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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Marginal characteristic function from a multivariate charateristic function

[I am posting this question here and not in Mathematics Stack Exchange because I will be using conventions as they are usually used in statistical physics (especially the notation is more involved ...
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Fringe movement analysis to extract displacement

I am using a Michelson interferometer with the interferogram recorded on a CMOS sensor. I have taken a video of the fringes moving when a displacement of 50 $\mu m$ is imposed to one of the mirrors. ...
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What is “Fourier Transformation induced Spectroscopy” really?

With regard to "Conventional and Advanced Characterization techniques", what is meant by, "Fourier Transformation Induced Spectroscopy "?A detailed answer to this is most welcome. ...
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How to emulate a drum sound?

I've been working in a personal project which the goal is to create a digital drum. During my conjectures, I was thinking what is the best way to simulate the sound, below I will explain a simple idea ...
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Fourier transform of the energy momentum tensor

I am trying to calculate the Fourier transform of the energy momentum tensor of a scalar field, in particular the first term: \begin{equation} T_{\mu \nu}(x) = \frac{1}{2} \partial_{\mu} \phi(x) \...
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Fourier transform of a real initial wavefunction

Consider the initial wavefunction given by: $$ \Psi (x,0) = \sin(k_0 x).$$ I've been taught that in order to time evolve a wavepacket one must first find the momentum space representation of the ...
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Fourier Transform of Green function

I have a question regarding the (discrete) Fourier Transform of the retarded Green function (I neglect hats on operators): $$G(i,j;t)=-i\theta(t)\langle \{c_i(t),c^\dagger_j \} \rangle $$ specifically ...
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Implement EPR Imaging deconvolution

I have bruker's EPRI data and I want to deconvolute it. I read some article, that says $$ f(r) = x(r) \circledast g(r) \\ F(ω) = X(ω) \times G(ω) \\ X(ω) = \frac{F(ω)}{G(ω)} $$ $f(r)$: imaging spectre,...
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How do optical prisms perform a Fourier-transform

I think I have a lackluster understanding of the time-frequency-uncertainty. I know that $\Delta f \cdot \Delta t \ge \frac{1}{2}$ where $\Delta f$ and $\Delta t$ are the uncertainty in frequency and ...
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An attempt at deriving canonical commutation relations

I am reviewing basic quantum mechanics since I feel like I am struggling with the fundamentals. I am not sure exactly how much I am "taking for granted" or whether or not my logic is clear. ...
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How to solve this wave equation using Fourier Transform?

I have the following wave equation: $$\frac{\partial}{\partial x}\left(\frac{1}{l(x)}\frac{\partial}{\partial x}V(x,t)\right)=c(x)\frac{\partial ^2}{\partial t^2}V(x,t)$$ where $l(x)$, $c(x)$ and $V(x,...
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What is the reality condition for an electric field polarization?

For example in Cohen-Tannoudji's book, it's given that for a propagating electromagnetic wave, you can define your polarization to have this property (all in Fourier space): $$E^*(k, t) = E(-k,t),$$ ...
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Issue with a sign in the commutator calculation of field operators for a real scalar field

In the derivation for the commutator (real scalar field, Klein-Gordon equation) $$[\phi(x),\phi(y)]=0$$ I have solved up to $$[\phi(x),\phi(y)]=\frac{1}{2(2\pi)^3}\int \frac{d^3 p}{\omega_p}[e^{ip(x-y)...
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Fourier transform of susceptibility example [closed]

I have not studied the Fourier transform (FT) in great detail, but came across a problem in electrodynamics in which I assume it is needed. The problem goes as follows: Evaluate $\chi (t)$ for the ...
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Why must $\omega^2 = k^2 c^2$?

From The Feynman Lectures: "The next subject we shall discuss is the interference of waves in both space and time. Suppose that we have two waves travelling in space. We know, of course, that we ...
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Obtain clear fringes from interferometer

I have just begun setting up an interferometer to take some measurements. I am trying to reproduce one paper I found in literature for 3 DoF homodyne interferometry. The detector is merly a CMOS ...
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Finding the free action in momentum space

I have the free action in position space $$S_0[J,\phi] = \int \! d^4x[\frac{1}{2} \phi (\partial_\mu \partial^\mu - m^2 + i\epsilon)\phi + \frac{\hbar}{i}J\phi].$$ Knowing that the Fourier transform ...
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Taking Fourier Transform to momentum space

I have a wave function $$\psi(x) = \frac{1}{\sqrt{\sigma\sqrt{\pi}}} \exp \left (\frac{-x^2}{2 \sigma^2} \right ) \exp \left (\frac{ipx}{\hbar} \right )$$ And I have to convert this to $Q(p)$, in ...
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Lorentz-invariant measure in Klein-Gordon field

What is the exact reason why the solution of the classical Klein-Gordon equation is written as a mode expansion with a Lorentz invariant measure and after that the coefficients are promoted to ...
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Can you expand a real scalar field $\phi(t,\mathbf{x})$ in terms of spherical harmonics?

A massless real scalar admits the expansion $$ \phi(t,\mathbf{x}) = \int \frac{d^3\mathbf{p}}{(2\pi)^{3/2} \sqrt{2|\mathbf{p}|}} \bigg( e^{ - i |\mathbf{p}| t + i \mathbf{p} \cdot \mathbf{x} } a_{\...
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Computing a second harmonic resistance out of a wave-like resistance signal

I want to reproduce the results of a paper, in which they measure the Anharmonic Hall Effect (AHE) resistance $R_{AHE}^{2\omega}$. There are several protocols for measuring it experimentally, but I'm ...
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Can concepts like “critical damping” or “resonant frequency” be applied to more complex systems than just a spring and damper in parallel?

I am trying to do some modeling analysis by representing materials with parallel systems of springs and dampers. In the simplest case with just one spring parallel to a damper, we have the traditional ...
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Sample functions of stochastic processes violate Dirichlet conditions. Why?

Suppose we have a real-valued periodic function that satisfies the Dirichlet conditions. In this case its Fourier series converges. With some expedients this concept can be extended also to non-...
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What motivates the trial solution of $\left[-\frac{\hbar^2\nabla^2}{2m}+\frac{e^2y^2B^2}{2m}-\frac{i\hbar eyB}{m}\right]\psi(x,y,z)=E\psi(x,y,z).$?

The time-independent Schrodinger equation for the problem of charged particles in an uniform magnetic field ${\vec B}=B{\hat k}$, in the Coulomb gauge ${\vec A}=(-yB,0,0)$, reduces to the following ...
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4-bit Quantum Fourier Transform Circuit with specific / custom Gates [closed]

I am studying the Quantum Fourier Transform and completing some exercises. I want to write the circuit for the 4-bit Quantum Fourier Transform in terms of specific gates: H, controlled-V, and C-NOT, ...
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Mixing Fourier Transform and Real Space for lattice with only one perodic direction

I have a 2D lattice consting of two different regions A and B and in real space a closed set of equations for a quantity $f_{\mu\nu}$ at every lattice site $\mu,\nu$. To solve this, I normally would ...
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Orthogonality of the annihilation operator in QFT?

In one of my QFT exercices it seems my teacher assumes $$\int dp \int dp' a^†(p)a^†(p')\exp(ipx)\exp(ip'x')=0.$$ Could someone tell me why that is?
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Harmonic gauge condition in Fourier space

I am currently working through Sean Carroll's notes on general relativity, specifically the weak fields and gravitational radiation section*. When discussing gravitational radiation emitted by an ...
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Plane wave basis in a box. Inconsistent terminology about Fourier series and transform

I will refer to the one-dimensional equivalent of the problem in the picture for simplicity. At the end of the picture, the author argues that the Fourier transform of the function $\psi(x)$ is the ...
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How does one calculate the Fourier transform of the correlation function for the ferromagnet?

I'm working through Huang's book and in chapter 16, it discusses the correlation function of the ferromagnet and its Fourier transform. In (16.8), the book states that $$\Gamma(\vec{r}) = \langle m(\...
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How to compute the Hamiltonian matrix elements in momentum basis? [closed]

I'm struggling to get it right, I'm having a lot of ideas but no one of them are making sense to me. So, the problem is as follows. "Evaluate: \begin{equation} \langle q'_t\mid\hat{H}\mid q_t\...
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Coherence time of beams from broadband FT instruments?

To determine whether incident and reflected waves interfere within a film, we have to consider whether the coherence time of the beam is less than the travel time of the beam in the film. (This is the ...
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Ewald summations and series convergence

Does anyone have a explanation of why the direct summation of the coulomb interactions between the ions of a crystal structure which might be conditionally convergent is mathematically allowed to be ...
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Confusion relating to discretizing a Lagrangian and the Fourier transform conventions thereof

My question boils down to how the Fourier transform is discretized when we discretize the field $\phi(x)$ in a Lagrangian $\mathcal{L}$. To put things on a concrete footing, consider the following ...
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Dirac delta in spherical coordinates. What I'm doing wrong?

I must show that the integral $$\frac{1}{(2\pi)^{3}}\int_{\vec{k}}d^{3}k\frac{\cos(\vec{k}\cdot\vec{x})}{\left({\sqrt{|\vec{k}|^2+m^{2}}}\right)^{s}}=\delta^{3}(\vec{x})$$ when $s=0$ by using ...
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Derivation of the Dirac spinor expressed as a Fourier transform

Introduction For the Klein Gordon Field, the equations of motion are described by the equation $$(\partial_{\mu}\partial^{\mu} + m^2)\phi(\vec{x},t)=0$$ Which when the field is expressed as a Fourier ...
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Fourier transform of linear response function

I was studying Linear Response Theory from 'A modern course in statistical physics' by Reichl, and some doubts came up. The response function is defined as $$<\alpha(t)>_{F} = \int_{-\infty}^{+\...
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Understanding Maxwell's Equations in a Box

Maxwell's equations for $n$ charged particles each with charge $e_j$ are known to be (in cgs) $$\begin{align} \nabla\cdot\textbf{E}(t,x)&=4\pi\rho(t,x)\\ \nabla\cdot\textbf{B}(t,x)&=0\\ \...
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Fourier transform of positive momentum states only?

Suppose I have the momentum-space eigenstates of a system $\psi_n(p)$. I write the time-evolved momentum states as $\Psi(p,t) = \sum c_n\psi_n(p)e^{-i E_n t}$ where $E_n$ are the corresponding energy ...
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Fourier-relation between the position space and the momentum space

I understand that in quantum mechanics every vector from the state space (e.g. $|\Psi(t) \rangle$) can be projected on to observables such as the momentum space basis vectors $|p\rangle$ or position ...
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What is the physical meaning of the operators of the positive and negative frequency components?

In quantum optics, after performing the quantization of the radiation field, the field operator $E$ is often split into the positive- and negative-frequency parts as $$ E(\mathbf{r},t) = E^{\left(+\...
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In what base should we express the transformation coefficient for a wavefunction?

We have, for example, that $\psi(p) = \int e^{ixp/\hbar} \psi(x)\, dx$, which isn't really hard to derive. But in what base should we express $e^{ixp/\hbar}$? In the $x$ or the $p$ basis? Also, what ...
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Polar molecule in external electric field - problem with the Fourier transform

I have a problem where I am considering a polar molecule with dipole vector $\vec{p}$ moving in a plane and exposed to an electric field, so that the interaction potential is: \begin{equation}\tag{1} ...
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How is Pauli's exclusion principle conserved in transformations from real space to $k$ space for a Fock space hopping Hamiltonian?

$$\hat{H}= -t\sum_{\langle i,j\rangle} c_{i\sigma}^{+}c_{j\sigma}+h.c$$ For this Hamiltonian a lattice with all sites doubly occupied would give 0 in real space and for single occupancy it gives $-4t^{...
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Evaluating general solution of Dirac equation

In Ashok Das Lecture on QFT book, pg. 40, the solution of the Dirac equation for the general motion of a free particle with mass $m$ along an arbitrary direction is given by $$\psi (x)=\int d^4p \ a(...
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Physical Interpretation of Field Operator in Quantum Field Theory and Mode Expansion

I'm struggling to understand the physical interpretation behind the field operators $ \phi(\mathbf x)$ and $\phi ^\dagger (\mathbf x)$ in quantum field theory. My understanding is $ \phi ^\dagger (\...
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Explanation of an worked example from N.Zettili Quantum Mechanics

In the book "Quantum Mechanics Concepts and Application" 2nd edition by N.Zettili the worked problem 1.11. Here I had to find the Fourier transform of the function, $$\phi(k) = \begin{cases}...
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Time evolution of position of a free particle with dirac delta momentum

If the initial state of a free particle is a delta function in momentum (it has a single specified momentum $k_{o}$), how does the position evolve with time? Thoughts: $$\Psi(x,t) = \frac{1}{\sqrt{2\...
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What does the overlap between momentum eigenstates mean?

I am studying quantum mechanics right now, and I learned that momentum eigenstates are plane waves, that is $$ \langle x \vert p\rangle = e^{ipx} $$ and $$ \langle p' \vert p\rangle = \int_{-\infty}^{\...
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Questions regarding 2D reciprocal lattices translation vectors

If I have a reciprocal net of a rectangular real space lattice with lattice parameters $a = 0.4$ nm and $b = 0.5 $ nm along $x$ and $y$. I am trying to find which are true (I have never had a ...

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