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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Complex fourier transformation function [duplicate]

can the FT function can be a complex function ? and if yes what does it mean because in all cases i came across till now the FT function only had a real part
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Meaning of complex Fourier transformation function

what does it mean, if the Fourier transformation function of an electromagnetic wave is complex? I know that normally the FT function $f(k)$ shows the wavenumbers that are involved in the wave. but i ...
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Fourier Transform of a short signal

if I have a sine wave signal for a duration of only a few seconds, the Fourier transform will show me, that this signal corresponds to a range of frequencies. Why is this the case? I do understand ...
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Determining Characteristics of Spatial Light Modulator (SLM)

im quite lost somehow. I know this is a really basic thing and I know I should be able to do it, but something is going on with me and I tend to have problems with everything these days. Anyway: I ...
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Asymptotic form of a Coulomb-like integral

I need to evaluate or work out the asymptotic scaling of the following integral: \begin{equation} I~=~\int_{\mathbb{R}^3} dq d^2p \frac{e^{i\vec{p}\cdot \vec{r}}e^{iq z}}{p^2 + \frac{1}{g^2}q^4} \end{...
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Propagators in position space

What are the possible applications of position-space representations of propagators? I'm speaking not only of the well-known free field case, but also of the particles in external fields and/or curved ...
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How does the structure factor reflect the characteristics of particle distribution?

The structure factor is defined as follows: $$S(\mathbf k)=\frac{1}{N}\sum_i\sum_je^{-\mathbf k\cdot\mathbf r_{ij}\sqrt{-1}}$$ It is related to the radial distribution function by Fourier transform: $$...
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Space-time Integral of Cubic Field(s)

Context Essentially, I am trying to analytically evaluate $$\tag{1}\int\zeta\phi^2$$ where $\zeta$ and $\phi$ are scalar fields. I have been using various formulae from Mikko Laine and Aleksi Vuorinen'...
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Does a lens always act as a Fourier transform?

I understand that putting a lens behind an aperture at the distance 1f, it will "get" the diffraction pattern to appear in the back focal plane. In this case the FT of the aperture plane ...
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Is there an optical element that can perform inverse Fourier transform?

We know that lenses perform Fourier transform of the incident wave-field distribution. Is there a similar optical element that can perform inverse Fourier transform such that when the output of a lens ...
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How can the Fourier delta function produce different matrices in single-pixel imaging?

The pattern in the masks used for single-pixel imaging are created applying equation $(1)$, $$P_\phi (x,y) = \frac{1}{2} \left[ 1 +| F^{-1} \{\delta_H (u,v) e^{i\phi}\}|\right], \tag{1}$$ in which the ...
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Poisson equation of point charge using Fourier transformation

i have problems applying the fourier transformation. all these integrals confuse me. so here is my calculation: $$ \Delta\phi(r)=-4\pi\delta(r) $$ $$ \text{Left hand side}:-\frac{1}{(2\pi)^{3/2}}\int_{...
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What is the meaning of the absolute of the Fourier transform in this context?

In a course talking about Fourrier transform and Nyquist frequency, there is this sentence that I can't understand: "If a signal has $|F(w)| \geq 0$ only for frequencies till f, than such ...
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Dualities in Physics and Fourier Transforms

In many articles, authors compare physical dualities to Fourier transforms. For example: Joseph Polchinski, in his article "String Duality" (hep-th/9607050v2), writes: "Weak/strong ...
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What is the one-sided Fourier transform of a constant? [migrated]

A definition of the Fourier transform commonly used is (I always forget which convention of normalization to use) \begin{align}f(\omega)=\int_{-\infty}^\infty e^{i \omega t}f(t) dt\end{align} For a ...
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A Problem on Tunneling in a dissipative environment

I am currently self-studying Many-Body Physics and I am using the textbook Condensed Matter Field Theory . I am currently trying to figure out the problem on page 151 (Chapter 3 and problem called &...
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Fourier transform of the state?

I'm trying to make sense of the following equation I saw on Wikipedia: https://en.wikipedia.org/wiki/Momentum_operator#Fourier_transform I saw on wikipedia that \begin{equation} \langle\psi|\hat{p}|\...
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How do I calculate the inverse Fourier transform of the delta function? [migrated]

In the context of single-pixel imaging, the following statement is given: "A Fourier basis pattern $P_F (x,y) $ can be obtained by applying an inverse Fourier transform $\delta_F (u, v, \phi)$to ...
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Two forms of spectral density, how to explain their equivalence?

This question was posted on 1 but got not reply. The mathematical definition of the spectrum of a stationary process $\mathbf{x}(t)$is to take the Fourier transform of a finite segment $$\hat{\mathbf{...
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Question on discrete commutation relation in QFT

Given the commutation relation $$\left[\phi\left(t,\vec{x}\right),\pi\left(t,\vec{x}'\right)\right]=i\delta^{n-1}\left(\vec{x}-\vec{x}'\right)$$ and define the Fourier transform as $$\tilde{\phi}(t,\...
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Confusions between Power spectrum and Matter Power spectrum and Relationship between the angular and 3D power spectra

From a previous post Relationship between the angular and 3D power spectra, I have got an demonstration making the link between the Angular power spectrum $C_{\ell}$ and the 3D Matter power spectrum $...
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Fourier Transform of a Signal [closed]

Mathematically if one computes FT of sine wave with one frequency f0 and amplitude A in time domain, then we get two peaks in frequency domain (mathematically, i.e it consist of difference of delta ...
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Poisson brackets of bosonic string oscillators

I'm reading string theory books and I'm stuck at the moment when we consider a Hamiltonian version of the classical string. Namely I don't understand how to derive the Poisson brackets for string ...
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Schrodingers equation for an electron in a periodic potential derivation

In Kittels Introduction to Solid State physics, when deriving schrodingers equation for an electron in a periodic potential, we begin by writing the wave function as a Fourier series $\psi = \sum_k C(...
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How does a transformation domain differ from a spatial domain?

In the context of single-pixel imaging, the following statement is given: "Global transformation has a property that each point (coefficient) in the transformation domain is a weighted sum of all ...
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Why won't sampling at time intervals shorter than the detector's response provide new information?

"When Fourier analysing a signal, sampling at time intervals shorter than the detector's response won't provide any new information, only smoother data." This was told to me, but I don't ...
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Fourier transform of sine function

While solving the Fourier transformation of a sine wave (say $h(t)=A\sin (2 \pi f_0 t)$) in time domain, we get two peaks in frequency domain in frequency space with a factor of $(A/2)j$ with ...
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Confusion about modes and quantum field theory

I'm learning quantum field theory from P&S and Srednicki. I'm having a lot of difficulties understanding the concept of a momentum state. In particular, I'm confused about how to interpret the ...
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Can the sound of infinitely long played music be decomposed in sine (or cosine) forms?

Every arbitrary waveform (except non-linear ones) can be decomposed in sine (or cosine) waveforms that spatially extend to infinity. That is if the waveform has a finite spatial extent. But say that I ...
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How to transform free field Hamiltonian from position to momentum space?

I'm reading Srednicki's Quantum Field Theory. The equation (3.1) says $$ H=\int\mathrm{d}^3xa^\dagger(\boldsymbol{x})\left(-\frac{1}{2m}\nabla^2\right)a(\boldsymbol{x}) $$ will be transformed $$ H=\...
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Path integral with initial definite momentum

Generally, in path integral formalism a propagator between two states with definite position is computed, something like, $$K(x_1,t_1;x_0,t_0)=\int_{x_0(t_0)}^{x_1(t_1)}\mathcal{D}x(t)\exp\left(\frac{...
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Accounts on the solutions of the Dirac equation

Consider the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi = 0$. As it is well known, there are different representations for the matrices $\gamma^{\mu}$, $\mu = 0,1,2,3$, the most famous ones ...
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Convert gaussian function from 3D configuration space into 3D momentum space

I know, If we convert a gaussian function from 1D position space into 1D momentum space, it will be again a gaussian function. But if we have a gaussian function in a 3D position space, how it will be ...
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How does the contancy of a function over a distance $a$ translates into its Fourier components?

In the book of Kardar, Statistical Physics of Fields, on page 20, it is given that It is important to emphasize that while $\mathbf{x}$ is treated as a continuous variable, the function $\vec{m}(\...
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Spatial derivative of Dirac wave function is zero at rest

Suppose $\psi$ is a solution of the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi = 0$. If the particle is at rest, the partial derivatives $\partial_{\mu}$, $\mu=1,2,3$ are supposed to be zero. ...
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Interpreting plane waves solutions of the Dirac equation

The Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi = 0$ has plane wave solutions of the form: $$\psi_{+,k}(t,x) = e^{-iE(k)t+ik\cdot x}u(k) \quad \mbox{and} \quad \psi_{-,k}(t,x) = e^{iE(k)t -ik\...
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Does this paragraph make sense? [closed]

Just as water remains water in both the solid, liquid and gaseous state, tension remains tension in both the energy’s rarefied and condensed state. The phase transition between one state to the other ...
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Why is the density of states required conceptually? Should it be seen as a mathematical trick related to Fourier series?

[edit]: My misunderstanding is more precisely asked here: Density of states and boundary conditions: how the density of states is physical if it depends on box size :it was suggested to open a new ...
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Fourier Decomposition of Schrödinger's Equation with a Potential ${V}{\left({x}\right)}=e^x$

Question: Can the equation ${\psi}_{{{t}}}-{i}{\psi}_{{{x}{x}}}={e}^{{{x}}}{\psi}$ be solved with a canonical Fourier transform? If it requires a Fokas transform or inverse scattering transform, how ...
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Fourier transform of sound wave

I am a high school student doing a project in Fourier transform. I was planning to learn Fourier transform and manually apply the calculations to a section of a song and see what I get. However, I ...
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First harmonic above fundamental in piano recording?

I am currently working on a project, the final aim of which is to see if one can classify which instrument a sound recording is coming from, by looking at the fourier transform of a note and comparing ...
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Where did this normalisation factor come from?

Just wondering where the normalisation factor $\frac{1}{\sqrt{2 \omega_{\mathrm{p}}}}$ comes from in this field operator expression? \begin{equation} \phi(\mathbf{x})=\int \frac{d^{3} p}{\left(2 \pi^{...
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Position operator explicit form & definition

I've got a question regarding the position operator. As far as I learned it, the position operator simply gets defined by saying: $\hat{x} \ \psi(x,t) = x \ \psi(x,t)$ And I also learned that the ...
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Calculating the Wigner transform of operators

Recently I started to study the formulation of quantum mechanics in the phase space. So I was introduced to the concept of Wigner function and Weyl transform. I learned that if F is an operator, then ...
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Dispersion relation with $(F^{\mu\nu}F_{\mu\nu})^2$ correction

This is in context of this paper by Nima et al where they derive positivity bounds using dispersion relations. Specifically, deriving the EOM from equation $(1)$ in the paper $$L=-\frac{1}{4}F^2+\frac{...
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Definition of RMS deviations from mean w.r.t. $|u(x,0)|^2$ and $|A(k)|^2$ in Jackson

In chapter 7 of Jackson’s Classical Electrodynamics (page 323) he speaks of ‘‘rms deviations from mean, $\Delta x$ and $\Delta k$, defined with respect to $|u(x,0)|^2$ and $|A(k)|^2$’’. I do not ...
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Confusion for Part of Bogolyubov Transform

Consider the operator (written in terms of a mode expansion) given by $$\hat P(u) = \int_0^\infty \frac{d\Omega}{\sqrt{2\pi}}\frac{1}{\sqrt{2\Omega}}\left(\hat b_\Omega e^{-i\Omega u} + \hat b_\Omega^\...
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White light interferometer

I am building white-light Michaelson interferometer and I am registering interference in frequency domain via spectrometer. I read an article, where the same is done (just Mach-Zender interferometer) ...
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Interference fringes movement analysis

I am registering interference fringes on CCD linear sensor from Michelson inteferometer runned by He-Ne laser. I am sampling every 15-20 ms. I made a program to follow fringe movement and to track ...
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Leakage of superconducting qubit: why it occurs with short driving pulses and not as well with long ones

My question in very short: Short pulses on superconducting qubit usually induce leakage. But a long pulse can be seen as a sequence of short pulses (imagine all of the short pulse as square pulses: ...

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