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:

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.