accessibility_new Kinesthetic

10 min.

##### Acting Out Charge Densities
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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
AIMS Maxwell AIMS 21

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.

AIMS Maxwell AIMS 21 Static Fields Winter 2021

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 Spring 2021

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.

computer Computer Simulation

30 min.

##### Approximating Functions with Power Series
AIMS Maxwell AIMS 21 Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021 Central Forces Spring 2021 Static Fields Winter 2021

Power Series Sequence (E&M)

Students use 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 a power series expansion in a previous activity, Calculating Coefficients for a Power Series.

group Small Group Activity

120 min.

##### Box Sliding Down Frictionless Wedge
Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021

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
AIMS Maxwell AIMS 21 Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021

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.

group Small Group Activity

30 min.

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

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 Winter 2021

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.

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
AIMS Maxwell AIMS 21 Central Forces Spring 2021 Static Fields Winter 2021

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
AIMS Maxwell AIMS 21 Central Forces Spring 2021 Static Fields Winter 2021

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.

assignment_ind Small White Board Question

10 min.

##### Dot Product Review
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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.

computer Mathematica Activity

30 min.

##### Effective Potentials
Central Forces Spring 2021 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
AIMS Maxwell AIMS 21 AIMS Maxwell AIMS 21 Static Fields Winter 2021

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.

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 a Parallel Plate Capacitor” 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 changes 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)
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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)
AIMS Maxwell AIMS 21 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). Students then evaluate the limiting cases of the potential on the axes of symmetry.

group Small Group Activity

30 min.

##### Electrostatic Potential Due to a Ring of Charge
AIMS Maxwell AIMS 21 AIMS Maxwell AIMS 21 Static Fields Winter 2021

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use the superposition principle $V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}$ 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.

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

30 min.

##### Energy and heat and entropy
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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

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 Fall 2020 Theoretical Mechanics Fall 2021

group Small Group Activity

5 min.

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

group Small Group Activity

30 min.

##### Expectation Values for a Particle on a Ring
Central Forces Spring 2021

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.

group Small Group Activity

30 min.

##### Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute
Quantum Fundamentals Winter 2021

group Small Group Activity

30 min.

##### Flux through a Cone
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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 Fall 2020 Energy and Entropy Fall 2021

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

60 min.

##### Going from Spin States to Wavefunctions

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.

assignment_ind Small White Board Question

10 min.

##### Gravitational and Electrostatic Potential
AIMS Maxwell AIMS 21

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

5 min.

##### Heat and Temperature of Water Vapor (Remote)

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

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.

group Small Group Activity

30 min.

##### Ideal Gas Model

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

group Small Group Activity

5 min.

##### Leibniz Notation
AIMS Maxwell AIMS 21 Static Fields Winter 2021 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

30 min.

##### Lorentz Transformation (Geometric)
Theoretical Mechanics Fall 2020

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
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use the Biot-Savart law $\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}$ to find an integral expression for the magnetic field, $\vec{B}(\vec{r})$, due to a spinning ring of charge.

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

group Small Group Activity

30 min.

##### Magnetic Vector Potential Due to a Spinning Charged Ring
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Power Series Sequence (E&M)

Ring Cycle Sequence

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

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

group Small Group Activity

30 min.

##### Mass is not Conserved
Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021

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?

biotech Experiment

60 min.

##### Microwave oven Ice Calorimetry Lab
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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.

group Small Group Activity

30 min.

##### Name the experiment
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021 Energy and Entropy Fall 2021 Students will design an experiment that measures a specific partial derivative.

group Small Group Activity

30 min.

##### Name the experiment (changing entropy)
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021 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.

##### Navigating a Hill
AIMS Maxwell AIMS 21

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 Winter 2021 Students are asked to:
• Test to see if one of the given functions is an eigenfunction of the given operator
• See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.

group Small Group Activity

30 min.

##### Outer Product of a Vector on Itself

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).

assignment_ind Small White Board Question

10 min.

##### Partial Derivatives from a Contour Map
AIMS Maxwell AIMS 21 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?

groups Whole Class Activity

10 min.

##### Pineapples and Pumpkins
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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.

assignment_ind Small White Board Question

10 min.

##### Possible Worldlines
Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021

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

group Small Group Activity

30 min.

##### Quantifying Change (Remote)

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 Winter 2021

face Lecture

5 min.

##### Quantum Reference Sheet
Central Forces Spring 2021 Central Forces Spring 2021

group Small Group Activity

120 min.

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

group Small Group Activity

30 min.

##### Right Angles on Spacetime Diagrams
Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021

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
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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.

accessibility_new Kinesthetic

10 min.

##### Spin 1/2 with Arms

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 Winter 2021 Static Fields Winter 2021

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 Spring 2021 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.

accessibility_new Kinesthetic

30 min.

##### The Distance Formula (Star Trek)
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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.

##### Thermodynamic States (Remote)

Little is needed. Some students might be bothered by thinking about entropy if it hasn't been mentioned at all in class. Try doing this activity as a follow-up to the “Changes in Internal Energy" about the first law of thermodynamics.

group Small Group Activity

30 min.

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

Quantum Ring Sequence

Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.

assignment_ind Small White Board Question

10 min.

##### Time Dilation
Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021

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 Winter 2021

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 Spring 2021

Quantum Ring Sequence

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

group Small Group Activity

30 min.

##### Total Charge
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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.

##### Using $pV$ and $TS$ Plots
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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

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

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)

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 Winter 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
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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.

assignment_ind Small White Board Question

10 min.

##### Vector Differential--Rectangular
AIMS Maxwell AIMS 21 Vector Calculus II Fall 2021 Vector Calculus II Summer 21 Static Fields Winter 2021

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
AIMS Maxwell AIMS 21

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 more sophisticated vector approach to find surface, and volume elements.

group Small Group Activity

30 min.

##### Visualization of Divergence
AIMS Maxwell AIMS 21 Vector Calculus II Fall 2021 Vector Calculus II Summer 21 Static Fields Winter 2021 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 Spring 2021

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.

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.

group Small Group Activity

30 min.

##### Working with Representations on the Ring
Central Forces Spring 2021

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.