Activities

flux electrostatics vector fields

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.

Small Group Activity

5 min.

Acting Out Flux

Static Fields 2023 (5 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.

Kinesthetic

10 min.

Acting Out the Gradient

Static Fields 2023 (7 years)

gradient vector fields electrostatics

central forces potential energy classical mechanics

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".

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.

Small White Board Question

5 min.

Angular Momentum

Central Forces 2023

Lecture

10 min.

Angular Momentum Commutation Relations: Lecture

Central Forces 2023 (3 years)

Small Group Activity

10 min.

Angular Momentum in Polar Coordinates

Central Forces 2023

Small Group Activity

30 min.

Applying the equipartition theorem

Contemporary Challenges 2021 (4 years)

equipartition theorem

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

Lecture

10 min.

Basics of heat engines

Contemporary Challenges 2021 (4 years) This brief lecture covers the basics of heat engines.

Small Group Activity

30 min.

Black space capsule

Contemporary Challenges 2021 (3 years)

stefan-boltzmann blackbody radiation

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 Simulation

30 min.

Blackbody PhET

Contemporary Challenges 2021 (4 years)

blackbody

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

Lecture

120 min.

Boltzmann probabilities and Helmholtz

Thermal and Statistical Physics 2020

ideal gas entropy canonical ensemble Boltzmann probability Helmholtz free energy statistical mechanics

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.

Small Group Activity

120 min.

Box Sliding Down Frictionless Wedge

Theoretical Mechanics (4 years)

Lagrangian Mechanics Generalized Coordinates Special Cases

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.

Small Group Activity

30 min.

Calculating Coefficients for a Power Series

Theoretical Mechanics (8 years)

Taylor series power series approximation

Thermo Internal Energy 1st Law of Thermodynamics

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

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

Students use this formula to compute the power series coefficients for a $\sin\theta$ (around both the origin and (if time allows) $\frac{\pi}{6}$). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up and in the follow-up activity: Visualization of Power Series Approximations.

Lecture

5 min.

Central Forces Introduction Lecture Notes

Central Forces 2022

Small Group Activity

30 min.

Changes in Internal Energy (Remote)

Completeness Relations Quantum States

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.

Small Group Activity

10 min.

Changing Spin Bases with a Completeness Relation

Quantum Fundamentals 2023 (3 years)

E&M Introductory Physics Electric Potential Electric Field

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

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.

Lecture

120 min.

Chemical potential and Gibbs distribution

Thermal and Statistical Physics 2020

chemical potential Gibbs distribution grand canonical ensemble statistical mechanics

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

Lecture

30 min.

Compare & Contrast Kets & Wavefunctions

Bra-Ket Notations Wavefunction Notation Completeness Relations Probability Probability Density

Thermo Multivariable Functions

In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.

Small Group Activity

30 min.

Conic Sections

Central Forces 2023 (3 years) Students are asked to explore the parameters that affect orbit shape using the supplied Maple worksheet or Mathematica notebook.

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.

Kinesthetic

10 min.

Curvilinear Basis Vectors

Static Fields 2023 (10 years)

symmetry curvilinear coordinate systems basis vectors

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).

Small White Board Question

10 min.

Curvilinear Coordinates Introduction

Static Fields 2023 (11 years)

Cylindrical coordinates spherical coordinates curvilinear coordinates

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.

Small Group Activity

30 min.

de Broglie wavelength after freefall

Contemporary Challenges 2021 (4 years)

de Broglie wavelength gravity

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.

Small Group Activity

30 min.

DELETE Navigating a Hill

Static Fields 2023 (5 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.

Small Group Activity

30 min.

Directional Derivatives

Vector Calculus I 2022

Directional derivatives

wave equation speed of sound

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.

Small White Board Question

10 min.

Dot Product Review

Static Fields 2023 (8 years)

dot product math

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.

Small Group Activity

30 min.

Earthquake waves

Contemporary Challenges 2021 (4 years)

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

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.

Small Group Activity

30 min.

Electric Field Due to a Ring of Charge

Static Fields 2023 (8 years)

coulomb's law electric field charge ring symmetry integral power series superposition

integration electrostatic potential surface charge density

Students work in small groups 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.

Computational Activity

120 min.

Electric field for a waffle cone of charge

Computational Physics Lab II 2022

electric field cone

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.

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: $\vec{F}=q\vec{E}$

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.

Computational Activity

120 min.

Electrostatic potential and Electric Field of a square of charge

Computational Physics Lab II 2023 (2 years)

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.

Small Group Activity

60 min.

Electrostatic Potential Due to a Pair of Charges (with Series)

Static Fields 2023 (6 years)

electrostatic potential multipole charge symmetry scalar field superposition coulomb's Law

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.

Small Group Activity

30 min.

Electrostatic Potential Due to a Pair of Charges (without Series)

Static Fields 2023 (4 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.

Small White Board Question

10 min.

Electrostatic Potential Due to a Point Charge

Static Fields 2023 (2 years)

electrostatic potential charge linear charge density taylor series power series scalar field superposition symmetry distance formula

Small Group Activity

30 min.

Electrostatic Potential Due to a Ring of Charge

Static Fields 2023 (8 years)

electrostatic potential python

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.

Computational Activity

120 min.

Electrostatic potential of four point charges

Computational Physics Lab II 2023 (2 years)

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.

Computational Activity

120 min.

Electrostatic potential of spherical shell

Computational Physics Lab II 2022

electrostatic potential spherical coordinates

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.

Small Group Activity

30 min.

Energy and Angular Momentum for a Quantum Particle on a Ring

central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum energy hermitian operators probability superposition representations notations degeneracy

thermodynamics statistical mechanics

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.

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.

Lecture

30 min.

Energy and heat and entropy

Energy and Entropy 2021 (2 years)

latent heat heat capacity internal energy entropy

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

Small Group Activity

30 min.

Energy radiated from one oscillator

Contemporary Challenges 2021 (4 years)

blackbody radiation

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

Lecture

120 min.

Entropy and Temperature

Thermal and Statistical Physics 2020

paramagnet entropy temperature statistical mechanics

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.

Lecture

30 min.

Equipartition theorem

Contemporary Challenges 2021 (4 years)

equipartition heat capacity

This lecture introduces the equipartition theorem.

Small Group Activity

120 min.

Equipotential Surfaces

E&M Quadrupole Scalar Fields

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.

Small Group Activity

60 min.

Establish Classroom Norms

Theoretical Mechanics (4 years)

Equity

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.

Small Group Activity

5 min.

Events on Spacetime Diagrams

Theoretical Mechanics 2021

Special Relativity Spacetime Diagrams Simultaneity Colocation

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.

Small Group Activity

60 min.

Expectation Value and Uncertainty for the Difference of Dice

Quantum Fundamentals 2023 (3 years)

Small Group Activity

30 min.

Expectation Values for a Particle on a Ring

Central Forces 2023 (3 years)

central forces quantum mechanics eigenstates eigenvalues hermitian operators quantum measurements degeneracy expectation values time dependence

Fermi level fermion boson Bose gas Bose-Einstein condensate ideal gas statistical mechanics phase transition

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.

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.

Small Group Activity

30 min.

Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute

Quantum Fundamentals 2023 (3 years)

Small Group Activity

30 min.

Flux through a Cone

Static Fields 2021 (4 years)

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

Small Group Activity

10 min.

Fourier Transform of a Derivative

Periodic Systems 2022

Small Group Activity

10 min.

Fourier Transform of a Gaussian

Periodic Systems 2022

Small Group Activity

5 min.

Fourier Transform of a Plane Wave

Periodic Systems 2022

Small Group Activity

5 min.

Fourier Transform of a Shifted Function

Periodic Systems 2022

Small Group Activity

5 min.

Fourier Transform of the Delta Function

Periodic Systems 2022

adiabatic expansion entropy temperature ideal gas

Students calculate the Fourier transform of the Dirac delta function.

Quiz

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.

Small Group Activity

10 min.

Gaussian Parameters

Periodic Systems 2022

Gibbs entropy information theory probability statistical mechanics

Students use an applet to explore the role of the parameters $N$, $x_o$, and $\sigma$ in the shape of a Gaussian \begin{equation} f(x)=Ne^{-\frac{(x-x_0)^2}{2\sigma^2}} \end{equation}

Small Group Activity

10 min.

Generalized Leibniz Notation

Static Fields 2023 (6 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.

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.

Small Group Activity

60 min.

Going from Spin States to Wavefunctions

Quantum Fundamentals 2023 (2 years)

Wavefunctions quantum states probability amplitude histograms matrix notation of quantum states Arms representation

Arms Sequence for Complex Numbers and Quantum States

Mechanics Gravitational Force Gravitational Potential Energy Derivatives Introductory Physics

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.

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.

Small Group Activity

60 min.

Gravitational Potential Energy

Mechanics Gravitational Potential Energy Zero of Potential Introductory Physics

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.

Small Group Activity

30 min.

Grey space capsule

Contemporary Challenges 2021 (4 years)

blackbody Stefan-Boltzmann Law

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

Small Group Activity

10 min.

Guess the Fourier Series from a Graph

Oscillations and Waves 2023 (2 years) The students are shown the graph of a function that is a superposition of three harmonic functions and asked to guess the harmonic terms of the Fourier series. Students then use prewritten Sage code to verify the coefficients from their guess. The program allows the students to enter functions of their own choice as well as the one that is preset.

Lecture

5 min.

Guide to Professional Typography in Physics

Contemporary Challenges 2021 (4 years)

typography

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

Small Group Activity

30 min.

Heat and Temperature of Water Vapor

Thermo Heat Capacity Partial Derivatives

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.

Small Group Activity

30 min.

Heat capacity of N₂

Contemporary Challenges 2021 (4 years)

equipartition quantum energy levels

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.”

Small Group Activity

30 min.

Hydrogen emission

Contemporary Challenges 2021 (5 years)

hydrogen atom photon energy

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

Small Group Activity

30 min.

Hydrogen Probabilities in Matrix Notation

Central Forces 2023 (2 years)

Small Group Activity

60 min.

Ice Calorimetry Lab

heat entropy water ice

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.

Lecture

120 min.

Ideal Gas

Thermal and Statistical Physics 2020

ideal gas particle in a box grand canonical ensemble chemical potential statistical mechanics

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.

Small Group Activity

30 min.

Ideal Gas Model

Ideal Gas surfaces thermo

Students consider whether the thermo surfaces reflect the properties of an ideal gas.

Lecture

30 min.

Introducing entropy

Contemporary Challenges 2021 (4 years)

entropy multiplicity heat thermodynamics

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.

Lecture

10 min.

Introduction to Central Forces

Central Forces 2023

Computational Activity

120 min.

Kinetic energy

Computational Physics Lab II 2022

finite difference hamiltonian quantum mechanics particle in a box eigenfunctions

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.

Lecture

30 min.

Lorentz Transformation (Geometric)

Theoretical Mechanics (3 years)

Special Relativity Lorentz Transformation Hyperbola Trig

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

Small Group Activity

30 min.

Magnetic Field Due to a Spinning Ring of Charge

Static Fields 2023 (7 years)

magnetic fields current Biot-Savart law vector field symmetry

Angular Momentum Spin Magnetic Moment Stern-Gerlach Experiments

Students work in small groups 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.

Small White Board Question

30 min.

Magnetic Moment & Stern-Gerlach Experiments

Quantum Fundamentals 2023 (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.

Small Group Activity

30 min.

Magnetic Vector Potential Due to a Spinning Charged Ring

Static Fields 2023 (6 years)

compare and contrast mathematica magnetic vector potential magnetic fields vector field symmetry

energy conservation mass conservation collision

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.

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?

Small Group Activity

10 min.

Matrix Representation of Angular Momentum

Central Forces 2023 (2 years)

Computational Activity

120 min.

Mean position

Computational Physics Lab II 2023 (2 years)

probability density particle in a box wave function quantum mechanics

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.

Experiment

60 min.

Microwave oven Ice Calorimetry Lab

Energy and Entropy 2021 (2 years)

heat entropy water ice thermodynamics

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.

Small Group Activity

30 min.

Name the experiment

Energy and Entropy 2021 (3 years)

partial derivatives experiment thermodynamics

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.

Small Group Activity

30 min.

Name the experiment (changing entropy)

Energy and Entropy 2021 (2 years)

thermodynamics entropy experiment derivative first law

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.

Small White Board Question

5 min.

Newton's 2nd Law SWBQ

Central Forces 2023 (2 years)

Small White Board Question

5 min.

Normalization of the Gaussian for Wavefunctions

Periodic Systems 2022

E&M Conservative Fields Surfaces

Students find a wavefunction that corresponds to a Gaussian probability density.

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.

Small Group Activity

30 min.

Operators & Functions

Quantum Fundamentals 2023 (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.

Small Group Activity

30 min.

Optical depth of atmosphere

Contemporary Challenges 2021 (4 years) In this activity students estimate the optical depth of the atmosphere at the infrared wavelength where carbon dioxide has peak absorption.

Small Group Activity

30 min.

Outer Product of a Vector on Itself

Quantum Fundamentals 2023 (2 years)

Projection Operators Outer Products Matrices

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).

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.

Small White Board Question

10 min.

Partial Derivatives from a Contour Map

Static Fields 2023 (4 years)

phase transformation Clausius-Clapeyron mean field theory thermodynamics

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?

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.

Whole Class Activity

10 min.

Pineapples and Pumpkins

Static Fields 2023 (6 years)

quantum mechanics operator matrix element particle in a box eigenfunction

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.

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.

Small White Board Question

10 min.

Possible Worldlines

Theoretical Mechanics (4 years)

Special Relativity Spacetime Diagrams Worldlines Postulates of Relativity

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

Small Group Activity

120 min.

Projectile with Linear Drag

Theoretical Mechanics (4 years)

Projectile Motion Drag Forces Newton's 2nd Law Separable Differential Equations

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.

Small Group Activity

10 min.

Proportional Reasoning

Static Fields 2023 (3 years) In this small group activity, students calculate a (linear) function to represent the charge density on a one-dimensional rod from a description of the charge density in words.

Small Group Activity

30 min.

Quantifying Change

Thermo Derivatives

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.

Small Group Activity

30 min.

Quantum Expectation Values

Quantum Fundamentals 2023 (3 years)

Small Group Activity

30 min.

Quantum Measurement Play

Quantum Fundamentals 2023 (2 years)

Quantum Measurement Projection Operators Spin-1/2

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

Lecture

5 min.

Quantum Reference Sheet

Central Forces 2023 (6 years)

Small Group Activity

60 min.

Raising and Lowering Operators for Spin

Central Forces 2023 (2 years)

Small Group Activity

120 min.

Representations of the Infinite Square Well

Quantum Fundamentals 2023 (3 years)

thermodynamics statistical mechanics

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.

Small Group Activity

30 min.

Right Angles on Spacetime Diagrams

Theoretical Mechanics (4 years)

Special Relativity

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.

Small Group Activity

30 min.

Scalar Surface and Volume Elements

Static Fields 2023 (7 years)

inner product wave function quantum mechanics particle in a box

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.

Small Group Activity

10 min.

Sequential Stern-Gerlach Experiments

Quantum Fundamentals 2023 (3 years)

Computational Activity

120 min.

Sinusoidal basis set

Computational Physics Lab II 2023 (2 years)

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.

Small Group Activity

5 min.

Spherical Harmonics

Central Forces 2023 (4 years)

Kinesthetic

10 min.

Spin 1/2 with Arms

Quantum Fundamentals 2023 (2 years)

Quantum State Vectors Complex Numbers Spin 1/2 Arms Representation

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.

Small Group Activity

120 min.

Spin-1 Time Evolution

Quantum Fundamentals 2023

Time Evolution Spin-1

Students do calculations for time evolution for spin-1.

Small Group Activity

5 min.

Static Fields Equation Sheet

Static Fields 2023 (5 years)

Lecture

30 min.

Statistical Analysis of Stern-Gerlach Experiments

Small Group Activity

30 min.

Superposition States for a Particle on a Ring

central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum hermitian operators probability superposition

distance formula coordinate systems dot product vector addition

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.

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.

Small Group Activity

60 min.

Systems of Equations Compare and Contrast

Lecture

10 min.

Systems of Particles Lecture Notes

Central Forces 2022 (2 years)

Kinesthetic

30 min.

The Distance Formula (Star Trek)

Static Fields 2023 (6 years)

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. \[\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}\]

Small Group Activity

30 min.

The Hill

Vector Calculus II 23 (7 years)

Gradient

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.

Small Group Activity

30 min.

The Hillside

Vector Calculus I 2022 (2 years)

Planck distribution blackbody radiation photon statistical mechanics

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).

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.

Small Group Activity

10 min.

Thermal radiation at twice the temperature

Contemporary Challenges 2021 (4 years)

Stefan-Boltzmann blackbody radiation

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.

Small Group Activity

30 min.

Time Dependence for a Quantum Particle on a Ring Part 1

Theoretical Mechanics (6 years)

central forces quantum mechanics eigenstates eigenvalues angular momentum time dependence hermitian operators probability degeneracy quantum measurements

Time Dilation Proper Time Special Relativity

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.

Small White Board Question

10 min.

Time Dilation

Theoretical Mechanics (4 years)

Students answer conceptual questions about time dilation and proper time.

Kinesthetic

5 min.

Time Dilation Light Clock Skit

Special Relativity Time Dilation Light Clock Kinesthetic Activity

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

Small Group Activity

30 min.

Time Evolution of a Spin-1/2 System

Quantum Fundamentals 2023 (3 years)

quantum mechanics spin precession time evolution

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.

Lecture

30 min.

Time Evolution Refresher (Mini-Lecture)

Central Forces 2023 (3 years)

schrodinger equation time dependence stationary states

charge charge density multiple integral scalar field coordinate systems differential elements curvilinear coordinates

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

Small Group Activity

30 min.

Total Charge

Static Fields 2023 (6 years)

quantum states complex numbers arms Bloch sphere relative phase overall phase

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

Small Group Activity

30 min.

Travelling wave solution

Contemporary Challenges 2021 (4 years)

travelling wave

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

Lecture

5 min.

Unit Learning Outcomes: Classical Mechanics Orbits

Central Forces 2023 This handout lists Motivating Questions, Key Activities/Problems, Unit Learning Outcomes, and an Equation Sheet for a Unit on Classical Mechanics Orbits. It can be used both to introduce the unit and, even better, for review.

Lecture

5 min.

Unit Learning Outcomes: Quantum Mechanics on a Ring

Central Forces 2023

Small Group Activity

30 min.

Using $pV$ and $TS$ Plots

Energy and Entropy 2021 (2 years)

work heat first law energy

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.

Kinesthetic

10 min.

Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems

Quantum Fundamentals 2023 (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).

Kinesthetic

10 min.

Using Arms to Represent Time Dependence in Spin 1/2 Systems

Quantum Fundamentals 2023 (2 years)

Arms Representation quantum states time dependence Spin 1/2

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.

Kinesthetic

10 min.

Using Arms to Visualize Complex Numbers (MathBits)

Lie Groups and Lie Algebras 23 (4 years)

arms complex numbers Argand diagram complex plane rectangular form exponential form complex conjugate math

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.

Kinesthetic

30 min.

Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)

Quantum Fundamentals 2021

arms complex numbers phase rotation reflection math

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.

Mathematica Activity

30 min.

Using Technology to Visualize Potentials

Static Fields 2023 (6 years)

electrostatic potential visualization

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 demonstrate several different ways of plotting the potential.

Small Group Activity

10 min.

Using Tinker Toys to Represent Spin 1/2 Quantum Systems

spin 1/2 eigenstates quantum states

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.

Small Group Activity

30 min.

Vector Differential--Curvilinear

Vector Calculus II 23 (11 years)

vector calculus coordinate systems curvilinear coordinates

vector differential rectangular coordinates math

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.

Small White Board Question

10 min.

Vector Differential--Rectangular

Vector Calculus II 23 (10 years)

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..

Small Group Activity

30 min.

Vector Integrals (Contour Map)

E&M Path integrals

Students explore path integrals using a vector field map and thinking about integration as chop-multiply-add.

Small Group Activity

30 min.

Vector Surface and Volume Elements

Static Fields 2023 (4 years)

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.

Small Group Activity

10 min.

Velocity and Acceleration in Polar Coordinates

Central Forces 2023 (3 years) Use geometry to find formulas for velocity and acceleration in polar coordinates.

Mathematica Activity

30 min.

Visualising the Gradient

Static Fields 2023 (7 years)

Taylor series power series approximation

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

Small Group Activity

30 min.

Visualization of Divergence

Vector Calculus II 23 (12 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 Simulation

30 min.

Visualization of Power Series Approximations

Theoretical Mechanics (13 years)

central forces quantum mechanics angular momentum probability density eigenstates time evolution superposition mathematica

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.

Mathematica Activity

30 min.

Visualization of Quantum Probabilities for a Particle Confined to a Ring

Central Forces 2023 (3 years)

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.

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 $m$.

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 Simulation

30 min.

Visualizing Flux through a Cube

Static Fields 2023 (6 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.

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.

Lecture

10 min.

Warm-Up Powerpoint