## Activity: Fourier Transform of a Plane Wave

Periodic Systems 2022
• This activity is used in the following sequences

Find the Fourier transform of a plane wave.

## Instructor's Guide

### Introduction

If students know about the Dirac delta function and its exponential representation, this is a great second example of the Fourier transform that students can work out in-class for themselves.

Students will need a short lecture giving the definition of the Fourier Transform \begin{equation} {\cal{F}}(f) =\tilde{f} (k)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx}\, f(x)\, dx \end{equation}

### Student Conversations

Students may ask what is meant by a plane wave. Help them figure out what is meant, from the context or give them the formula if time is tight.

Keep the time dependence in or leave it out depending on how much time you have to deal with a little extra algebraic confusion.

### Wrap-up

This example is (almost) the inverse of Fourier Transform of the Delta Function. If you really want the inverse problem, change the prompt to “Find the inverse Fourier transform of a plane wave.”
• 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.
• group Fourier Transform of a Shifted Function

group Small Group Activity

5 min.

##### Fourier Transform of a Shifted Function
Periodic Systems 2022

Fourier Transforms and Wave Packets

• assignment Fourier Transform of Cosine and Sine

assignment Homework

##### Fourier Transform of Cosine and Sine
Periodic Systems 2022
1. Find the Fourier transforms of $f(x)=\cos kx$ and $g(x)=\sin kx$.
2. Find the Fourier transform of $g(x)$ using the formula for the Fourier transform of a derivative and your result for the Fourier transform of $f(x)$. Compare with your previous answer.
3. In quantum mechanics, the Fourier transform is the set of coefficients in the expansion of a quantum state in terms of plane waves, i.e. the function $\tilde{f}(k)$ is a continuous histogram of how much each functions $e^{ikx}$ contributes to the quantum state. What does the Fourier transform of the function $\cos kx$ tell you about which plane waves make up this quantum state? Write a sentence or two about how this makes sense.
• group Fourier Transform of a Derivative

group Small Group Activity

10 min.

##### Fourier Transform of a Derivative
Periodic Systems 2022

Fourier Transforms and Wave Packets

• group Fourier Transform of a Gaussian

group Small Group Activity

10 min.

##### Fourier Transform of a Gaussian
Periodic Systems 2022

Fourier Transforms and Wave Packets

• format_list_numbered Fourier Transforms and Wave Packets

format_list_numbered Sequence

##### Fourier Transforms and Wave Packets
This is a unit that introduces the Fourier transform and its properties and then applies the Fourier transform to free particle wave packets in non-relativistic quantum mechanics. The activities and homework are listed here. Appropriate text materials for mini-lectures can be found in the chapter Fourier Transforms and Wave Packets in the free online textbook The Geometry of Mathematical Methods.
• accessibility_new Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)

accessibility_new Kinesthetic

30 min.

##### Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)
Quantum Fundamentals 2021

Arms Sequence for Complex Numbers and Quantum States

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.
• group Operators & Functions

group Small Group Activity

30 min.

##### Operators & Functions
Quantum Fundamentals 2023 (3 years) Students are asked to:
• Test to see if one of the given functions is an eigenfunction of the given operator
• See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.
• group Visualizing Plane Waves

group Small Group Activity

60 min.

##### Visualizing Plane Waves

Each small group of 3-4 students is given a white board or piece of paper with a square grid of points on it.

Each group is given a different two-dimensional vector $\vec{k}$ and is asked to calculate the value of $\vec{k} \cdot \vec {r}$ for each point on the grid and to draw the set of points with constant value of $\vec{k} \cdot \vec{r}$ using rainbow colors to indicate increasing value.

• assignment Using Canvas Discussions

assignment Homework

##### Using Canvas Discussions
Static Fields 2023 The question is meant to get you used to using the Canvas Discussion Board. Please go to the course Canvas page and find the Discussions tab on the left hand side. Find the Discussion titled Random and add one of the following:
1. A random physics fact.
2. One thing you like about physics.
3. One question you have for Jeff.

Learning Outcomes