# 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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### Fourier optics, classical optics and complex field?

I am studying fourier optics on Goodman bible. For example one of the most useful formula is this one (pag. 96): where Uf is the final field distribution,d is the distance of the input from lens, f ...
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### The Ward-Takahashi identity in Peskin and Schroeder (page 311)

I'm working on the Ward-Takahashi identity in Peskin (page 311), but I canʻt obtain Eq.(9.105) from Eq.(9.103) According to Eq.(9.103) \begin{align} &i \partial_{\mu}\left\langle 0\left|T j^{\mu}(...
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### Duffing equation

I have a differential equation: $$\ddot{\omega}+2k\dot{\omega}+2k^2{\omega}-2{\omega}^3=0$$ As I understand it's a Duffing equation, but I can't find the first integral. How can I do it? I didn't find ...
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### Quantum Field theory question just conceptually grasping from Sean Carroll's “Biggest Ideas in the Universe”!

Quantum Field Theory from Sean Carroll's Biggest Ideas in the Universe. I’m just checking to see if I’m on the right track of what he's explaining. He talks about a free field (non-interacting field), ...
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### Fourier Transform of $1/k^4$

I am dealing with a higher derivative theory problem and I have to perform the following integral, \begin{equation} \int \dfrac{d^3k}{(2\pi)^3}\dfrac{e^{i{\bf k}\cdot {\bf r}}}{k^4} \end{equation} ...
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### Transverse displacement profile of the string using Fourier series

A string of length $L$ fixed at $x=0$ and $x=L$ and released at time time $t=0$, the transverse displacement at a position $x$ along the string is given by: $y(x,0)=Ax(L-x)$ Assuming that the string ...
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### What are the real world applications involving Laplace transforms? [closed]

Laplace transform is really interesting. Speaking about Fourier transform, there are many real world applications like we use in removal of noise and Laplace transform is again the extension of ...
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### Fourier Optics: Far Field Image

I have a question about computing the far field diffraction pattern of a laser beam: If $L_{1}$ is large enough, then at $z=L_{1}$ we see the Fourier transform of the input $f(x, y)$. If $L_{2}$ is ...
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### Normalization for the Fourier series of a Gaussian Electric field

I have a function $f(x) = \exp(-\frac{x^2}{\sigma^2})$ on a box of size $L$. I have the given relation between the size of the box and the width of the Gaussian $\sigma = \sqrt{\frac{2}{\pi}}L$. I use ...
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### Trouble solving partial differential equation with Laplacian squared

I am working in extensions of General Relativity Theory and at the moment of taking the Newtonian limit of this extension theory (essentialy, mathematically speaking, this is just linearizing the ...
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### Why doesn't Gaussian wavepacket broadening in position mean there will be a shortening in momentum?

Many sources that say in free broadening of a Gaussian wavepacket, the momentum uncertainty (I think defined in terms of the range of 'significant' momentum amplitudes) is time invariant even as the ...
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### Can I distinguish hexagonal close packing (HCP) from face centred cubic (FCC) arrangement based on Fourier transform

First of all I would say that I'm not a physicist, but I have recently been given the task of distinguishing a hexagonal close packing (HCP) from a face centred cubic (FCC) arrangement in a set of 3D ...
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### Mechanical impedance and dynamic stiffness of a mass, spring, damper system including Coulomb friction

I'm trying to understand the concepts of mechanical impedance and dynamic stiffness, what do they mean and if/how they differ. Consider the very simple system below: Image curtesy of Joshua ...
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### Group velocity in solid state physics force equation

When reading about electron holes here https://en.wikipedia.org/wiki/Effective_mass_(solid-state_physics), the group velocity is introduced as the reciprocal space gradient of the dispersion relation. ...
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### Continuous spectra and measurement

Suppose I have a particle whose momentum I measure to be $p$ with uncertainty $\delta p$. Right after the measurement we know that its wave function is given by $\psi(x)=\int g(p)e^{ipx/\hbar}dp$ (...
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### Evaluating Fourier transform of $f(t) = \sinh ^{-q}(a t) \left(1-\frac{1-a t \coth (a t)}{B}\right)$

I am interested in evaluating Fourier transform of the following function analytically, $$f(t) = \sinh ^{-q}(a t) \left(1-\frac{1-a t \coth (a t)}{B}\right)$$ where $a, B, q$ are some real parameters ...
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### Fourier expansions of Klein Gordon field not Lorentz invariant?

I’m working in Peskin and Schroeders book on QFT and noticed that they expanded a solution to the Klein Gordon equation in a manner that seems to me not to be be Lorentz invariant even though the ...
I am wondering if it's possible to compute the derivative of the Dirac Delta function using the definition obtained from Fourier transformation: $\delta(x-x')=\frac{1}{\sqrt{2\pi}}\int e^{-ik(x-x')}dk$...