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## Visualizing Complex Two Component Vectors: Instructor's Guide

### Main Ideas

- Different forms of complex numbers ($x+iy$ and $re^{i\phi}$)
- Use of Argand diagrams
- Linear transformations

### Students' Task

*Estimated Time: 15 minutes*

Students use their left arm as an Argand diagram and work in pairs to demonstrate transformations of complex two-component vectors.

### Prerequisite Knowledge

- Argand diagrams

### Props/Equipment

- Pairs of students

### Activity: Introduction

Students use the embodied experience of using their left arm as an Argand diagram where the left shoulder is the origin and practice using this to represent various two-component complex vectors. The same convention should be chosen across the classroom; typically the partner on the left side represents the top component, and the one on the right side represents the bottom component.

### Activity: Student Conversations

Each example is chosen to highlight a particular aspect of the representation.

- Representing two component vectors in pairs, where one person is the top component and the other is the bottom component. These simple examples help to orient the students to representing two component vectors in pairs.

$$\pmatrix{1\\i} \rm{and} \pmatrix{1\\-i}$$

- Transformations of two component vectors. This example highlights the idea that what is meant by an eigenvalue is that both components have the same thing happen to them and that if the eigenvalue is complex, it means a rotation in the complex plane.

$$e^{i\pi/4}\pmatrix{1\\i}=\pmatrix{e^{i\pi/4}\\ie^{i\pi/4}}=\pmatrix{e^{i\pi/4}\\e^{i3\pi/4}}$$

- This provides an example of a transformation where different things happen to each component.

$$\pmatrix{i&0\\0&1}\pmatrix{1\\i}=\pmatrix{i\\i}$$

- This is an example of a transformation where both components stay real, but the total length (which stays constant) sloshes back and forth between the two components.

$$\pmatrix{\cos\theta&\sin\theta\\-\sin\theta&\cos\theta}\pmatrix{1\\0}=\pmatrix{\cos\theta\\-\sin\theta}=\dfrac{1}{\sqrt{2}}\pmatrix{1\\-1}$, where $\theta=\dfrac{\pi}{4}$$

- This is the same rotation in the complex plane and can be used to highlight the connection between transformations and eigenvalues and eigenvectors.

$$\pmatrix{\cos\theta&\sin\theta\\-\sin\theta&\cos\theta}\pmatrix{1\\i}=\pmatrix{\cos\theta+i\sin\theta\\-\sin\theta+i\cos\theta}=\dfrac{1}{\sqrt{2}}\pmatrix{1+i\\-1+i}=\pmatrix{e^{i\pi/4}\\e^{i3\pi/4}}=e^{i\pi/4}\pmatrix{1\\i}$$

### Activity: Wrap-up

After each example, a short wrap-up should be done by having one or more groups discuss what they did and why while highlighting the relative features of the particular example.

### Extensions

This activity is the second activity in a sequence addressing Visualizing Complex Numbers in the context of quantum mechanics.

- Preceding activity:
- Visualizing Complex Numbers: This kinesthetic activity introduces students to the rectangular and exponential forms of complex numbers.

- Follow-up activity:
- Visualizing Complex Time Dependence for Spin 1/2 Systems: This kinesthetic activity is designed to help students visualize complex time dependence of the spin-1/2 system by using their arms, in pairs, to represent the different time dependencies of the eigenstates and the superposition states.