accessibility_new Kinesthetic

30 min.

##### Time Evolution of a Quantum Particle on a Ring with Arms
Students use their arms to act out stationary and non-stationary states of a quantum particle on a ring.

• Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

group Small Group Activity

30 min.

##### Mass is not Conserved

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?

• Found in: Theoretical Mechanics course(s)

None

##### Lines in Polar Coordinates

(Algebra involving trigonometric functions) Purpose: Practice with polar equations.

The general equation for a straight line in polar coordinates is given by: $$r(\phi)=\frac{r_0}{\cos(\phi-\delta)}$$ where $r_0$ and $\delta$ are constant parameters. Find the polar equation for the straight lines below. You do NOT need to evaluate any complicated trig or inverse trig functions. You may want to try plotting the general polar equation to figure out the roles of the parameters.

1. $y=3$
2. $x=3$
3. $y=-3x+2$

• Found in: Central Forces course(s)

None

##### Events on Spacetime Diagrams
1. Which pairs of events (if any) are simultaneous in the unprimed frame?

2. Which pairs of events (if any) are simultaneous in the primed frame?

3. Which pairs of events (if any) are colocated in the unprimed frame?

4. Which pairs of events (if any) are colocated in the primed frame?

1. For each of the figures, answer the following questions:
1. Which event occurs first in the unprimed frame?

2. Which event occurs first in the primed frame?

• Found in: Theoretical Mechanics course(s)

group Small Group Activity

5 min.

##### Events on Spacetime Diagrams
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.

• Found in: Theoretical Mechanics course(s)

None

##### Energy of a relativistic Fermi gas

For electrons with an energy $\varepsilon\gg mc^2$, where $m$ is the mass of the electron, the energy is given by $\varepsilon\approx pc$ where $p$ is the momentum. For electrons in a cube of volume $V=L^3$ the momentum takes the same values as for a non-relativistic particle in a box.

1. Show that in this extreme relativistic limit the Fermi energy of a gas of $N$ electrons is given by \begin{align} \varepsilon_F &= \hbar\pi c\left(\frac{3n}{\pi}\right)^{\frac13} \end{align} where $n\equiv \frac{N}{V}$ is the number density.

2. Show that the total energy of the ground state of the gas is \begin{align} U_0 &= \frac34 N\varepsilon_F \end{align}

• Found in: Thermal and Statistical Physics course(s)

group Small Group Activity

30 min.

##### Visualization of Curl
Students predict from graphs of simple 2-d vector fields whether the curl is positive, negative, or zero in various regions of the domain using the definition of the curl of a vector field at a point as the maximum circulation per unit area through an infinitesimal box surrounding that point. Optionally, students can use computer algebra to verify their predictions.
• Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s)

face Lecture

30 min.

##### Determining $|\pm_x\rangle$ and $|\pm_y\rangle$ in the $S_z$ basis

• Found in: Quantum Fundamentals course(s)

group Small Group Activity

30 min.

##### Visualization of Divergence
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 geometric definition of the divergence of a vector field at a point as flux per unit volume (here: area) through an infinitesimal box surrounding that point. Optionally, students can use computer algebra to verify their predictions.
• Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s)

None

##### Diatomic hydrogen

At low temperatures, a diatomic molecule can be well described as a rigid rotor. The Hamiltonian of such a system is simply proportional to the square of the angular momentum \begin{align} H &= \frac{1}{2I}L^2 \end{align} and the energy eigenvalues are \begin{align} E_{\ell m} &= \hbar^2 \frac{\ell(\ell+1)}{2I} \end{align}

1. What is the energy of the ground state and the first and second excited states of the $H_2$ molecule? i.e. the lowest three distinct energy eigenvalues.

2. At room temperature, what is the relative probability of finding a hydrogen molecule in the $\ell=0$ state versus finding it in any one of the $\ell=1$ states?
i.e. what is $P_{\ell=0,m=0}/\left(P_{\ell=1,m=-1} + P_{\ell=1,m=0} + P_{\ell=1,m=1}\right)$

3. At what temperature is the value of this ratio 1?

4. At room temperature, what is the probability of finding a hydrogen molecule in any one of the $\ell=2$ states versus that of finding it in the ground state?
i.e. what is $P_{\ell=0,m=0}/\left(P_{\ell=2,m=-2} + P_{\ell=2,m=-1} + \cdots + P_{\ell=2,m=2}\right)$

• Found in: Energy and Entropy course(s)

accessibility_new Kinesthetic

10 min.

##### Spin 1/2 with Arms
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.

• Found in: Quantum Fundamentals course(s) Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

group Small Group Activity

30 min.

##### Directional Derivatives
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.

• Found in: Vector Calculus I course(s) Found in: Gradient Sequence sequence(s)

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.

None

##### Electric Field of a Finite Line

Consider the finite line with a uniform charge density from class.

1. Write an integral expression for the electric field at any point in space due to the finite line. In addition to your usual physics sense-making, you must include a clearly labeled figure and discuss what happens to the direction of the unit vectors as you integrate.Consider the finite line with a uniform charge density from class.
2. Perform the integral to find the $z$-component of the electric field. In addition to your usual physics sense-making, you must compare your result to the gradient of the electric potential we found in class. (If you want to challenge yourself, do the $s$-component as well!)

None

##### Unknowns Spin-1/2 Brief
With the Spins simulation set for a spin 1/2 system, measure the probabilities of all the possible spin components for each of the unknown initial states $\left|{\psi_3}\right\rangle$ and $\left|{\psi_4}\right\rangle$.
1. Use your measured probabilities to find each of the unknown states as a linear superposition of the $S_z$-basis states $\left|{+}\right\rangle$ and $\left|{-}\right\rangle$.
2. Articulate a Process: Write a set of general instructions that would allow another student in next year's class to find an unknown state from measured probabilities.
3. Compare Theory with Experiment: Design an experiment that will allow you to test whether your prediction for each of the unknown states is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results.
4. Make a Conceptual Connection: In general, can you determine a quantum state with spin-component probability measurements in only two spin-component-directions? Why or why not?
• Found in: Quantum Fundamentals course(s)

accessibility_new Kinesthetic

10 min.

##### Curvilinear Basis Vectors
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).

• Found in: Static Fields, Central Forces, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Curvilinear Coordinate Sequence sequence(s)

accessibility_new Kinesthetic

30 min.

##### The Distance Formula (Star Trek)
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. $\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}$

• Found in: Static Fields, AIMS Maxwell course(s) Found in: E&M Ring Cycle Sequence sequence(s)

group Small Group Activity

60 min.

##### The Park

This is the first activity relating the surfaces to the corresponding contour diagrams, thus emphasizing the use of multiple representations.

Students work in small groups to interpret level curves representing different concentrations of lead.

• Found in: Vector Calculus I, Surfaces/Bridge Workshop, Problem-Solving course(s)

group Small Group Activity

30 min.

##### Energy and Angular Momentum for a Quantum Particle on a Ring
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.

• Found in: Quantum Ring Sequence sequence(s)

keyboard Computational Activity

120 min.

##### Position operator
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.

• Found in: Computational Physics Lab II course(s)