format_list_numbered Sequence

Quantum Ring Sequence
Students calculate probabilities and expectation values for a quantum mechanical particle confined to a circular ring in bra/ket, matrix, and wave function representations and compare the different calculation methods. Several different graphical representations of the time dependence for both states with special symmetry and arbitrary states are explored in a Mathematica notebook. Compared to the analogous particle-in-a-box, this quantum system has a new feature---degenerate energy eigenstates.

format_list_numbered Sequence

Arms Sequence for Complex Numbers and Quantum States
“Arms” is an engaging representation of complex numbers in which students use their left arms to geometrically represent numbers in the complex plane (an Argand diagram). The sequence starts with pure math activities in which students represent a single complex number (using prompts in both rectangular and exponential forms), demonstrate multiplication of complex numbers in exponential form, and act out a number of different linear transformation on pairs of complex numbers. Later activities, relevant to spin 1/2 systems in quantum mechanics, explore overall phases, relative phases, and time dependence. These activities can be combined and sequenced in many different ways; see the Instructor's Guide for the second activity for ideas about how to introduce the Arms representation the first time you use it.

group Small Group Activity

60 min.

Raising and Lowering Operators for Spin
Central Forces 2023 (2 years)

assignment Homework

Ring Table
Central Forces 2023 (3 years)

Attached, you will find a table showing different representations of physical quantities associated with a quantum particle confined to a ring. Fill in all of the missing entries. Hint: You may look ahead. We filled out a number of the entries throughout the table to give you hints about what the forms of the other entries might be. pdf link for the Table or doc link for the Table

group Small Group Activity

30 min.

Working with Representations on the Ring
Central Forces 2023 (3 years)

accessibility_new Kinesthetic

10 min.

Spin 1/2 with Arms
Quantum Fundamentals 2022 (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.

accessibility_new Kinesthetic

10 min.

Using Arms to Visualize Complex Numbers (MathBits)
Quantum Fundamentals 2022 (2 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.

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.

groups Whole Class Activity

10 min.

Air Hockey
Central Forces 2023 (3 years)

central forces potential energy classical mechanics

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.

keyboard 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.

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.

accessibility_new 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.

group 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.

assignment Homework

Circle Trigonometry
trigonometry cosine sine math circle Quantum Fundamentals 2022 (2 years)

On the following diagrams, mark both \(\theta\) and \(\sin\theta\) for \(\theta_1=\frac{5\pi}{6}\) and \(\theta_2=\frac{7\pi}{6}\). Write one to three sentences about how these two representations are related to each other. (For example, see: this PHET)

assignment Homework

Series Convergence

Power Series Sequence (E&M)

Static Fields 2022 (5 years)

Recall that, if you take an infinite number of terms, the series for \(\sin z\) and the function itself \(f(z)=\sin z\) are equivalent representations of the same thing for all real numbers \(z\), (in fact, for all complex numbers \(z\)). This is not always true. More commonly, a series is only a valid, equivalent representation of a function for some more restricted values of \(z\). The technical name for this idea is convergence--the series only "converges" to the value of the function on some restricted domain, called the “interval” or “region of convergence.”

Find the power series for the function \(f(z)=\frac{1}{1+z^2}\). Then, using the Mathematica worksheet from class (vfpowerapprox.nb) as a model, or some other computer algebra system like Sage or Maple, explore the convergence of this series. Where does your series for this new function converge? Can you tell anything about the region of convergence from the graphs of the various approximations? Print out a plot and write a brief description (a sentence or two) of the region of convergence. You may need to include a lot of terms to see the effect of the region of convergence. Keep adding terms until you see a really strong effect!

Note: As a matter of professional ettiquette (or in some cases, as a legal copyright requirement), if you use or modify a computer program written by someone else, you should always acknowledge that fact briefly in whatever you write up. Say something like: “This calculation was based on a (name of software package) program titled (title) originally written by (author) copyright (copyright date).”

assignment_ind Small White Board Question

10 min.

Dot Product Review
Static Fields 2022 (6 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.

group 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

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.

group Small Group Activity

10 min.

Matrix Representation of Angular Momentum
Central Forces 2023 (2 years)

group Small Group Activity

30 min.

Thermodynamic States (Remote)

Thermo

Try doing this activity as a follow-up to the Changes in Internal Energy (Remote) about the first law of thermodynamics.

assignment_ind Small White Board Question

10 min.

Electrostatic Potential Due to a Point Charge
Static Fields 2022

Warm-Up