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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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Missing $i$ in Feynman Propagator Minkowski Space

I'm Trying to resolve the following equation in Minkowski Space $$ \left(\Delta_{x}-m^2\right)G(x^{\mu},y^{\mu})= - \frac{\delta^{(D)}(x^{\mu}-y^{\mu})}{\sqrt{g(x)}}$$ where $$ \Delta_{x}=\frac{...
Oswaldo RO's user avatar
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Phonon operator in Electron-Phonon Interaction [closed]

I'm going through Mahan's many body physics (3rd Ed), and I'm having some trouble understanding a step on p.28. He has the electron phonon coupling term in terms of a Fourier Transform (Eq 1.225) $$...
Redcrazyguy's user avatar
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Finding out the number of minima for a fourier expansion [migrated]

Suppose I have a Fourier series f(x) = $\sum_{n=1}^N t_n cos(nx)$ defined in the domain $(-\pi,\pi]$. we need to prove that mathematically we can ' at most' have N minima points excluding the boundary ...
ANIMESH GHOSH's user avatar
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Momentum space representaion of an electron-phonon coubling Hamiltonian

I am facing a problem transforming the following Hamiltonian into momentum space: \begin{align}\hat{H} = -\gamma \sum_\alpha\sum_{i=1}^2 \hat{X}_{i,\alpha} \hat{c}_{i,\alpha}^+\hat{c}_{i,\alpha} +t\...
elfarhan's user avatar
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Retarded Green's function in Peskin & Schroeder

In an Introduction to Quantum Field Theory by M. E. Peskin & D. V. Schroeder (eq. 2.56 on page 30) the following relation for the retarded Green's function was established: $$(\partial^2 + m^2) ...
Volodymyr's user avatar
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Alternative way to compute expectation value of momentum? [closed]

This might be ridiculously incorrect, but is it possible to find the expectation value of momentum like this? In the position space: $$\langle x | \psi \rangle = \psi(x)$$ $$\langle \hat{A} \rangle_{x\...
Aryan MP's user avatar
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How to compute the diffraction efficiency of a thin phase grating with arbitrary groove shape?

From Magnusson and Gaylord (1978), the wave amplitudes, $S_i(z)$, of $p$-polarized light for a thin, arbitrary phase grating are given by the equation $$ \frac{\partial S_i}{\partial z} + \gamma \sum_{...
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Confusion about EM spectrum and the Fourier transform

Since courses on signal analysis and electromagnetism I have become confused about what the spectrum of electromagnetic radiation really means. I know light is when electric and magnetic fields become ...
Jelle 3.0's user avatar
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Fourier Transform of periodic Signals

If $f(x)$ is a periodic signal with a period $A$ then the Fourier transform of the signal $F(k)$ is zero, unless $k A$= $2 \pi n$ where $n$ is an integer. How can there Be a Fourier transform of a ...
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Feynman claimed "The ear is not very sensitive to the relative phases of the harmonics." Is that true?

In The Feynman Lectures on Physics, Dr. Richard Feynman claimed that the ear (I assume he meant the human ear) is not sensitive to the relative phases of harmonics. However, I was asked to test ...
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Fourier transform of spin Matsubara Green function

Is the spin Matsubara Green function of a generic spin operator (or product of spin operators) bosonic? How can one obtain its frequency decomposition?
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Chain rule with functional derivative?

I posted the same question on math exchange but no answer yet, so I post it also here: "I'd like to make the functional derivative of the functional $S[\phi(x)]$ with respect to the Fourier ...
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Mode expansions of canonical quantized fields in QFT

I have been self-studying QFT Quantum Field Theory for the Gifted Amateur by Stephen Blundell and Tom Lancaster. They devote Chapters 11, 12, 13 to the subject of canonically quantizing any given ...
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How to derive momentum-space Feynman rules from position-space ones?

I was reading Peskin-Schroeder's QFT text (P-S) and came across Equation (4.47) stating the vertex factor when four lines meet. P-S says the $z$-dependent factors of the diagram is: $$ \int d^4z\,e^{-...
math-physicist's user avatar
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Trick for integration into a plane wave basis

I'm reading the article https://arxiv.org/abs/hep-th/9705200 and part of it has left me very confused. In order to speak about their equation (1) the authors make the following statement: \begin{align*...
João Paulo Melo's user avatar
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Impulse Response And convolution

What is a impulse Response Function ? Can it be called as a Resolution function? And in my math text book the resolution function is defined as the probability density function which gives the ...
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Resolution and delta function

Any attempt to measure the value of a physical quantity is limited, by the finite resolution of the measuring apparatus used. on the one hand the physical quantity we wish to measure will be in ...
Hello's user avatar
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Fourier transform of Light impulse [closed]

A function in the time domain and the frequency domain has inverse relationship in standard deviation. So the less standard deviation the function has in time domain the more spread out in the ...
Hello's user avatar
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1 answer
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"Fourier Transformation" of angle spinors to twistor variables

This relates to the derivation of equation (5.15) if Elvang and Huang's textbook. The idea is to transform the spinor helicity variables we are using, $(|i\rangle_{\dot{a}},[j|^a)$ to go into twistor ...
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Does the exponential representation of Dirac delta function depend on choice of Fourier convention?

Is it always true that $$\delta(\omega) = \frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{i \omega t} dt , $$ regardless of your Fourier convention? For example, if I choose to use the Fourier convention ...
photonica's user avatar
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Particle Creation by a Classical Source (on-shell mass momenta)

It is noted in Peskin and Schroeder's QFT text that the momenta used in the evaluation of the field operator $\phi(x)$ are "on mass-shell": $p^2=m^2$. Specifically, this is in relation to ...
Albertus Magnus's user avatar
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Fraunhofer temporal far-field condition for dispersive Fourier transform (DFT) technique

I am trying to understand the dispersive Fourier transform (DFT) technique for spectral characterization of pulses. In the literature, I found this far-field condition from Fraunhofer approximation in ...
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The Klein-Gordon Propagator According to Peskin and Schroeder (Derivation of *Retarded* Green's Function)

On page 29 of Peskin and Schroeder's An Introduction to Quantum Field Theory, the authors write that the propagator is given by: $$\begin{align} \langle 0|[\phi(x),\phi(y)]|0\rangle&=\int{d^3p\...
Albertus Magnus's user avatar
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Restrict Fourier Transform to a Hypersurface

Assume a field evolves according to the Klein-Gordon equation $$(\Box+m^2)\phi=0$$ The general solution to this is given by Fourier mode decomposition $$\phi(t,\vec{x})=\int\frac{\text{d}^3p}{(2\pi)^3}...
go_science's user avatar
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How to Perform Fourier Transform on a Quantum State of Spin-1/2 Particle?

I am currently studying quantum mechanics and need help understanding how to perform the Fourier transform of a particular state. I have a spin-1/2 particle whose momentum and spin state at time $t=0$ ...
bougab's user avatar
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Fourier transform of the Gaussian action for the real scalar bosonic field

In my current homework, we have to get familiar with quadratic theory in order to reach $\phi^4$-theory. So the starting point is $$Z = \int Dx e^{-S[\phi]}$$ with the action for the real scalar ...
Johnny_T's user avatar
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How to find the image field of a Gaussian beam impinging on a positive lens? I am trying to calculate the impulse response and find the image field

How to get the image field for the setup in the following question , where a Gaussian beam is incident on a positive lens with focal length f z0 and z1 are the object and image distance? Can anyone ...
condensedvaddi's user avatar
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When can I commute the 4-gradient and the "space-time" integral?

Let's say I have the following situation $$I = \dfrac{\partial}{\partial x^{\alpha}}\int e^{k_{\mu}x^{\mu}} \;d^4k$$ Would I be able to commute the integral and the partial derivative? If so, why is ...
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Do solutions to the time-independent Schrödinger equation always (for any $V$) form a basis for solutions to the time-dependent equation?

Griffith's "Intro to Quantum Mechanics" shows that for $V(x)=x^2$ and $V(x)=0$, solutions to the SE can be constructed as a linear combination of stationary solutions. But is there a theorem ...
user56834's user avatar
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1 answer
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The relation between spectral function, Green function and particle number in many-body systems

The spectral function is defined as the imaginary part of Green function multiplied by $-2$ (Ref. Mahan. Many-Particle Physics 3ed. Kluwer Academic, 1990.), $$ A(\mathbf{k},\mathrm{i} \omega_n) = -2 \...
ZQW's user avatar
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2 votes
2 answers
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Dirac delta of operators multiplying matrix element

In playing around with the Wigner-Weyl correspondence, I found myself needing to perform an integral of exponential operators, which I am confused about. TLDR: help to evaluate $$\int d{x}dy\ \delta(x\...
Landuros's user avatar
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2 answers
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Anomalous dimension must be positive in Ginzburg-Landau $\phi^4$-like theories?

I am trying to understand/find the argument behind a claim made in this paper (page 3, column 1): that the anomalous dimension/exponent $\eta$ of a continuous phase transition in Ginzburg-Landu $\phi^...
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2 answers
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Basic confusion about evolution of wave function of a free particle

I am going through Griffith's introduction to quantum mechanics. An example for a free particle is given where $$\Psi(x,0) = \begin {cases}A \quad \text{if } x\in [-a,a]\\ 0\quad \text{otherwise}\end{...
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How can one "encode" momentum into the wave-equation of a QM harmonic oscillator? [duplicate]

I am learning about Quantum Mechanics using Griffiths book and after reading the section about the quantum harmonic oscillator, I was left wondering how one can construct a solution to the Schrodinger ...
Mantabit's user avatar
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How to extract the "matter fluctuation amplitude" from the CMB power spectrum?

How do you convert the value listed in Planck 2018 results. VI. Cosmological parameters, $A_s = 2.101\times10^{-9}$ to the value of the matter fluctuation amplitude $\sigma_8=0.8111$? I tried ...
Finerichmen's user avatar
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Fraunhofer diffraction from composite apertures

I'm trying to calculate the fraunhofer diffraction pattern from a aperture composed of multiple simple shapes, but I've ran into some trouble when trying to simulate the results and I really dont ...
Fotondetektor's user avatar
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How are the gauge transformations of $\epsilon(\mu)$ and $A^\mu$ related?

To find a local field description of massless spin-1 particles that is Lorentz invariant, we can identify $\epsilon^\mu_{\pm}(k)$ with $\epsilon^\mu_{\pm}(k)+\alpha(k)k^\mu$. As $A^\mu$ and $\epsilon^\...
IGY's user avatar
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-1 votes
1 answer
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A simple question in quanum mechanics on position and momenum eigenstates

The eigenfunctions (eigenstates) for the momentum of a particle are given by the plane waves $$\phi(x,t) = \sin(kx - \omega t)$$ If we sum a large number of these waves in a range from $0$ to $k_m$, ...
Anky Physics's user avatar
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2 answers
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I don't understand intuitively why the instantaneous frequency is obtained by calculating the time derivative of the phase

I don't understand intuitively why the instantaneous frequency is obtained by calculating the time derivative of the phase
krunker.io's user avatar
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Dispersion relation for non-harmonic waves

This question is related to my previous one. The entire linear theory of waves is built on dispersion relations, which represent the algebraic dependence of frequency on wave number. That is we ...
shamil khal's user avatar
3 votes
1 answer
286 views

How does the Planck constant enter into the uncertainty principle?

In Stein & Shakarchi's Fourier Analysis, the Fourier transform of a Schwartz function $\psi$ is defined to be $$\hat{\psi}(\xi) = \int_{-\infty}^\infty \psi(x) e^{-2\pi i x \xi} dx$$ which gives ...
Drake's user avatar
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How to find the expectation value of momentum operator? [closed]

I have struggled with the following steps in finding the expectation value of the momentum operator $\hat{p}$. $$\left \langle \hat{p} \right \rangle=\int_{-\infty}^{\infty}\psi^{*}(x)\hat{p}\psi{x}dx=...
SaaN's user avatar
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Why don't we see the transverse waves with a phonon lattice dynamic while we see them with linear elasticity theory?

Let's say that we have a lattice with particles sitting on the nodes. Each particle $n$ has neighbors $\bar n$: $$\dfrac{d^2\boldsymbol u_n}{dt^2}=-K\sum_{\bar n}\boldsymbol u_{\bar n}\tag{1}$$ The ...
Syrocco's user avatar
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Is there a class of phenomena the description of which is not limited to the study of the properties of individual harmonic waves?

There are lots of examples of oscillatory phenomena in nature the description of which boils down to simple harmonic behavior, i.e. to Cosine/Sine/Complex Exp. This answer explains that we use sines ...
shamil khal's user avatar
1 vote
1 answer
63 views

Gaussian wave packet with complex coefficients [closed]

I am trying to obtain a representation of the momentum-space wavefunction $<p'|\alpha>$ Its position space wavefunction is given as $$ <x'|\alpha> = N \exp [-(a+ib)x'^2 +(c+id)x'] $$ where ...
raccoon's user avatar
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Dirichlet’s Theorem and Solutions to Laplace Equation in Cartesian Coordinates

I have been reading Introduction to Electrodynamics - Griffiths about solving Laplace equation in cartesian coordinates, and in that book, I saw this statement: The functions $\sin(n\pi y/a)$ are ...
Sanjay's user avatar
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3 votes
1 answer
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Is there a momentum representation of the atomic stationary states?

In standard quantum mechanics, the atomic orbitals are represented by the following wave functions (where $u = 2 \mathrm{Z} r / n a$): $$\tag{1} \psi_{n l m}(r, \theta, \varphi) = \phi_{n l}(r) \, Y_{...
Cham's user avatar
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Fourier transform of $\phi$ [closed]

I was reading through David Tong's Lectures Notes on Quantum Field Theory and I was wondering how, on page 22, he derives that the Fourier transform of $\phi(\vec{x}, t), \tilde{\phi}(\vec{p}, t)$, ...
vesta's user avatar
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1 vote
1 answer
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Do wavenumber pairs exist for standing waves?

The equation for a travelling wave is usually taken in the form of, $$y = A\mathrm{e}^{i(kx - \omega t)}$$ When a standing wave is formed by the interference of two counter-propagating waves, then it ...
Alucard Nosferatu's user avatar
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1 answer
87 views

Is complex integration needed when normalizing a wave function?

I have to find $\phi_0$ from following wave function in the momentum space: \begin{equation} \phi(k) = \phi_0 \text{exp}\bigg(-\frac{(k-k_0)^2}{2\kappa^2}\bigg) \end{equation} I know that I have to ...
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