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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Why the General Lomb-Scargle results are doubled?

I got a problem when computing the GLS on MATLAB for my master deegree thesis. I need to do a comparison between the PSD and amplitude spectrum of a signal computed with three different metods, the ...
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Numerical solution of 2D Wave Equation using Fourier Transform and Finite Difference

This is the 2D Wave equation: $$ u_{tt} = u_{xx} + u_{yy} $$ using initial condition: $u(x,y,0)=f(x,y), \:\: u_{t}(x,y,0) = 0$. The inverse Fourier transform used is: $$ u(x,y,t) = \int\int \hat{u}(\...
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Beats and Raman effect

I have some doubts on Raman effect: if we have a diatomic molecule and we want to see Raman effect, we can send an electric field $E(t)=E_0cos(\omega_pt)$ on the molecule which has, due to vibrational ...
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Wick's theorem: From operators to fields

I understand Wick's Theorem when operators are involved to be, $$\mathcal{N}(f(a,a^\dagger) = :\!\sum\textbf{All contractions}\!:$$ But I'm getting slightly confused when this is expanded to fields, I'...
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Why sum of squares of the magnitudes of Fourier coefficients in Infinite Square Well equals one but it is not so in regular Fourier analysis?

My question is basically this.. In regular math, Fourier Coefficients give the "amount" a particular frequency is available in any periodic signal. The squares of sum of coefficients is not ...
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Stationary Schrödinger Equation in Momentum space

Given the time dependent equation: $$\partial_t\,\hat{\psi}(p,t) = \dfrac{p^2}{2\,m}\,\hat{\psi}(p,t) + \hat{\psi}(p,t)\star{\hat{V}(p)}$$ and forcing through some kind of separation: $\hat{\psi}(p,t) ...
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Calculating Fourier Coefficients for this 50% bipolar square wave [closed]

I'm getting stuck trying to derive an equation for the Fourier coefficients of this waveform. Following along from this example: https://lpsa.swarthmore.edu/Fourier/Series/ExFS.html It is clear that ...
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Fourier Transform of a Wave Packet [closed]

In the analysis of coherence and interference, I encountered the following expression: $$F(t)=\Re\int_0^\infty\mathrm d\omega e^{-i\omega t}H(\omega)$$, where $\Re$ denotes the real part of the ...
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Is this Fourier derivation of Heisenberg principle true?

I noticed that I can derive Heisenberg principle very easy. $\begin{aligned}p=\dfrac{h}{\lambda }=\dfrac{h\nu }{\lambda \nu }=\dfrac{h\nu }{c}=h\nu \left( c=1\right) \\ Fourier\Delta v\cdot \Delta x &...
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Derivation of Wigner Function of cosine with phase

The Wigner Function of $x(t)$ is $$ W(t,f) = \frac{1}{2\pi}\int x\Big(t+\frac{\tau}{2}\Big)x^*\Big(t-\frac{\tau}{2}\Big) e^{-j2\pi f\tau}\;d\tau $$ I know how to get the $W(t,f)$ of $$x(t)=\cos(2\pi ...
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Finding coefficients for wave function when Fourier transform is not possible

I am looking at a wave function moving towards a potential step with potential $V_0$ for $x>0$ while having a total energy that is smaller than $V_0$. I already know how you can find the unbound ...
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Superposition of momentum plane waves for a WF: discrete of continuous?

On the one hand there is a theorem that states that any reasonable wave function $\Psi$ can be written as a superposition of eigenstates of $\hat Q$ (a hermitian operator). So if $\Psi _i$ are the ...
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Fourier-Transforming the Schrödinger Equation in order to solve it?

In Quantum all time favourites equation is given by: $$-\dfrac{\hbar^2}{2\,m}\,{\partial_{x}}^2\psi(x,t) = i\,\hbar\,\partial_t\,\psi(x,t)$$ What happens if you were to apply a Fourier-Transform on ...
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How is this Fourier transform done?

Given a Hermitian operator $\hat{\phi}(x,t)$, why we can write it in terms of Fourier transformations as $$\hat{\phi}(x,t)=\int^\infty_{-\infty}\frac{dk}{(2\pi)(2\omega)}[\hat{a}(k)e^{ikx-i\omega t}+\...
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How can we ignore the diverging term $e^\infty$ in the integral?

In Question (2.20) of Griffiths' Quantum Mechanics book, they have given this Solution. In the Solution of question 2.20(b), they omitted $e^{(ik-a) \infty}$ (or may have considered $e^{(ik-a) \infty}=...
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Deriving the momentum Feynman rule for a vertex with a derivative of the field

Consider the following modification of the klein gordon lagrangian: $$S = \int\left(\frac{1}{2}(\partial \phi)^2 - \frac{m^2}{2}\phi^2 + \frac{\delta Z}{2}(\partial \phi)^2 - \frac{\delta m^2}{2}\phi^...
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Schrödinger equation obtain $ψ(x,t)$ from $ψ(x,0)$

In this answer of the post "Wave packet expression and Fourier transforms" it is said that for the S.E. we have this property: If we start with an initial profile $ψ(x,0)=e^{ikx}$, then the ...
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Commutator of $[\hat{x}, \hat{k}]$

We define the operators $$\hat{x} |x\rangle = x |x\rangle\tag{1}$$ and $$\hat{k} |k\rangle = k |k\rangle\tag{2}$$ where $$\sqrt{2\pi} \Psi(k)=\int dxe^{-ikx}\Psi(x)\tag{3}$$ and $$\langle x|k\rangle = ...
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Time-independent amplitude to go from one point to another in Feynman lectures (free particle)

In the third chapter of Feynman Lectures Volume III, I found this expression Suppose a particle with a definite energy is going in empty space from a location $\boldsymbol{r_1}$ to a location $\...
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Understanding the Fourier transform of a signal already with amplitude and phase information

I'm from an image processing background and am working my way into optics. I'm working on phase retrieval problems, trying to understand the Gerchberg-Saxton algorithm. I found this video which is ...
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Momentum integral yielding $\delta$ function

I am reading the paper Asymptotic conditions and infrared divergences in quantum electrodynamics by P. P. Kulish & L. D. Faddeev (the paper is not important for the question I think, but I will ...
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Magnitude of an Electric field as superposition of plane waves

I need to show the magnitude $u(x,y,z)$ of an arbitrary electric field can be written as a superposition of infinite number of plane waves travelling along different directions. Can someone provide ...
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Doubt regarding the calculation of first Born approximation of Yukawa potential

The expression for the scattering amplitude upto first Born approximation for Yukawa potential is $$f^{1}(\mathbf{k},\mathbf{k'})=\int_0^\infty r^2 dr \int_0^{2\pi} d\phi \int_0^{\pi}\sin{\theta}\frac{...
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How can we show that a lens is a low pass filter?

I understand from the derivation in Goodman Chapter 6 that a lens Fourier transforms light from the front focal plane onto the back focal plane, ignoring aperture effects. I've also read that a lens ...
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How to deduce the energy of a pair of vortices the classical XY model?

Consider a pair of oppositely charged vortices with unit strength, we estimate the energy of a pair of vortices as: $$ E_{\text {pair }}-E_{0} \cong \frac{J}{2} \int d^{2} r(\nabla \theta)^{2}=\frac{J}...
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Why are CFTs not usually studied in momentum space?

Conformal symmetry in QFT has been extremely useful for physics. However, while most of QFT is usually done in momentum space, CFTs are usually studied in position space or in terms of Mellin ...
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Units in Actions (Functional Field Integrals)

When one rewrites the partition function of a grand-canonical ensemble (quantum version) as functional field integral $$ Z = \operatorname{Tr}_{ \mathscr{F}} \mathrm{e}^{ - \beta \left( H - \mu N \...
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Is there a name for this type of wavevector?

When we say wavevector we often mean a vector $\mathbf k$ that is related to the direction and wavelength of a plane wave, given by $e^{i\,\mathbf k\cdot\mathbf x}$. I have to write something about ...
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Generating a spectrum from relaxation times

I have some relaxation times that span several orders of magnitude. I'd like to plot the time spectrum and look at the frequency via a Fourier transform. I imagine there should be an appropriate ...
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How can I compute $\langle x|e^{-a \hat{p}^{2}}| x^{\prime}\rangle$? [closed]

I want to compute $$\langle x|e^{-a \hat{p}^{2}}| x^{\prime}\rangle$$ I think $$e^{-a \hat{p}^{2}}=\sum_{n=0}^{\infty}\frac{\left[-a \hat{p}^{2}\right]^{n}}{n!}$$ and I know that $\langle x|\hat{p}| x^...
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Intuition for momentum operator in position space

The derivation of the momentum operator in position space. But, several assumptions are usually made that a) we are dealing with the particle in free space or b) that the two representations are ...
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Klein-Gordon Hamiltonian in terms of Fourier transformed variables

The Klein-Gordon Hamiltonian density is a function of four complex variables $\psi , \psi ^* , \pi , \pi ^*$. Suppose we make the change to Fourier transformed variables. Then the Fourier expansions ...
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Why do most of the book represent Plane waves by considering a single sine or cosine wave? There should be many, right? Isn't it misrepresentation?

This Image is from Electrodynamics by Griffiths. Here also a monochromatic electromagnetic wave is considered.
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Physical interpretation of FFT frequencies

I need to calculate the PSD of a discrete signal and want to compare it to other processes. By Nyquist theorem, I only can account half of the frequencies. Assume I have a signal of length $N=100$, ...
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Euler-Bernoulli equation for a periodically supported static beam

The Euler-Bernoulli equation for a homogeneous beam is $$ EI w^{(4)}(x) = q(x),$$ where $w$ is beam height and $q$ is load density. Inspired by the deflection in a multi-support cantilever bridge ...
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Integral of the intensity curve over path difference

Suppose the intensity amplitude of an elementary beam is $i_1 (k_0)$ where $k_0$ is the wavenumber. Let $k_0 = 2\pi/\lambda_0$ be the wavenumber corresponding a particular "desired monochromatic ...
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Understanding the 'Truncated DFT' for Moiré Phase Analysis

I am researching the use of a Moiré sampling algorithm in order to determine the two-dimensional displacement field of an object. In the paper of Wang and Ri (2022), the algorithm is described. ...
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How does the Fourier picture relate to non-commutativity?

A compelling video by 3Blue1Brown visualizes the uncertainty principle with Fourier transforms. The gap I'm trying to bridge is between this "Fourier picture" and the matrix-based statement $...
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Is it possible to explain the Ultraviolet Catastrophe as a manifestation of the Riemann-Lebesgue Lemma?

Is it possible to explain the Ultraviolet Catastrophe as a manifestation of the Riemann-Lebesgue Lemma? I don't fully understand any of both topics, but reading about the Ultraviolet Catastrophe on ...
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Inconsistent? Hamiltonians for a scalar field in the presence of a classical source

Consider a real scalar field \begin{equation} \phi(x)=\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}}\left(a_pe^{-ip\cdot x}+a_p^{\dagger}e^{ip\cdot x} \right). \end{equation} In Chapter 2 of Peskin, ...
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Understanding the relationship of displacement and electrical field of nonlinear material using fourier

**I posted this question first in math.stackexchange but was told it might be more suitable here. I have a general formula of $$D = P_0 + \epsilon_1 E + \epsilon_2 E^2 + \epsilon_3 E^3 + ...$$ with $D$...
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Why can we arbitrarily set the expectation value of a field operator by representing the field state as a product of coherent states?

In the paper "Unusual Transitions Made Possible by Superoscillations", the author begins by solving for a coherent state \begin{equation}|\alpha\rangle\end{equation} such that \begin{...
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Convergence of an oscillatory integral to real number

I have a physical model, where the long-time behavior of the system can be described by $$C(t)=\frac{1}{2\pi}\int_{-\pi}^\pi\mathrm{d}k\,\mathrm{e}^{-t\omega(k)}$$ with $\omega(k)\geq0$ and $\omega(t)\...
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In calculation of Hamiltonian of real scalar field (Quantum field theory Srednicki)

I'm now reading the Mark Srednicki, Quantum field theory, p.27 I'm now trying to understand the Third step in the calculation of $H$. Through the integration over $k'$ involving the delta functions $...
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Uniform radial distribution function

I am attempting to properly account for excluded volume effects on the radial distribution function for a fluid. A correction for these effects has been proposed back in 2000 in this paper by Hartnig ...
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Is the uncertainity principle explained by disturbances or only by the Fourier picture?

Qualitatively, the tradeoff in uncertainty between two non-commuting observables $\hat{x}$ and $\hat{y}$, could be explained by... the Fourier picture where the more one variable is defined (i.e., ...
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Fourier transform in the exponent with a multiplication operator

I am going through Dr Frederic Schuller's course on Quantum Theory. In the 19th Lecture on The Schrodinger Operator, where F is a Fourier Transform and P squared is maximally defined real ...
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Why is this acoustic scattering integral a Fourier transform?

I am puzzeled about this part in the book "Acoustic Absorbers and Diffusors" from T. J. Cox, P. and D'Antonio. It describes the scattered pressure from a surface of a pressure source in ...
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Fourier transform of Wick rotated functions

I am learning the imaginary time formalism of thermal field theory / reviewing the Euclidean formalism of quantum field theory. One thing that appears to be left implicit in many treatments is a ...
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Integrating $\int \frac{e^{ikx}}{x} dk$ by parts to get delta function derivative, how to handle undefined boundary terms?

I'm going through Sergio Dutra's Cavity Electrodynamics: The Strange Theory of Light in a Box. In equation (2.31) he computes: $$\begin{aligned}\langle x|\hat{p}|x'\rangle&=i\hbar\int\frac{dk}{2\...
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