## Activity: Using Tinker Toys to Represent Spin 1/2 Quantum Systems

Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases ($x$, $y$, and $z$). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.
• This activity is used in the following sequences
What students learn
• The eigenstates of any operator can be written, in matrix notation, in many different bases;
• Each eigenstate looks like the standard basis in the basis in which the operator is diagonal;
• The states are specified by the relative lengths of the two complex numbers and their relative phase;
• The overall phase of the two complex numbers does not affect the state.
• Media

## Instructor's Guide

### Prerequisite Knowledge

Students should already have some experience with the $S_x$, $S_y$, and $S_z$ eigenstates for a spin 1/2 system, all written in the $z$ basis.

### Introduction

Draw the complex plane on a board at the front of the room and provide a list of the $S_x$, $S_y$, and $S_z$ eigenstates for a spin 1/2 system, all written in the $z$ basis.

\begin{align*} S_z: \qquad\qquad \begin{pmatrix} 1\\0 \end{pmatrix} \qquad &\begin{pmatrix} 0\\1 \end{pmatrix}\\ S_x: \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\1 \end{pmatrix} \qquad &\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\-1 \end{pmatrix}\\ S_y: \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\i \end{pmatrix} \qquad &\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\-i \end{pmatrix}\\ \end{align*}

Using a center piece (for the origin) and a long straight piece, demonstrate how a complex number can be represented with Tinker Toys.

Now ask the student groups to connect two center pieces with a short connector through their centers. Then have them build a representation of the $S_x$, $S_y$, and $S_z$ eigenstates.

### Student Conversations

Here is an image of the three sets of eigenstates: Tinker Toy models of the $S_z$, $S_x$, and $S_y$ eigenstates, all expressed in the $z$-basis. Spin up is on the left and spin down is on the right. The top complex number is represented by the top level of the Tinker Toy model, the bottom complex number is the bottom level of the model.

The complex numbers are in the standard orientation (positive real axis to the right).

The short (green) connector has no physical/geometric meaning.

### Wrap-up

Discuss how you can tell that each model represents a different state: i.e. they all have a different relative phase between the two complex numbers.

Discuss how the models can represent the overall phase independence of the state: i.e. any rotation of the model around its vertical axis represents the same state.

• accessibility_new Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems

accessibility_new Kinesthetic

10 min.

##### Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems
Quantum Fundamentals 2023 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).
• group Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute

group Small Group Activity

30 min.

##### Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute
Quantum Fundamentals 2023 (3 years)
• group Applying the equipartition theorem

group Small Group Activity

30 min.

##### Applying the equipartition theorem
Contemporary Challenges 2022 (4 years)

Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature $T$.
• 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.
• group Quantum Measurement Play

group Small Group Activity

30 min.

##### Quantum Measurement Play
Quantum Fundamentals 2023 (2 years)

The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.
• 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?
• assignment Matrix Elements and Completeness Relations

assignment Homework

##### Matrix Elements and Completeness Relations

Completeness Relations

Quantum Fundamentals 2023 (3 years)

Writing an operator in matrix notation in its own basis is easy: it is diagonal with the eigenvalues on the diagonal.

What if I want to calculate the matrix elements using a different basis??

The eigenvalue equation tells me what happens when an operator acts on its own eigenstate. For example: $\hat{S}_y\left|{\pm}\right\rangle _y=\pm\frac{\hbar}{2}\left|{\pm}\right\rangle _y$

In Dirac bra-ket notation, to know what an operator does to a ket, I needs to write the ket in the basis that is the eigenstates of the operator (in order to use the eigenvalue equation.)

One way to do this to stick completeness relationships into the braket: \begin{eqnarray*} \left\langle {+}\right|\hat{S_y}\left|{+}\right\rangle = \left\langle {+}\right|(I)\hat{S_y}(I)\left|{+}\right\rangle \end{eqnarray*}

where $I$ is the identity operator: $I=\color{blue}{\left|{+}\right\rangle _{yy}\left\langle {+}\right|}\;+\;\color{blue}{\left|{-}\right\rangle _{yy}\left\langle {-}\right|}$. This effectively rewrite the $\left|{+}\right\rangle$ in the $\left|{\pm}\right\rangle _y$ basis.

Find the top row matrix elements of the operator $\hat{S}_y$ in the $S_z$ basis by inserting completeness relations into the brakets. (The answer is already on the Spins Reference Sheet, but I want you do demonstrate the calculation.)

• group Raising and Lowering Operators for Spin

group Small Group Activity

60 min.

##### Raising and Lowering Operators for Spin
Central Forces 2023 (2 years)
• assignment Phase 2

assignment Homework

##### Phase 2
quantum mechanics relative phase overall phase measurement probability Quantum Fundamentals 2023 (3 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 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.

Author Information
Corinne Manogue
Keywords
spin 1/2 eigenstates quantum states
Learning Outcomes