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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.

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Finding normalization constant of a wave function with definite momentum

I try to read Sakurai's Modern Quantum Mechanics but I stuck at this point, $$\delta(x^{'}-x^{''})=|N|^{2}\int dp^{'}\exp\Biggl({ip^{'}(x^{'}-x^{''})2\pi\over h}\Biggr)$$ This is an expression for ...
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Physical Interpretation of Energy-wavenumber Graphs

Consider an energy-wavenumber graph, typical in solid state physics, like the one below. I can follow the mathematics in the derivations with a KP model. But I don't understand the physical ...
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Fourier transform of some discrete, finite, non-uniform signal

Suppose I have some finite signal $x(t)$ of $N$ data points. This signal is produced by some program or some experiment and so is discrete and the difference in time between each data point $\delta t$ ...
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Examples of non-sine waves? [closed]

What would be a non-sine wave? AFAIK, all sound is a sine wave, equally to waves on the sea. What would be a common example of something in nature that's a wave but not a sine wave? Or, would we have ...
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The action of a lens on a point dipole (Fourier optics)

I want to model the action of a lens on radiation from a point dipole. I have found examples but am struggling to interrupt them. My knowledge of Fourier optics is that from the Helmholtz equation, $...
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Properties of Dirac delta function in Integral

I was reading commutation relation of canonical momentum in KG Field from Lectures of Quantum Field Theory by Ashok Das. In page 179, He has used Integration to derive the result where he expressed ...
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Not understanding about the sign of Fourier transform of power spectrum [migrated]

I don't understand with the below equations the affirmation that $FT(\Delta(\vec{k})$ is the Fourier transform of $\Delta(\vec{r})$ : $$\left\{\begin{array}{l}{\Delta(\vec{r})=\frac{V}{(2 \pi)^{3}} \...
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Fourier transform of the wave equation

In string theory, as the one dimensional string propagates in time, it sweeps out a two-dimensional surface known as the string worldsheet. The spacetime coordinates are taken to be functions $X = X(x,...
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Path integral calculations $e^{i\omega 0^+}$

When computing correlation functions using the path integral formulation, I often need to compute integrals such as $$ \int_{-\infty}^\infty \frac{d\omega}{2\pi} \frac{1}{i\omega -\epsilon} $$ ...
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Can any solution to the three-dimensional wave equation be written as a superposition of plane waves?

Can any solution to the three-dimensional wave equation, $$\nabla^2f = \frac{1}{v^2}\frac{\partial^2 f}{\partial t^2},$$ be written as a superposition of sinusoidal plane waves? In "Introduction to ...
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Calculating the charge density from a form factor

For an atomic form factor $F(\textbf q)$, the corresponding charge density distribution is given by $$ \rho(\textbf r) = \frac{1}{(2\pi)^3}\int\text{d}^3 \textbf q \,F(\textbf q)\,\text{e}^{-\text{i}\...
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CFT in momentum space

Is there a way to see the conformal symmetry in momentum space in a CFT? I mean if I can recover the conformal group in some way in momentum space.
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Fourier transformation of a discrete function; conclusion of having a finite sum [migrated]

For a function $g$ defined at discrete points $x_n = n a$ with $n \in \{0, \ldots, N\}$ with periodicity $g(x_0) = g(x_N)$ the discrete Fourier transformation reads $g(x_n) = \frac{1}{Na} \sum_{q \in ...
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Field operator commutation: If two operators commute, then their fourier transforms also commute?

Im doing this in the context of field operators $$\psi(x)=\sum_k a_k e^{ikx},$$ $$\psi^T(y)=\sum_k a_k^T e^{-iky},$$ and their being defined as the fourier transform of the creation/annihilation ...
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Fourier transform of the amplitude of a laser evolving along $x$-axis: how the cross-section appear?

Consider the vector potential describing the field of a laser defined for $x>0$. It has a cross-section $L^2$. The field propagates along the $x$-axis. In the book "Quantum measurement theory and ...
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What this quantum field operator represents $ b_{in}(t) = \frac{1}{\sqrt{2 \pi}} \int e^{-i \omega t} b(\omega)$

In Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation Physics Department, Uniuersity of Waikato, Hamilton, ¹tuZealand (Received 29 ...
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Why EHT have to use Fourier transform to process the image?

I read that Fourier transform is a mathematical tool to deconstruct a wave taken from a source into basic sine and cosine waves, since visible light coming from the M87 accretion disk will be obscured ...
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Is there any nonlinear equations depending on Fourier coefficients?

A nonlinear partial differential equation is an expression depending on derivatives of $u$ $$f(x,t,u,u_x,u_t,\cdots)=0,$$ where the derivatives of $u$ can be obtained from the Taylor series of $u$. ...
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Structure factor in a homogenous system

I want to calculate the structure factor for a homogenous system. The system that I am dealing with is the results of a Vicsek type model simulation. The structure factor is defined as : $$S(q) = \...
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Error in FFT (Sectrum) analyzer Instrument

Recently I performed an experiment on finding the natural frequency of vertical cantilever. I used accelerometer and FFT analyzer instrument for measuring it. First I kept the accelerometer near the ...
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Why do people study plane wave in wave physics?

I have recently been studying a structure for high sound absorption. There are a lot of literature on similar design, where all of them are using the plane wave as an input to the structure. I have ...
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Functions constant on boundary and topology of underlying manifold

Here are my thoughts: Say I have two manifolds $M$ (one dimensional in my thoughts) and $\mathbb{R}$. Thinking in physical terms; $M$ I imagine as my space of states: of possible configurations of my ...
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Symmetry Operation in Reciprocal Space

I have a set of k points spanning the entire Brillouin zone and I want to reduce it to the irreducible BZ. So for reduction, I use the point group symmetry operation. To verify, I use quantum espresso ...
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How does the green function for the wave equation in three dimensions preserve the ordering of noises between a speaker and a listener

I was provided with the following equation in class for the Green's function of a three dimensional wave equation: However, I am confused as to how this form of the Greens function preserves the ...
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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 ...
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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]{...
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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 ...
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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 ...
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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 ...
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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: ...
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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)=\...
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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.
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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:
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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 ...
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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?
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \...
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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 ...
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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 $$ \Psi (x) = <x| \Psi>,\tag{2.83}$$ ...
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$\Delta x$ 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) ...
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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 $f(\vec{x})=f(\vec{x}+\vec{r})$...
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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 ...
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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') \ \...