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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How to understand this Dirac delta function?

I am reading this paper about quantization of the electromagnetic field, and there is a point where the author imposes the fundamental commutation relation between the vector potential and its ...
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Is the spectrum of discrete Fourier transform (DFT) of discrete signal continuous or in turn discrete, and is it periodic? [closed]

I've read in "Signals and Systems" (by Alan V. Oppenheim, Alan S. Willsky, with S. Hamid) that the DFT of signal is periodic with period $\:2\pi\:$ and it is clearly shown that the plot of ...
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Why are the distances in real space and Fourier space inverses of each other?

I just came across a paragraph in a set of physics notes where they implicitly claim that imposing a cut-off $k<\Lambda$ to the modes in Fourier space is equivalent to smoothing the field in real ...
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Green's function of screened Coulomb interaction using partial Fourier transform

The solution of the differential equation (DGL) $$ (-\epsilon_0\nabla^2 + l^{-2} )G(\vec{r},\vec{r}') = \delta(\vec{r},\vec{r}') $$ is given by a screened Coulomb potential $$ G(\vec{r},\vec{r}') = \...
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In trouble with QFT field operators

I'm stuck on the contents of a side box in "QFT for the Gifted Amateur", chapter 4, dealing about field operators. The side box sets the scene about the use case the section will be ...
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Wave packet for particle in a potential

In the book Quantum Mechanics by Cohen-Tannoudji, the author explains that the solutions of the Schrodinger equation of a free particle in one dimension are plane waves: $$\psi (x, t) = A e^{i(kx-\...
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Interpreting $4D$ massive scalar momentum space action as a gauge-field action in 1D?

Consider the following action for massive scalar as follows $$S = \int d^4x \left(-\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\frac{1}{2}m^2\phi^2\right) \tag{1}$$ with Minkowski signature $(-,+,+...
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A simple calculation in the XXX spin chain

I am currently studying the XXX Heisenberg spin chain using the Bethe ansatz. I am working in the string hypothesis and I am having troubles deriving a simple expression for Fourier transformation of ...
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Fourier expansion of the potential vector

I am studying the Fourier expansion (Mandel&Wolf; Pag. 467.) of the potential vector $A(\textbf{r},t)$ with respect to its space variables $x$, $y$, $z$. We have confined the field in a cubic ...
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Relationship between position kets in different topologies

Consider a particle moving on a ring $\mathcal{S}^1 \sim \mathbb{R} / \mathbb{Z}$ of circumference $L$. Due to periodic boundary conditions, $$ \langle x\mid p_n\rangle=\frac{1}{\sqrt{L}}e^{ip_{n}x/\...
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Another Fourier transformation but with a $\sqrt{\mathbf{q}^2 + m^2}$ term now

Just as the title proclaims, I have a Fourier transformation I am trying to determine. Here is the Fourier transformation in its full form: \begin{equation} \int\frac{d^3q}{(2\pi)^3}e^{i\mathbf{q}\...
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How to construct a quantum circuit for quantum Fourier transform in a prime dimensional Hilbert space?

Consider a Hilbert space of dimension $p$ where $p$ is a prime number. Quantum Fourier transform (QFT) in this space is defined as $$ |j\rangle \rightarrow \frac{1}{\sqrt{p}} \sum_{k=0}^{p-1}e^{\frac{...
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Discrete Schrödinger equation and Fourier transformation

I am currently working on an exercise involving the discretized version of Schrödinger's equation in an infinite potential well. The problem involves a well with a width of 1 and assumes $$\frac{\hbar}...
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Solving Wave Equation in 1+1D via Fourier Transforms with Dirac Delta function initial condition

I'm trying to use the Fourier transform method to solve the following PDE: This is a an infinite string with a pulse for it's initial condition. (At $t=0$, the string is stricken sharply so that the ...
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Canonical quantization of relativistic particle using Fourier transform

Suppose I want to quantize the Hamiltonian of a relativistic particle on space-time $\mathbb{R}^{4}$. Setting $c=1$ for simplicity, the energy of the particle is given by $w(p) = \sqrt{|p|^{2}+m^{2}}$,...
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A tricky derivation accompanied by delta function

I have been reading a book on Thermal Field theory by Michel Le Bellac During the reading I have come into a seemingly trivial but indeed tricky derivation. On page 26(2.47), we are supposed too prove ...
quantumology's user avatar
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Can I use Fourier transform of Matsubara Green's functions for imaginary time-ordered polarizability function?

I am learning Matsubara Green's functions using Henrik Bruus, Karsten Flensberg, Many-Body Quantum Theory in Condensed Matter Physics, An Introduction (2016). There, the authors calculated the ...
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The more precisely we want to locate a particle in space, the more energy we will need to expend

The more precisely we want to locate a particle in space, the more energy we will need to expend. does this statement have a mathematical formulation? When studying quantum mechanics, I often found ...
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How to prove the equivalence of Wigner distribution function expressions?

I'm currently going through Goodman's Introduction to Fourier Optics, Fourth Edition and I'm at the Wigner distribution function section, where he states the definition: $$W_{g}(x,y;f_{x},f_{y}) = \...
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Action of a Scalar Field in Path Integral Formulation Peskin & Schroeder (Pag. 285)

I'm really confused on the discretization stuff on this chapter of P&S. My question is related to the computation of the Action in scalar field theory done in page 285. When they compute the ...
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How can I get sound spectrum data of instruments?

I am trying to generate a piano noise by a python function. ...
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The ground state energy of the 1D XY model with staggered magnetic fields

I am calculating the ground state energy of the 1D XY model with staggered magnetic fields, \begin{align*} H=J\sum_n \left[S^x_nS^x_{n+1}+S^y_nS^y_{n+1}\right]+H\sum_n(-1)^nS^z_n \end{align*} By ...
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Setting up a wall with a slit inside the domain of a 2D wave being solved using the fourier-spectral method

The wave equation is given by: $$ \frac{\partial^2 u}{\partial t^2} = \alpha^2 \cdot \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right)$$ If we assume the solution to ...
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Fourier Transform the Total Interaction Energy for a Coulomb System

It seems that many Posts have solved the Fourier transform of the Coulomb interaction $V(r)=1/r$ which is $v(k) = 4\pi / k^2$. This is not my question. I have come across the Fourier transform (Hansen ...
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Integration step when deriving position-space expression of Feynman propagator

I am looking at the various ways to derive explicit position-space expressions of the Green's function with certain boundary conditions (Feynman propagator) of the Klein-Gordon equation. In this ...
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Show that $i/2m\int d^3\vec x\hat\pi(\vec x)\partial^2_i\hat\phi(\vec x)=1/(2\pi)^3\int d^3\vec p E(\vec p)\hat a(\vec p)^\dagger\hat a(\vec p)$ [closed]

Show that the quantum field for the Hamiltonian, $$\hat H=\frac{i}{2m}\int d^3 \vec x\hat{\pi}(\vec x)\partial^2_i\hat{\phi}(\vec x)\tag{1}$$ can be written as $$\int \frac{d^3\vec p}{(2\pi)^3}E(\vec ...
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Corresponding Measurement in Quantum Mechanics

I recently started with R.Shankar Principles of Quantum Mechanics. There he discuss the part of measurement in Quantum mechanics which I didn't understood. For example I want to know the the momentum ...
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Momentum and position operator in momentum basis [closed]

I was recently studying Quantum mechanics from R.Shankar's Principles of Quantum Mechanics. I recently encountered improper vectors, and function and infinte-dimensional vectors. But I got confused at ...
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What is the position-space form of the photon propagator in axial gauge?

I'm interested in the form of the photon propagator in position space, when expressed in an axial gauge $ n \cdot A =0$, where in the case I am interested in, $n^\mu = \{1,0,0 \dots, 0\}$ (for a $D$-...
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Where can I learn Fourier analysis and complex signal processing for quantum mechanics?

I'm requesting resourses to dive deep and get a good grip on complex signals, Fourier analysis and connections to information theory and information encoded by complex signals. My background: 3rd year ...
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Fourier transformation of $\log(\mathbf{q}^2)/\mathbf{q}^4$ in $d=3$

(Note: I posted the exact same question in the math StackExchange, but I am trying to get more people to view it (I posted it here: same question on math StackExchange). The Fourier transformation has ...
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Why temporal and spatial term is multiplied to get the wave equation?

When deriving the wave equation we have: $$y_{spatial} = A\exp(ikx)$$ $$y_{temporal} = A\exp(-i\omega t).$$ Then the overall wave equation is obtained by multiplying both time and spatial component. ...
chaos24's user avatar
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An electromagnetic wave can be decomposited into sum of many monochromatic waves, so is the total energy of this EMW the sum of these componen waves?

we know that a waveform of electromagnetic wave can be decomposited into sum of a series of monochromatic waves, so is the total energy of this EMW also equal to the sum of energy of these ...
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Fourier coefficients of fluid velocity in Cosmology

I start with the following equation, that can be obtained by taking the curl of the continuity equation, and in which $\omega=\vec{\nabla}\times\vec{v}$ is the curl of the velocity field of the ...
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Getting equation (16.159) in Ashok Das' QFT textbook

I am having difficulty in getting equation (16.159), page 730, in the book "Lectures on Quantum Field Theory", 1st edition, by Ashok Das. (The equation and page number is slightly different ...
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Fourier transform of Green function using residue theroem

I want to compute the Fourier transform of a Green function in $k$-space : $$ G^R_{n,m}(\omega)=\int_0^{2\pi}\frac{dk}{2\pi}\frac{e^{ik(n-m)}}{\omega+i\eta-\epsilon_k} $$ By substituting $\omega$ and ...
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Fourier transform of velocities of individual particles vs coarse-grained velocities

In fluid dynamics, we are interested in $E(k,t)$ the energy in mode $k$. From the velocity field $v(x, t)$, $E(k, t)$ can be computed by simply taking the Fourier transform of $v(x, t)$, squaring, etc....
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Is the Fourier Transform a reliable way to infer the physical phenomena producing an RF signal?

Consider the following thought experiments: Scenario 1: A person standing far away shines 3 light beams at you, each beam having a narrow spectral distribution centered around different frequencies F1,...
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Can I recover the physical wavelengths present in a light source from a time-series measurement of its amplitude?

Consider a mixture of different wavelengths being emitted from the same point (ex: a star). This light consists of a mixture of wavelengths and intensities at each wavelength. When measuring the ...
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Would the "FFT" of a light source be a reliable indicator of perceived color?

Paraphrasing from here: A purely monochromatic 575nm wavelength light would be "perceived" as yellow, as would a light that has equal components in red and green (but no yellow). However, ...
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Fourier transform of the Heisenberg antiferromagnetic model

I have a short question about the Fourier transform of the antiferromagnetic Heisenberg model. The Hamiltonian, written in terms of bosonic operators, is: $$ \widehat{H} = -NJ\hbar^2s^2 + J\hbar^2s \...
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Is the scalar-field Feynman propagator at the origin ($x=0$) equal to 1?

I was reading about Feynman rules for scalar field in $\phi^4$ theory in section 4.6, pages 113-114 of Peskin & Schroeder, and, calculating amplitudes for processes, the authors show that Feynman ...
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How do you represent a plane wave propagating at an angle $\theta$ w.r.t. $z$-axis? [closed]

There is a plane wave $\exp(i\mathbf{k}\cdot\mathbf{r}-i\omega t)$, where $\mathbf{k}$ is the wave vector. Suppose this wave propagates at an angle $\theta$ w.r.t. $z$-axis. What will be the wave ...
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Marginal of Wigner Function calculation

I am reading on the Wigner function from the Gerry and Knight book. It defines it as: $$ W(q, p) \equiv \frac{1}{2 \pi \hbar} \int_{-\infty}^{\infty}\left\langle q+\frac{1}{2} x|\hat{\rho}| q-\frac{1}{...
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How to calculate the Green function of 1D Kitaev chain?

After performing Jordan-Wigner transformation, a uniform transverse Ising model becomes a 1D Kitaev chain as $\hat{H}_{p=0,1} = -J\sum_{j=1}^{L}{(\hat{c}_{j}^{\dagger}\hat{c}_{j+1}+\hat{c}_{j}^{\...
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How do I compute the diffraction efficiency of the orders of an arbitrary grating in the Fraunhofer regime?

Suppose I have an arbitrary grating and it has some transmission profile $t(x)$. I want to compute the diffraction efficiency of the grating's $n^{th}$ order in the Fraunhofer regime. Let the spatial ...
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How to solve Fresnel diffraction integral of vortex beam

Solving Fresnel integral for vortex beam diffraction is necessary and difficult. Without the necessary integral formula, this isn't easy to derive. Therefore, I need some means to calculate it ...
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How to write a general state in Fock space in the form of a series of products of $b_k^\dagger$s on a vacuum state?

Consider the state in bosonic system with $N$ sites with periodic boundary condition $$|n_1,n_2,\cdots,n_{N/2}>=\left(\prod_{i\in\{0,\pm 1,\dots,\pm N/2\}}\frac{(b^\dagger_i)^{n_i}}{\sqrt{n_i!}}\...
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Inverse of an operator [closed]

I want to understand how to find the Inverse of an operator. I know it involves the use of Green's function but I can't seem to figure out how. Here is the actual problem: On page 302 of Peskin&...
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The equation for linear combinations of sinusoidal waves

I was reading Introduction to Electrodynamics by D.J Griffith and in the chapter of Electromagnetic waves, it gives an equation for the representation of any wave in terms of sinusoidal waves. This is ...
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