## Activity: Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)

Quantum Fundamentals 2021
Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.
• This activity is used in the following sequences
What students learn
• Linear transformations can act on complex-valued vectors;
• Eigenvalues can be complex;
• Linear transformations can have complex eigenvectors even when they have no real eigenvectors.

## Instructor's Guide

### Introduction

Students use their left arm to represent a complex number in the complex plane (Argand diagram) where their left shoulder is the origin. They practice using this representation to explore various transformations of 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.

### 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 rotation, a transformation where both components stay real and the total length stays constant. The relative magnitude of the two components depends on the value of $\theta$. $\pmatrix{\cos\theta&\sin\theta\\-\sin\theta&\cos\theta}\pmatrix{1\\0}=\pmatrix{\cos\theta\\-\sin\theta}$ $\rightarrow\dfrac{1}{\sqrt{2}}\pmatrix{1\\-1}$ when $\theta=\dfrac{\pi}{4}$
• This next example is the same rotation, now acting on a two-component complex vector and can be used to highlight the connection between transformations and eigenvalues and eigenvectors. $\pmatrix{\cos\theta&\sin\theta\\-\sin\theta&\cos\theta}\pmatrix{1\\i}=\pmatrix{\cos\theta+i\sin\theta\\-\sin\theta+i\cos\theta}=\dfrac{1}{\sqrt{2}}\pmatrix{1+i\\-1+i}=\pmatrix{e^{i\pi/4}\\e^{i3\pi/4}}=e^{i\pi/4}\pmatrix{1\\i}$

### Wrap-up

After each example, a short wrap-up should be done by having one or more groups discuss what they did and why. Highlight the relative features of the particular example.
• group Fourier Transform of the Delta Function

group Small Group Activity

5 min.

##### Fourier Transform of the Delta Function
Periodic Systems 2022

Fourier Transforms and Wave Packets

Students calculate the Fourier transform of the Dirac delta function.
• format_list_numbered Arms Sequence for Complex Numbers and Quantum States

format_list_numbered Sequence

##### Arms Sequence for Complex Numbers and Quantum States
“Arms” is an engaging representation of complex numbers in which students use their left arms to geometrically represent numbers in the complex plane (an Argand diagram). The sequence starts with pure math activities in which students represent a single complex number (using prompts in both rectangular and exponential forms), demonstrate multiplication of complex numbers in exponential form, and act out a number of different linear transformation on pairs of complex numbers. Later activities, relevant to spin 1/2 systems in quantum mechanics, explore overall phases, relative phases, and time dependence. These activities can be combined and sequenced in many different ways; see the Instructor's Guide for the second activity for ideas about how to introduce the Arms representation the first time you use it.
• 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.
• accessibility_new Using Arms to Visualize Complex Numbers (MathBits)

accessibility_new Kinesthetic

10 min.

##### Using Arms to Visualize Complex Numbers (MathBits)
Lie Groups and Lie Algebras 23 (4 years)

Arms Sequence for Complex Numbers and Quantum States

Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.
• group Using Tinker Toys to Represent Spin 1/2 Quantum Systems

group Small Group Activity

10 min.

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

Arms Sequence for Complex Numbers and Quantum States

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.
• 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).
• assignment Circle Trig Complex

assignment Homework

##### Circle Trig Complex
Complex Numbers Exponential Form Rectangular Form Polar Form Unit Circle Quantum Fundamentals 2023 (2 years)

Find the rectangular coordinates of the point where the angle $\frac{5\pi}{3}$ meets the unit circle. If this were a point in the complex plane, what would be the rectangular and exponential forms of the complex number? (See figure.)

• assignment Phase

assignment Homework

##### Phase
Complex Numbers Rectangular Form Exponential Form Square of the Norm Overall Phase Quantum Fundamentals 2023 (3 years)
1. For each of the following complex numbers $z$, find $z^2$, $\vert z\vert^2$, and rewrite $z$ in exponential form, i.e. as a magnitude times a complex exponential phase:
• $z_1=i$,

• $z_2=2+2i$,
• $z_3=3-4i$.
2. In quantum mechanics, it turns out that the overall phase for a state does not have any physical significance. Therefore, you will need to become quick at rearranging the phase of various states. For each of the vectors listed below, rewrite the vector as an overall complex phase times a new vector whose first component is real and positive. $\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix}$
• group Spin-1 Time Evolution

group Small Group Activity

120 min.

##### Spin-1 Time Evolution
Quantum Fundamentals 2023

Students do calculations for time evolution for spin-1.
• accessibility_new Using Arms to Represent Time Dependence in Spin 1/2 Systems

accessibility_new Kinesthetic

10 min.

##### Using Arms to Represent Time Dependence in Spin 1/2 Systems
Quantum Fundamentals 2023 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.

Author Information
Corinne Manogue
Learning Outcomes