## Activity: Fourier Transform of the Delta Function

Periodic Systems 2022
Students calculate the Fourier transform of the Dirac delta function.
• This activity is used in the following sequences
What students learn
1. The derivative of the Dirac delta function is a constant. (The Fourier transform of the “spikiest” function that exists is the most “spread out” function that exists.)
2. Even though the Dirac delta function is real-valued, its Fourier transform is complex.

Find the Fourier Transform of the delta function.

## Instructor's Guide

### Introduction

If students know about the Dirac delta function, this is a great first 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 $${\cal{F}}(f) =\tilde{f} (k)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx}\, f(x)\, dx$$ We strongly suggest the convention of putting the exponential to the left of the function in the integrand to highlight the relationship between the Fourier Transform and the quantum mechanics notion of finding the projection of a quantum wavefunciton $f(x)$ onto a plane wave (complete with the complex conjugate of the plane wave that turns the ket into a bra).

We always use the convention of putting a factor of $\frac{1}{\sqrt{2\pi}}$ into the definitions of each of the Fourier Transform and its inverse to make the operations symmetric in this way. Warn students that this convention is NOT universal. They should use caution when using other resources.

### Student Conversations

Students may ask where the peak of the delta function should be. If so, this is a great opportunity to (again) highlight the advantages of choosing a parameter for an unknown value so that you are doing many cases at once. Use the case $x_0=0$ as a limiting case.

### Wrap-up

This example highlights the inherent complex number nature of Fourier transforms. Because of the factor $e^{-ikx}$, the Fourier transform of a real function is typically NOT real.

The answer here is trying to be the constant number $\frac{1}{\sqrt{2\pi}}$, but this number is augmented by a complex phase $e^{-ikx_0}$ that depends on $x_0$m where the peak of the delta function is.

You might want to ask students to act out the Fourier transform of the delta function using the arms representation.

This example is (almost) the inverse of Fourier Transform of a Plane Wave. If you really want the inverse problem, change one of the prompts to “Find the inverse Fourier transform of ...”

• 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

• group Fourier Transform of a Plane Wave

group Small Group Activity

5 min.

##### Fourier Transform of a Plane Wave
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

• 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.
• 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

• 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 Guess the Fourier Series from a Graph

group Small Group Activity

10 min.

##### Guess the Fourier Series from a Graph
Oscillations and Waves 2023 (2 years) The students are shown the graph of a function that is a superposition of three harmonic functions and asked to guess the harmonic terms of the Fourier series. Students then use prewritten Sage code to verify the coefficients from their guess. The program allows the students to enter functions of their own choice as well as the one that is preset.
• group Energy and Angular Momentum for a Quantum Particle on a Ring

group Small Group Activity

30 min.

##### Energy and Angular Momentum for a Quantum Particle on a Ring

Quantum Ring Sequence

Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.
• assignment Fourier Series for the Ground State of a Particle-in-a-Box.

assignment Homework

##### Fourier Series for the Ground State of a Particle-in-a-Box.
Oscillations and Waves 2023 (2 years) Treat the ground state of a quantum particle-in-a-box as a periodic function.
• Set up the integrals for the Fourier series for this state.

• Which terms will have the largest coefficients? Explain briefly.

• Are there any coefficients that you know will be zero? Explain briefly.

• Using the technology of your choice or by hand, calculate the four largest coefficients. With screen shots or otherwise, show your work.

• Using the technology of your choice, plot the ground state and your approximation on the same axes.

Learning Outcomes