group Small Group Activity

10 min.

Sequential Stern-Gerlach Experiments
Quantum Fundamentals 2022 (3 years)

group Small Group Activity

30 min.

Quantum Measurement Play
Quantum Fundamentals 2022 (2 years)

Quantum Measurement Projection Operators Spin-1/2

The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.

group Small Group Activity

30 min.

Expectation Values for a Particle on a Ring
Central Forces 2023 (3 years)

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

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.

Time Evolution of a Spin-1/2 System
Quantum Fundamentals 2022 (3 years)

quantum mechanics spin precession time evolution

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.

group Small Group Activity

30 min.

Superposition States for a Particle on a Ring

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

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

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.

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.

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.

assignment Homework

Phase 2
quantum mechanics relative phase overall phase measurement probability Quantum Fundamentals 2022 (2 years) Consider the three quantum states: \[\left\vert \psi_1\right\rangle = \frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle\] \[\left\vert \psi_2\right\rangle = \frac{4}{5}\left\vert +\right\rangle- i\frac{3}{5} \left\vert -\right\rangle\] \[\left\vert \psi_3\right\rangle = -\frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle\]
  1. For each of the \(\left|{\psi_i}\right\rangle \) above, calculate the probabilities of spin component measurements along the \(x\), \(y\), and \(z\)-axes.
  2. Look For a Pattern (and Generalize): Use your results from \((a)\) to comment on the importance of the overall phase and of the relative phases of the quantum state vector.

group Small Group Activity

30 min.

Quantum Expectation Values
Quantum Fundamentals 2022 (3 years)

assignment Homework

Wavefunctions
Quantum Fundamentals 2022 (2 years)

Consider the following wave functions (over all space - not the infinite square well!):

\(\psi_a(x) = A e^{-x^2/3}\)

\(\psi_b(x) = B \frac{1}{x^2+2} \)

\(\psi_c(x) = C \;\mbox{sech}\left(\frac{x}{5}\right)\) (“sech” is the hyperbolic secant function.)

In each case:

  1. normalize the wave function,
  2. plot the wave function using Mathematica or other computer plotting tool (be sure to include the code you used and label your plots/axes appropriately),
  3. find the probability that the particle is measured to be in the range \(0<x<1\).

assignment Homework

Unknowns Spin-1/2 Brief
Quantum Fundamentals 2022 (2 years) 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?

group Small Group Activity

60 min.

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

assignment Homework

Spin Fermi Estimate
Quantum Fundamentals 2022 The following two problems ask you to make Fermi estimates. In a good Fermi estimate, you start from basic scientific facts you already know or quantities that you can reasonably estimate based on your life experiences and then reason your way to estimate a quantity that you would not be able guess. You may look up useful conversion factors or constants. Use words, pictures, and equations to explain your reasoning:
  1. Imagine that you send a pea-sized bead of silver through a Stern-Gerlach device oriented to measure the z-component of intrinsic spin. Estimate the total z-component of the intrinsic spin of the ball you would measure in the HIGHLY improbable case that every atom is spin up.
  2. Protons, neutrons, and electrons are all spin-1/2 particles. Give a (very crude) order of magnitude estimate of the number of these particles in your body.

group Small Group Activity

120 min.

Representations of the Infinite Square Well
Quantum Fundamentals 2022 (3 years)

Warm-Up

group Small Group Activity

60 min.

Expectation Value and Uncertainty for the Difference of Dice
Quantum Fundamentals 2022 (3 years)

group Small Group Activity

30 min.

Finding if \(S_{x}, \; S_{y}, \; and \; S_{z}\) Commute
Quantum Fundamentals 2022 (3 years)

group Small Group Activity

30 min.

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

accessibility_new Kinesthetic

10 min.

Using Arms to Represent Time Dependence in Spin 1/2 Systems
Quantum Fundamentals 2022 (2 years)

Arms Representation quantum states time dependence Spin 1/2

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.

group Small Group Activity

30 min.

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