Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [fourier-transform]

A unitary linear operator which resolves a function on $\mathbb{R}^N$ into a linear superposition of "plane wave functions". Most often used in physics for calculating the response of a time shift invariant linear system as the sum of its response to time harmonic excitation or for transforming a quantum state in position co-ordinates into one in momentum co-ordinates and contrawise. There is also a discrete, fast Fourier transform for discretised data.

1
vote
0answers
25 views

Fourier transforms and convolution for finding Fraunhofer diffraction of compound objects: how does it work?

I've been exposed to this notion in multiple classes (namely math and physics) but can't find any details about how one would actually calculate something using this principle: Diffraction in optics ...
2
votes
1answer
45 views

Non-commutative Fourier transform of an operator

Wigner-Weyl transform relates an operator to its distribution function in phase space through an operator Fourier transform which is said to be non-commutative. $$ \hat{\rho} \xrightarrow[non-comm]{...
1
vote
1answer
93 views

Fourier transform of cross-spectral density space matrix elements

In order to derive phase space like equation of motion (e.g. the equation of motion for the Wigner function of a single particle in one-dimension), it is an advantage to work with the Fourier ...
0
votes
2answers
60 views

Multiple frequencies [closed]

could someone please inform me how it is possible to send multiple frequencies down one wire? I’m referring specifically to a communication protocol known as HART. It seems they send a 4-20mA signal ...
0
votes
1answer
28 views

Why don't we use Leibniz integral rule when solving Diffusion equation using the Fourier transform?

My question concerns the solution to the diffusion equation: $$\frac{\partial{p(x,t)}}{\partial{t}}=D\frac{\partial^2{p(x,t)}}{\partial{x}^2}~.\tag{1}\label{1}$$ I have a question about the solution ...
2
votes
1answer
65 views

Fourier transform property in Feynman 1986 Dirac Memorial Lecture

In his famous 1986 Dirac Memorial Lecture, Feynman refers to a Fourier transforms theorem holding in case F(w) satisfies "certain properties", while being restricted to positive frequencies only: ...
2
votes
0answers
59 views

Expressing position wave function in momentum space for a single state

I am doing an introductory quantum mechanics course, I have been told that the momentum space wave function is essentially the Fourier transform of the position-space wave function. I.e. $$ \Phi(p,t)=\...
1
vote
1answer
43 views

Fourier transform of derivative of plane wave [closed]

What is the Fourier transform $F(k)$ of: $$ f(y) = A \, ik \, e^{iky} $$ If you calculate it with Wolfram Alpha, it says that there are no results found in terms of standard functions.
-1
votes
1answer
23 views

Need some help to show this relationship using parseval's theorem [closed]

Use Parseval’s theorem for the Fourier series and take L → ∞ to show that:
0
votes
0answers
32 views

Fourier transform of variable in path integral

In Sredinicki's QFT given below, he changed the integration variables in eq(174). This step confuses me. I only know some basics about path integral. In my opinion, when he used fourier transform of ...
0
votes
0answers
37 views

How does one calculate Fourier transform of Feynman propagator?

I am struggling with calculating the following integral on Sredinicki: How did he get the second line of (10.6)? That is, how did he calculate the Fourier transform of Feynman propagator?
1
vote
2answers
39 views

Density of states in Frequency Space vs K Space

Why do we use the density of states in frequency space when the density of states in k space is one state per unit k cubed (in 3 dimensions0?
1
vote
1answer
20 views

To isolate a particular mode of vibration in a standing wave on a string

Suppose a string bound between two rigid end-points is vibrating and it is a combination of a number of normal modes of vibration, is it possible to isolate a particular mode of vibration in wave by a ...
0
votes
0answers
26 views

Why is this version of Fourier deconvolution justiifed?

As I understand it, one typically performs Fourier deconvolution for the data produced by some instrument of a linear system by: Taking the observation of some unit impulse signal $m$, with the ...
0
votes
2answers
62 views

Dispersion relations in solid state physics

Could you please explain what exactly is the relevant information that is conveyed through a dispersion relation? Edit 1: Sorry about being vague. I am currently trying to understand the dispersion ...
0
votes
1answer
15 views

Wavelength of Intermittent Laser vs Prism

The following YouTube video from Sixty Symbols ( What is the maximum Bandwidth? - Sixty Symbols https://www.youtube.com/watch?v=0OOmSyaoAt0 )states that only a laser which always shines truly shines ...
5
votes
0answers
54 views

What does it mean to Fourier transform a ladder operator (in the input-output formalism)?

I am currently trying to get my head around the input-output formalism. In describing the input-output formalism (link) , Gardiner and Collett take ladder operators in the Heisenberg picture and ...
0
votes
1answer
28 views

Using FFT for spins in a non-cubic crystal lattice

Classical Ising/XY/Heisenberg models on a crystal lattice are commonly used to model magnetic materials. These can be studied using Monte Carlo simulations on a computer. Magnetic systems are often ...
0
votes
1answer
37 views

How to find wavepacket time dependence from the $k$-wavefunction?

I am trying to code the time dependence of a gaussian wavepacket using the Fourier transform techniques. I began with constructing a wavepacket (real parts only at the moment) at $t=0$ by multiplying ...
6
votes
3answers
210 views

Understanding the statement of the bandwidth theorem

I know that the Bandwidth Theorem (BT) and the Heisenberg Uncertainty Principle (HUP) are basically the same thing, and stem from the fact that for operators $A,B$, we have: $$\Delta A \Delta B \geq \...
3
votes
1answer
82 views

Does there exist some other type of electromagnetic waves?

When I learned about electromagnetic waves, I was told that some accelerating charge, specifically oscillating, produces electromagnetic waves, in a way like this: It produces a changing magnetic ...
0
votes
0answers
37 views

QM limit of QFT in Schwartz [duplicate]

In Matthew Schwartz's QFT text, he derives the Schrodinger Equation in the low-energy limit. I got lost on one of the steps. First he mentions that $$ \Psi (x) = <x| \Psi>,\tag{2.83}$$ ...
0
votes
1answer
53 views

$\Delta x$ and $\Delta k$ for purely harmonic wave [closed]

For a purely harmonic wave, composed of a single frequency/ wavelength. Within the context of Fourier analysis, what are $\Delta x$ and $\Delta k$ (corresponding to the shape of wave packets) ...
1
vote
1answer
52 views

A computation problem on reciprocal lattice

I am reading David Tong’s lecture notes on Application of Quantum Mechanics. My confusion is about the following paragraph: Consider a function $f(\vec{x}) $, suppose $f(\vec{x})=f(\vec{x}+\vec{r})$...
0
votes
1answer
72 views

Drawing reciprocal lattice structures

I am currently trying to understand why and how reciprocal lattice relates to the diffraction plane, so I did some research on Wiki about reciprocal lattice, but I seem to be stuck as I would like to ...
0
votes
1answer
69 views

Trying to first understand position and momentum bases in Quantum Mechanics

In my lectures, I am told: $$\langle x \mid \psi \rangle = \psi (x)$$ Which can only be valid if the overlap integral is: $$\langle x \mid \psi \rangle = \int_{-\infty}^{\infty} \delta (x-x') \ \...
1
vote
1answer
23 views

Optical convolution processor

I was reading this paper: But I'm not agree with its conclusion (I think it is wrong). Indeed, we have (I use a slightly different notation, hope it's clear): $$U(x,y,2f)=kost \ F[g(x_{1};y_{1})]_{(\...
0
votes
0answers
31 views

How does one construct the $n$th Brillouin Zone in a rectangular lattice geometrically?

What are the general rules of constructing the Brillouin Zone in a rectangular lattice geometrically? While the construction of the 1st and 2nd Brillouin zone is rather simple, starting to construct ...
1
vote
0answers
37 views

Fourier Coefficients

Suppose i've a two voice samples v1 and v2. Comparatively voice v1 is louder than the v2. If both the voice is spoken by the same person.(Spoken normally as he speaks) Is it good to state the ...
1
vote
2answers
66 views

Schrodinger equation in momentum representation with position-dependent effective mass

I'm trying to convert Schrodinger equation with position-dependent effective mass (PDEM) to momentum representation, and I'm not sure how to apply the kinetic energy operator. In position ...
2
votes
1answer
41 views

Quantisation of $z$-angular momentum eigenvalues

Consider the eigenvalue equation for the $\hat{l}_z$ angular momentum operator: $$\hat{l}_zY_{lm_l}(\theta,\phi)=m\hbar Y_{lm_l}(\theta,\phi)$$ with separable solution $$Y_{lm_l}(\theta,\phi)=\Theta_{...
0
votes
0answers
42 views

Dispersion of light

I've been taking digital signal processing course. It's pretty interesting for me. One thought came to my mind while I've been practicing numerical problem on Fourier transform. So my question is ...
0
votes
0answers
35 views

Quantization of EM field

Usually the first step to quantization of EM field is the Fourier expansion of vector potential: $$ A = \sum_k A_k e^{jkr} .$$ For example, in book "The classical theory of fields" by Landau, ...
0
votes
0answers
24 views

Fourier Transform of the Lienard Wiechert Fields and the retardation condition

If the Fourier Transform of the field as a function of space at a specific moment of time, $\vec{E}(\vec r , t)$, with respect to time gives us the field as a function of space at a specific frequency ...
2
votes
1answer
41 views

Fourier transform of a scatering potential

The Fermi golden rules states $$ \Gamma(\vec{k},\vec{k}') = \frac{2\pi}{\hbar} \left| \left \langle \vec{k}|V|\vec{k}' \right \rangle \right|^2 \delta(E(\vec{k})-E(\vec{k}')) \, .$$ Many places (for ...
0
votes
1answer
34 views

Phase conventions for phasors and Fourier transforms

There is a natural isomorphism between the complex plane and the set of sine waves, but this isomorphism is ambiguous up to a rotation and/or flip of the plane. This ambiguity seems to be related to ...
1
vote
1answer
53 views

Greiner's Green's function for diffusion

I am reading Greiner's "Quantum Electrodynamics". In example 1.5 he derives the Green's function for diffusion. I am stuck on a step in the derivation. He has the defining differential equation as $$ ...
1
vote
0answers
39 views

Decoupling of degrees of freedom in Klein-Gordon equation

In David Tong's notes in QFT he states that the degrees of freedom decouple in momentum space for the Klein-Gordon eq. He writes that this can be seen by using the Fourier transform (see picture below)...
0
votes
1answer
46 views

Back Focal Plane After A Convergent Lens

Acutually this is related to computer simulation, especially MATLAB. I am very confused, assume a plane wave passes through a convengent lens, where the propagation direction is parallel to paraxial ...
1
vote
2answers
32 views

Translating between momentum eigenstate and position eigenstate

Say a particle is in an eigenstate of momentum: $$\phi_{\mathbf p} = N e^{i \mathbf p x}$$ Then, according to my lecture notes, apparently a position measurement renders $$\psi = \delta ^{(3)} (\...
3
votes
1answer
136 views

Feynman $i\varepsilon$-prescription in path integral by adding an imaginary part to time

It is known that the well-definiteness of the path integral leads to the Feynman's $i\varepsilon$-prescription for the field propagator. I've found many ways of showing this in the literature, but it ...
1
vote
0answers
124 views

Fraunhofer diffraction problem in Python: How to interpret discrete Fourier transform (DFT) spectrum?

I have a periodic phase grating consisting of lenslets along the x-direction, invariant in y. I want to use python to calculate the far-field (Fraunhofer) diffraction pattern that one gets when ...
0
votes
2answers
129 views

What is a “Fourier transform limited pulse”?

I have some doubts about the definiton of a Fourier transform limited pulse. For example if I consider a generic pulse: $$E(z,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}A(\omega)e^{i(-\beta(\...
0
votes
1answer
80 views

How do you expand a wavefunction in the basis of eigenfunctions of the free particle?

If we have an initial state given by $ \Psi(x,0) $ and we want to find $ \Psi(x,t) $, we would expand the function in the basis of eigenstates of the Hamiltonian, $\{\psi_n\}$: $ \Psi(x,t)=\sum _nC_n ...
2
votes
3answers
209 views

How is the envelope of a wave derived from the wave equation?

I'm reading this book about electrical properties of materials where the electron is introduced as a wave. Using the equation of a wave, they bring about the "envelope" of a wave. So here is how the ...
2
votes
1answer
72 views

What is the relationship between directions in reciprocal and real space of a photonic crystal?

I am reading "Photonic crystals - molding the flow of light" by Joannopoulos et al. (available on-line). The figures below are reproduced from there. This is a diagram of a triangular lattice of air ...
1
vote
1answer
41 views

Question about Mode expansion of free compact boson

$(1+1)$-Dim free compact boson, Lagrangian is $$\mathcal{L}= \frac{1}{2}(\partial_\mu\phi(\sigma,t))^2$$ with $\phi(x,t)\sim\phi(x,t)+2\pi r$ and periodic boundary condition along $x$, i.e. $\phi(\...
0
votes
0answers
33 views

On the integration for the electromagnetic density

Upon integrating the electromagnetic hamiltonian density \begin{equation} \mathcal{H}=\left|\textbf{E}\right|^{2}+\left|\textbf{B}\right|^{2}, \end{equation} on a cube of lengths L and volume V we ...
0
votes
0answers
37 views

Why does the integration over momentum has normalization constant of volume?

If I Fourier transform a wave function in position space, integration carries no normalization constant: $$\displaystyle{\phi(k) \equiv \langle k|\psi\rangle = \sum\limits_x \langle k|x\rangle\langle ...
1
vote
2answers
68 views

What are the allowed wavenumbers in the finite size system?

Usually, we introduce wavenumber $\textbf{q}$ by Fourier transform, for example, an operator $A_{\textbf{q}}=1/\sqrt{N}*\sum_{i}e^{i \textbf{q}\cdot \textbf{r}_{i}}A_{i}$, where $N$ is number of sites,...