*group* Small Group Activity

30 min.

Cartesian Basis $S_z$ basis completeness normalization orthogonality basis

Student explore the properties of an orthonormal basis using the Cartesian and \(S_z\) bases as examples.*face* Lecture

30 min.

*group* Small Group Activity

30 min.

thermodynamics intensive extensive temperature volume energy entropy

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.

density charge density mass density linear density uniform idealization

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.

Steady current current density magnetic field idealization

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*group* Small Group Activity

5 min.

*accessibility_new* Kinesthetic

10 min.

*groups* Whole Class Activity

10 min.

*group* Small Group Activity

30 min.

*face* Lecture

10 min.

*group* Small Group Activity

30 min.

*computer* Computer Simulation

30 min.

*face* Lecture

120 min.

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.*group* Small Group Activity

120 min.

*group* Small Group Activity

30 min.

Taylor series power series approximation

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.

*group* Small Group Activity

30 min.

*group* Small Group Activity

10 min.

*group* Small Group Activity

30 min.

*face* Lecture

120 min.

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.*face* Lecture

30 min.

Bra-Ket Notations Wavefunction Notation Completeness Relations Probability Probability Density

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

30 min.

*accessibility_new* Kinesthetic

10 min.

*assignment_ind* Small White Board Question

10 min.

Cylindrical coordinates spherical coordinates curvilinear coordinates

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.

*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

*assignment_ind* Small White Board Question

10 min.

*group* Small Group Activity

30 min.

*computer* Mathematica Activity

30 min.

*group* Small Group Activity

30 min.

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

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.

*keyboard* Computational Activity

120 min.

*group* Small Group Activity

30 min.

- 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
- objects with like charge repel and opposite charge attract,
- object tend to move toward lower energy configurations
- The potential energy of a charged particle is related to its charge: \(U=qV\)
- The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)

*group* Small Group Activity

30 min.

*group* Small Group Activity

60 min.

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.*group* Small Group Activity

30 min.

*assignment_ind* Small White Board Question

10 min.

*group* Small Group Activity

30 min.

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

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.

*keyboard* Computational Activity

120 min.

*keyboard* Computational Activity

120 min.

electrostatic potential python

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.

*group* Small Group Activity

30 min.

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

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.

thermodynamics statistical mechanics

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.

latent heat heat capacity internal energy entropy

This short lecture introduces the ideas required for Ice Calorimetry Lab or Microwave oven Ice Calorimetry Lab.*group* Small Group Activity

30 min.

*face* Lecture

120 min.

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.*face* Lecture

30 min.

*group* Small Group Activity

120 min.

*group* Small Group Activity

60 min.

*group* Small Group Activity

5 min.

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.*group* Small Group Activity

60 min.

*group* Small Group Activity

30 min.

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

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

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.

*group* Small Group Activity

30 min.

*group* Small Group Activity

5 min.

*grading* Quiz

60 min.

*group* Small Group Activity

10 min.

*group* Small Group Activity

10 min.

*face* Lecture

120 min.

Gibbs entropy information theory probability statistical mechanics

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.

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

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.

Mechanics Gravitational Force Gravitational Potential Energy Derivatives Introductory Physics

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.

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.*group* Small Group Activity

30 min.

*face* Lecture

5 min.

*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

*group* Small Group Activity

60 min.

*face* Lecture

120 min.

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.*group* Small Group Activity

30 min.

*face* Lecture

30 min.

entropy multiplicity heat thermodynamics

This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between*keyboard* Computational Activity

120 min.

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.
*face* Lecture

30 min.

*group* Small Group Activity

30 min.

magnetic fields current Biot-Savart law vector field symmetry

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.

*assignment_ind* Small White Board Question

30 min.

Angular Momentum Spin Magnetic Moment Stern-Gerlach Experiments

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.

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

Students work in small groups to use the superposition principle \[\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}\] to find an integral expression for the magnetic vector potential, \(\vec{A}(\vec{r})\), due to a spinning ring of charge.

In an optional extension, students find a series expansion for \(\vec{A}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

*group* Small Group Activity

30 min.

energy conservation mass conservation collision

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.

*keyboard* Computational Activity

120 min.

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.*biotech* Experiment

60 min.

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.*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

*assignment_ind* Small White Board Question

5 min.

*group* Small Group Activity

30 min.

E&M Conservative Fields Surfaces

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*group* Small Group Activity

30 min.

- 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.

*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

- 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.

*face* Lecture

120 min.

phase transformation Clausius-Clapeyron mean field theory thermodynamics

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.

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.

quantum mechanics operator matrix element particle in a box eigenfunction

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.

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.*group* Small Group Activity

120 min.

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.*group* Small Group Activity

10 min.

*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

*group* Small Group Activity

60 min.

*group* Small Group Activity

120 min.

*face* Lecture

30 min.

thermodynamics statistical mechanics

These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

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.

*keyboard* Computational Activity

120 min.

inner product wave function quantum mechanics particle in a box

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.*accessibility_new* Kinesthetic

10 min.

Quantum State Vectors Complex Numbers Spin 1/2 Arms Representation

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

30 min.

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

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.

*accessibility_new* Kinesthetic

30 min.

distance formula coordinate systems dot product vector addition

A short improvisational role-playing skit based on the*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

*face* Lecture

120 min.

Planck distribution blackbody radiation photon statistical mechanics

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.

*group* Small Group Activity

30 min.

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

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.

*accessibility_new* Kinesthetic

5 min.

Special Relativity Time Dilation Light Clock Kinesthetic Activity

Students act out the classic light clock scenario for deriving time dilation.*group* Small Group Activity

30 min.

*face* Lecture

30 min.

*group* Small Group Activity

30 min.

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

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.

*group* Small Group Activity

30 min.

*accessibility_new* Kinesthetic

10 min.

quantum states complex numbers arms Bloch sphere relative phase overall phase

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.

*accessibility_new* Kinesthetic

10 min.

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

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.

*computer* Mathematica Activity

30 min.

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*group* Small Group Activity

10 min.

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.*group* Small Group Activity

30 min.

vector calculus coordinate systems curvilinear coordinates

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 coordinates math

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.

*group* Small Group Activity

30 min.

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.

*group* Small Group Activity

30 min.

*computer* Computer Simulation

30 min.

Taylor series power series approximation

Students use prepared*computer* Mathematica Activity

30 min.

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

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.

*computer* Mathematica Activity

30 min.

*computer* Computer Simulation

30 min.

*group* Small Group Activity

60 min.

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.

*face* Lecture

5 min.

*group* Small Group Activity

30 min.

*face* Lecture

120 min.

work heat engines Carnot thermodynamics entropy

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.

*group* Small Group Activity

30 min.