computer Mathematica Activity

30 min.

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

keyboard Computational Activity

120 min.

Mean position
Computational Physics Lab II 2022

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.

computer Mathematica Activity

30 min.

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

group Small Group Activity

60 min.

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

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

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.

computer Mathematica Activity

30 min.

Visualization of Quantum Probabilities for a Particle Confined to a Ring
Central Forces 2023 (3 years)

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

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.

assignment Homework

Ring Function
Central Forces 2023 (3 years) Consider the normalized wavefunction \(\Phi\left(\phi\right)\) for a quantum mechanical particle of mass \(\mu\) constrained to move on a circle of radius \(r_0\), given by: \begin{equation} \Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)} \end{equation} where \(N\) is the normalization constant.
  1. Find \(N\).

  2. Plot this wave function.
  3. Plot the probability density.
  4. Find the probability that if you measured \(L_z\) you would get \(3\hbar\).
  5. What is the expectation value of \(L_z\) in this state?

group Small Group Activity

30 min.

Earthquake waves
Contemporary Challenges 2022 (4 years)

wave equation speed of sound

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

group Small Group Activity

30 min.

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

face Lecture

30 min.

Review of Thermal Physics
Thermal and Statistical Physics 2020

thermodynamics statistical mechanics

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

assignment Homework

Visualization of Wave Functions on a Ring
Central Forces 2023 (3 years) Using either this Geogebra applet or this Mathematica notebook, explore the wave functions on a ring. (Note: The Geogebra applet may be a little easier to use and understand and is accessible if you don't have access to Mathematica, but it is more limited in the wave functions that you can represent. Also, the animation is pretty jumpy in some browsers, especially Firefox. Imagine that the motion is smooth.)
  1. Look at graphs of the following states \begin{align} \Phi_1(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +\left|{-2}\right\rangle )\\ \Phi_2(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle -\left|{-2}\right\rangle )\\ \Phi_3(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +i\left|{-2}\right\rangle ) \end{align} Write a short description of how these states differ from each other.
  2. Find a state for which the probability density does not depend on time. Write the state in both ket and wave function notation. These are called stationary states. Generalize your result to give a characterization of the set of all possible states that are stationary states.
  3. Find a state that is right-moving. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are right-moving.
  4. Find a state that is a standing wave. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are standing waves.

group Small Group Activity

30 min.

Time Dependence for a Quantum Particle on a Ring
Theoretical Mechanics (6 years)

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

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.

group Small Group Activity

30 min.

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

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.

face 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.

assignment Homework

Ideal gas in two dimensions
Ideal gas Entropy Chemical potential Thermal and Statistical Physics 2020
  1. Find the chemical potential of an ideal monatomic gas in two dimensions, with \(N\) atoms confined to a square of area \(A=L^2\). The spin is zero.

  2. Find an expression for the energy \(U\) of the gas.

  3. Find an expression for the entropy \(\sigma\). The temperature is \(kT\).

keyboard 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.

assignment Homework

Symmetry Arguments for Gauss's Law
Static Fields 2022 (4 years)

Instructions for 2022: You will need to complete this assignment in a 15 minute appointment on Zoom or in person with one of the members of the teaching team between 1/21 and 10 pm on 1/26. Here is a link to a sign-up page.

You are required to watch a sample video for how to make symmetry arguments here. As demonstrated in the video you should bring with you to the meeting a cylinder, an observer, and a vector.

Use good symmetry arguments to find the possible direction for the electric field due to a charged wire. Also, use good symmetry arguments to find the possible functional dependence of the electric field due to a charged wire. Rather than writing this up to turn in, you should find a member of the teaching team and make the arguments to them verbally.

group Small Group Activity

30 min.

Magnetic Vector Potential Due to a Spinning Charged Ring
Static Fields 2022 (5 years)

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

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.

Electrostatic Potential Due to a Ring of Charge
Static Fields 2022 (7 years)

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

Power Series Sequence (E&M)

Ring Cycle Sequence

Warm-Up

Students work in groups of three to use the superposition principle \[V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}\] to find an integral expression for the electrostatic potential, \(V(\vec{r})\), everywhere in space, due to a ring of charge.

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

face Lecture

120 min.

Fermi and Bose gases
Thermal and Statistical Physics 2020

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.

Magnetic Field Due to a Spinning Ring of Charge
Static Fields 2022 (6 years)

magnetic fields current Biot-Savart law vector field symmetry

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.