identifying whether events on spacetime diagrams are simultaneous, colocated, or neither for different observers. Then students decide which of two events occurs first in two different reference frames.

group Small Group Activity

60 min.

##### Expectation Value and Uncertainty for the Difference of Dice
Quantum Fundamentals 2022 (3 years)

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.

face Lecture

120 min.

##### Fermi and Bose gases
Thermal and Statistical Physics 2020

These lecture notes from week 7 of Thermal and Statistical Physics apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.

group Small Group Activity

30 min.

##### Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute
Quantum Fundamentals 2022 (3 years)

group Small Group Activity

30 min.

##### Flux through a Cone
Static Fields 2022 (4 years)

Integration Sequence

Students calculate the flux from the vector field $\vec{F} = C\, z\, \hat{z}$ through a right cone of height $H$ and radius $R$ .

60 min.

##### Free expansion
Energy and Entropy 2021 (2 years)

Students will determine the change in entropy (positive, negative, or none) for both the system and surroundings in three different cases. This is followed by an active whole-class discussion about where the entropy comes from during an irreversible process.

group Small Group Activity

10 min.

##### Generalized Leibniz Notation
Static Fields 2022 (4 years) This short small group activity introduces students to the Leibniz notation used for partial derivatives in thermodynamics; unlike standard Leibniz notation, this notation explicitly specifies constant variables. Students are guided in linking the variables from a contextless Leibniz-notation partial derivative to their proper variable categories.

face Lecture

120 min.

##### Gibbs entropy approach
Thermal and Statistical Physics 2020

These lecture notes for the first week of Thermal and Statistical Physics include a couple of small group activities in which students work with the Gibbs formulation of the entropy.

group Small Group Activity

60 min.

##### Going from Spin States to Wavefunctions
Quantum Fundamentals 2022 (2 years)

Arms Sequence for Complex Numbers and Quantum States

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.

group Small Group Activity

30 min.

##### Gravitational Force

Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.

group Small Group Activity

60 min.

##### Gravitational Potential Energy

Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.

group Small Group Activity

30 min.

##### Grey space capsule
Contemporary Challenges 2022 (4 years)

In this small group activity, students work out the steady state temperature of an object absorbing and emitting blackbody radiation.

face Lecture

5 min.

##### Guide to Professional Typography in Physics
Contemporary Challenges 2022 (4 years)

This is really a handout, which gives students guidelines on how to type up physics content.

group Small Group Activity

30 min.

##### Heat and Temperature of Water Vapor

In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.

group Small Group Activity

30 min.

##### Heat capacity of N2
Contemporary Challenges 2022 (4 years)

Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”

group Small Group Activity

30 min.

##### Hydrogen emission
Contemporary Challenges 2022 (5 years)

In this activity students work out energy level transitions in hydrogen that lead to visible light.

group Small Group Activity

30 min.

##### Hydrogen Probabilities in Matrix Notation
Central Forces 2023 (2 years)

group Small Group Activity

60 min.

##### Ice Calorimetry Lab

The students will set up a Styrofoam cup with heating element and a thermometer in it. They will measure the temperature as a function of time, and thus the energy transferred from the power supply, from which they compute changes in entropy.

face Lecture

120 min.

##### Ideal Gas
Thermal and Statistical Physics 2020

These notes from week 6 of Thermal and Statistical Physics cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.

group Small Group Activity

30 min.

##### Ideal Gas Model

Students should be familiar with the basic definitions of pressure, volume, temperature, and internal energy.

face Lecture

30 min.

##### Introducing entropy
Contemporary Challenges 2022 (4 years)

This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between changes in entropy and heat.

keyboard Computational Activity

120 min.

##### Kinetic energy
Computational Physics Lab II 2022

Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.

face Lecture

30 min.

##### Lorentz Transformation (Geometric)
Theoretical Mechanics (3 years)

In this lecture, students see a geometric derivation of the Lorentz Transformation on a spacetime diagram.

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.

assignment_ind Small White Board Question

30 min.

##### Magnetic Moment & Stern-Gerlach Experiments
Quantum Fundamentals 2022 (3 years)

Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.

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.

group Small Group Activity

30 min.

##### Mass is not Conserved
Theoretical Mechanics (4 years)

Groups are asked to analyze the following standard problem:

Two identical lumps of clay of (rest) mass m collide head on, with each moving at 3/5 the speed of light. What is the mass of the resulting lump of clay?

group Small Group Activity

10 min.

##### Matrix Representation of Angular Momentum
Central Forces 2023 (2 years)

keyboard Computational Activity

120 min.

##### Mean position
Computational Physics Lab II 2022

Students compute probabilities and averages given a probability density in one dimension. This activity serves as a soft introduction to the particle in a box, introducing all the concepts that are needed.

biotech Experiment

60 min.

##### Microwave oven Ice Calorimetry Lab
Energy and Entropy 2021 (2 years)

In this remote-friendly activity, students use a microwave oven (and optionally a thermometer) to measure the latent heat of melting for water (and optionally the heat capacity). From these they compute changes in entropy. See also Ice Calorimetry Lab.

group Small Group Activity

30 min.

##### Name the experiment
Energy and Entropy 2021 (3 years)

Student groups design an experiment that measures an assigned partial derivative. In a compare-and-contrast wrap-up, groups report on how they would measure their derivatives.

group Small Group Activity

30 min.

##### Name the experiment (changing entropy)
Energy and Entropy 2021 (2 years)

Students are placed into small groups and asked to create an experimental setup they can use to measure the partial derivative they are given, in which entropy changes.

group Small Group Activity

30 min.

##### Number of Paths

Student discuss how many paths can be found on a map of the vector fields $\vec{F}$ for which the integral $\int \vec{F}\cdot d\vec{r}$ is positive, negative, or zero. $\vec{F}$ is conservative. They do a similar activity for the vector field $\vec{G}$ which is not conservative.

group Small Group Activity

30 min.

##### Operators & Functions
Quantum Fundamentals 2022 (3 years) 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 Small Group Activity

30 min.

##### Optical depth of atmosphere
Contemporary Challenges 2022 (4 years) In this activity students estimate the optical depth of the atmosphere at the infrared wavelength where carbon dioxide has peak absorption.

group Small Group Activity

30 min.

##### Outer Product of a Vector on Itself
Quantum Fundamentals 2022 (2 years)

Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).

group Small Group Activity

30 min.

##### Paramagnet (multiple solutions)
• Students evaluate two given partial derivatives from a system of equations.
• Students learn/review generalized Leibniz notation.
• Students may find it helpful to use a chain rule diagram.

assignment_ind Small White Board Question

10 min.

##### Partial Derivatives from a Contour Map
Static Fields 2022 (3 years)

In this sequence of small whiteboard questions, students are shown the contour graph of a function of two variables and asked to find the derivative. They discover that, without a function to differentiate, they must instead think of the derivative as a ratio of small changes. This requires them to pick two nearby points. Which two?

face Lecture

120 min.

##### Phase transformations
Thermal and Statistical Physics 2020

These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.

groups Whole Class Activity

10 min.

##### Pineapples and Pumpkins
Static Fields 2022 (5 years)

Integration Sequence

There are two versions of this activity:

As a whole class activity, the instructor cuts a pumpkin in order to produce a small volume element $d\tau$, interspersing their work with a sequence of small whiteboard questions. This version of the activity is described here.

As a small group activity, students are given pineapple rounds and pumpkin wedges to explore area volume elements in cylindrical and spherical coordinate systems. In this version of the activity, the fruit is distribued to the students with appropriate children's pumpkin cutting equipment, as part of activities Vector Differential--Curvilinear, Scalar Surface and Volume Elements, or Vector Surface and Volume Elements.

keyboard Computational Activity

120 min.

##### Position operator
Computational Physics Lab II 2022

Students find matrix elements of the position operator $\hat x$ in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.

assignment_ind Small White Board Question

10 min.

##### Possible Worldlines
Theoretical Mechanics (4 years)

Student consider several curves on a spacetime diagram and have to judge which curves could be worldlines for an object.

group Small Group Activity

120 min.

##### Projectile with Linear Drag
Theoretical Mechanics (4 years)

Students consider projectile motion of an object that experiences drag force that in linear with the velocity. Students consider the horizontal motion and the vertical motion separately. Students solve Newton's 2nd law as a differential equation.

group Small Group Activity

30 min.

##### Quantifying Change

In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.

group Small Group Activity

30 min.

##### Quantum Expectation Values
Quantum Fundamentals 2022 (3 years)

group Small Group Activity

30 min.

##### Quantum Measurement Play
Quantum Fundamentals 2022 (2 years)

The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.

face Lecture

5 min.

##### Quantum Reference Sheet
Central Forces 2023 (6 years)

group Small Group Activity

60 min.

##### Raising and Lowering Operators for Spin
Central Forces 2023 (2 years)

group Small Group Activity

120 min.

##### Representations of the Infinite Square Well
Quantum Fundamentals 2022 (3 years)

Warm-Up

face Lecture

30 min.

##### Review of Thermal Physics
Thermal and Statistical Physics 2020

These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.

group Small Group Activity

30 min.

##### Right Angles on Spacetime Diagrams
Theoretical Mechanics (4 years)

Students take the inner product of vectors that lie on the spacetime axis to show that they are orthogonal. To do the inner product, students much use the Minkowski metric.

group Small Group Activity

30 min.

##### Scalar Surface and Volume Elements
Static Fields 2022 (6 years)

Integration Sequence

Students use known algebraic expressions for length elements $d\ell$ to determine all simple scalar area $dA$ and volume elements $d\tau$ in cylindrical and spherical coordinates.

This activity is identical to Vector Surface and Volume Elements except uses a scalar approach to find surface, and volume elements.

group Small Group Activity

10 min.

##### Sequential Stern-Gerlach Experiments
Quantum Fundamentals 2022 (3 years)

keyboard Computational Activity

120 min.

##### Sinusoidal basis set
Computational Physics Lab II 2022

Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.

group Small Group Activity

5 min.

##### Spherical Harmonics
Central Forces 2023 (3 years)

accessibility_new Kinesthetic

10 min.

##### Spin 1/2 with Arms
Quantum Fundamentals 2022 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.

group Small Group Activity

5 min.

##### Static Fields Equation Sheet
Static Fields 2022 (3 years)

face Lecture

30 min.

##### Statistical Analysis of Stern-Gerlach Experiments

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 Small Group Activity

10 min.

##### Survivor Outer Space: A kinesthetic approach to (re)viewing center-of-mass
Central Forces 2023 (3 years) A group of students, tethered together, are floating freely in outer space. Their task is to devise a method to reach a food cache some distance from their group.

group Small Group Activity

60 min.

face Lecture

10 min.

##### Systems of Particles Lecture Notes
Central Forces 2023 (3 years)

accessibility_new Kinesthetic

30 min.

##### The Distance Formula (Star Trek)
Static Fields 2022 (5 years)

Ring Cycle Sequence

A short improvisational role-playing skit based on the Star Trek series in which students explore the definition and notation for position vectors, the importance of choosing an origin, and the geometric nature of the distance formula. $\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}$

group Small Group Activity

30 min.

##### The Hill
Vector Calculus II 2022 (4 years)

In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.

group Small Group Activity

30 min.

##### The Hillside
Vector Calculus I 2022

Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).

face Lecture

120 min.

##### Thermal radiation and Planck distribution
Thermal and Statistical Physics 2020

These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.

group Small Group Activity

10 min.

##### Thermal radiation at twice the temperature
Contemporary Challenges 2022 (4 years)

This small group activity has students reasoning about how the Planck distribution shifts when the temperature is doubled. This leads to a qualitative argument for the Stefan-Boltzmann law.

group Small Group Activity

30 min.

##### Thermodynamic States (Remote)

Try doing this activity as a follow-up to the Changes in Internal Energy (Remote) about the first law of thermodynamics.

group Small Group Activity

30 min.

##### Time Dependence for a Quantum Particle on a Ring
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.

assignment_ind Small White Board Question

10 min.

##### Time Dilation
Theoretical Mechanics (4 years)

accessibility_new Kinesthetic

5 min.

##### Time Dilation Light Clock Skit

Students act out the classic light clock scenario for deriving time dilation.

group Small Group Activity

30 min.

##### Time Evolution of a Spin-1/2 System
Quantum Fundamentals 2022 (3 years)

In this small group activity, students solve for the time dependence of two quantum spin 1/2 particles under the influence of a Hamiltonian. Students determine, given a Hamiltonian, which states are stationary and under what circumstances measurement probabilities do change with time.

face Lecture

30 min.

##### Time Evolution Refresher (Mini-Lecture)
Central Forces 2023 (3 years)

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.

group Small Group Activity

30 min.

##### Total Charge
Static Fields 2022 (5 years)

Integration Sequence

In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.

group Small Group Activity

30 min.

##### Travelling wave solution
Contemporary Challenges 2022 (4 years)

Students work in a small group to write down an equation for a travelling wave.

group Small Group Activity

30 min.

##### Using $pV$ and $TS$ Plots
Energy and Entropy 2021 (2 years)

Students work out heat and work for rectangular paths on $pV$ and $TS$ plots. This gives with computing heat and work, applying the First Law, and recognizing that internal energy is a state function, which cannot change after a cyclic process.

accessibility_new Kinesthetic

10 min.

##### Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems
Quantum Fundamentals 2022 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).

accessibility_new Kinesthetic

10 min.

##### Using Arms to Represent Time Dependence in Spin 1/2 Systems
Quantum Fundamentals 2022 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.

accessibility_new Kinesthetic

10 min.

##### Using Arms to Visualize Complex Numbers (MathBits)
Quantum Fundamentals 2022 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.

accessibility_new Kinesthetic

30 min.

##### Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)
Quantum Fundamentals 2021

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.

computer Mathematica Activity

30 min.

##### Using Technology to Visualize Potentials
Static Fields 2022 (5 years)

Begin by prompting the students to brainstorm different ways to represent a three dimensional scalar field on a 2-D surface (like their paper or a whiteboard). The students use a pre-made Sage code or a Mathematica worksheet to visualize the electrostatic potential of several distributions of charges. The computer algebra systems demonstrates several different ways of plotting the potential.

group Small Group Activity

10 min.

##### Using Tinker Toys to Represent Spin 1/2 Quantum Systems

Arms Sequence for Complex Numbers and Quantum States

Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases ($x$, $y$, and $z$). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.

group Small Group Activity

30 min.

##### Vector Differential--Curvilinear
Vector Calculus II 2022 (8 years)

Integration Sequence

In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.

assignment_ind Small White Board Question

10 min.

##### Vector Differential--Rectangular
Static Fields 2022 (7 years)

Integration Sequence

In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.

This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..

group Small Group Activity

30 min.

##### Vector Integrals (Contour Map)

group Small Group Activity

30 min.

##### Vector Surface and Volume Elements
Static Fields 2022 (3 years)

Integration Sequence

Students use known algebraic expressions for vector line elements $d\vec{r}$ to determine all simple vector area $d\vec{A}$ and volume elements $d\tau$ in cylindrical and spherical coordinates.

This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.

computer Mathematica Activity

30 min.

Static Fields 2022 (5 years)

Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.

group Small Group Activity

30 min.

##### Visualization of Divergence
Vector Calculus II 2022 (8 years) Students predict from graphs of simple 2-d vector fields whether the divergence is positive, negative, or zero in various regions of the domain using the definition of the divergence of a vector field at a point: The divergence of a vector field at a point is flux per unit volume through an infinitesimal box surrounding that point. Optionally, students can use a Mathematica notebook to verify their predictions.

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.

computer Mathematica Activity

30 min.

##### Visualization of Quantum Probabilities for the Hydrogen Atom
Central Forces 2023 (3 years) Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of $n$, $\ell$, and $m$.

computer Mathematica Activity

30 min.

##### Visualizing Combinations of Spherical Harmonics
Central Forces 2023 (3 years) Students observe three different plots of linear combinations of spherical combinations with probability density represented by color on the sphere, distance from the origin (polar plot), and distance from the surface of the sphere.

computer Computer Simulation

30 min.

##### Visualizing Flux through a Cube
Static Fields 2022 (5 years) Students explore the effects of putting a point charge at various places inside, outside, and on the surface of a cubical Gaussian surface. The Mathematica worksheet or Sage activity shows the electric field due to the charge, then plots the the flux integrand on the top surface of the box, calculates the flux through the top of the box, and the value of the flux through the whole cube.

group Small Group Activity

60 min.

##### Visualizing Plane Waves

Each small group of 3-4 students is given a white board or piece of paper with a square grid of points on it.

Each group is given a different two-dimensional vector $\vec{k}$ and is asked to calculate the value of $\vec{k} \cdot \vec {r}$ for each point on the grid and to draw the set of points with constant value of $\vec{k} \cdot \vec{r}$ using rainbow colors to indicate increasing value.

face Lecture

10 min.

##### Warm-Up Powerpoint

Warm-Up

The attached powerpoint articulates the possible paths through the curriculum for new graduate students at OSU.

group Small Group Activity

30 min.

##### Wavefunctions on a Quantum Ring
Central Forces 2023 (2 years)

face Lecture

5 min.

##### Wavelength of peak intensity
Contemporary Challenges 2022 (3 years)

This very short lecture introduces Wein's displacement law.

group Small Group Activity

30 min.

##### Work By An Electric Field (Contour Map)

Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.

face Lecture

120 min.

##### Work, Heat, and cycles
Thermal and Statistical Physics 2020

These lecture notes covering week 8 of Thermal and Statistical Physics include a small group activity in which students derive the Carnot efficiency.

group Small Group Activity

30 min.

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

group Small Group Activity

30 min.

##### “Squishability” of Water Vapor (Contour Map)

Students determine the “squishibility” (an extensive compressibility) by taking $-\partial V/\partial P$ holding either temperature or entropy fixed.