## 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.
• group Small Group Activity schedule 10 min. build Tinker Toys
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• 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.
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## 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.

Author Information
Corinne Manogue
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Learning Outcomes