## Activity: Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute

Quantum Fundamentals 2023 (3 years)

Commutation Relations for Spin Operators

A commutator of two observables is defined as:

$[\hat{A},\hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$

Determine the results of the following commutators:

1. $[\hat{S}_x,\hat{S}_y]$
2. $[\hat{S}_y,\hat{S}_z]$
3. $[\hat{S}_z,\hat{S}_x]$
4. $[\hat{S}_y,\hat{S}_x]$
5. $[\hat{S}_z,\hat{S}_y]$
6. $[\hat{S}_x,\hat{S}_z]$

Remember that the matrix representation of the spin operators written in the $S_z$ basis is: \begin{align*} \hat{S}_x \doteq \frac{\hbar}{2}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \quad \hat{S}_y \doteq \frac{\hbar}{2}\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \quad \hat{S}_z \doteq \frac{\hbar}{2}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \end{align*}

## Activity: Introduction

Divide students into groups to work out whether the spin operators commute.

## Activity: Wrap-up

Groups should find that none of the quantum operators commute and therefore do not share the same basis for their respective eigenvectors. Because of this, it provides mathematical evidence for many properties that have so far been only observed. Since none of them commute, none of them have the same base, nor can the spin operators be measured simultaneously.

• group Time Evolution of a Spin-1/2 System

group Small Group Activity

30 min.

##### Time Evolution of a Spin-1/2 System
Quantum Fundamentals 2023 (3 years)

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 Angular Momentum Commutation Relations: Lecture

face Lecture

10 min.

##### Angular Momentum Commutation Relations: Lecture
Central Forces 2023 (3 years)
• group Outer Product of a Vector on Itself

group Small Group Activity

30 min.

##### Outer Product of a Vector on Itself
Quantum Fundamentals 2023 (2 years)

Completeness Relations

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 Unknowns Spin-1/2 Brief

assignment Homework

##### Unknowns Spin-1/2 Brief
Quantum Fundamentals 2023 (3 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 Quantum Expectation Values

group Small Group Activity

30 min.

##### Quantum Expectation Values
Quantum Fundamentals 2023 (3 years)
• accessibility_new Spin 1/2 with Arms

accessibility_new Kinesthetic

10 min.

##### Spin 1/2 with Arms
Quantum Fundamentals 2023 (2 years)

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 Changing Spin Bases with a Completeness Relation

group Small Group Activity

10 min.

##### Changing Spin Bases with a Completeness Relation
Quantum Fundamentals 2023 (3 years)

Completeness Relations

Students work in small groups to use completeness relations to change the basis of quantum states.
• assignment Spin Three Halves Time Dependence

assignment Homework

##### Spin Three Halves Time Dependence
Quantum Fundamentals 2023 A spin-3/2 particle initially is in the state $|\psi(0)\rangle = |\frac{1}{2}\rangle$. This particle is placed in an external magnetic field so that the Hamiltonian is proportional to the $\hat{S}_x$ operator, $\hat{H} = \alpha \hat{S}_x \doteq \frac{\alpha\hbar}{2}\begin{pmatrix} 0 & \sqrt{3} & 0 & 0\\ \sqrt{3} & 0 & 2 & 0\\ 0 & 2 & 0 & \sqrt{3} \\ 0 & 0 & \sqrt{3} & 0 \end{pmatrix}$
1. Find the energy eigenvalues and energy eigenstates for the system.
2. Find $|\psi(t)\rangle$.
3. List the outcomes of all possible measurements of $S_x$ and find their probabilities. Explicitly identify any probabilities that depend on time.
4. List the outcomes of all possible measurements of $S_z$ and find their probabilities. Explicitly identify any probabilities that depend on time.
• group Expectation Value and Uncertainty for the Difference of Dice

group Small Group Activity

60 min.

##### Expectation Value and Uncertainty for the Difference of Dice
Quantum Fundamentals 2023 (3 years)
• group Matrix Representation of Angular Momentum

group Small Group Activity

10 min.

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

Learning Outcomes