# 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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### 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 ...
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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\...
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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*...
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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 ...
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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 ...
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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 ...
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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 ...
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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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### 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 $t=0$ ...
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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 ...
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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 ...
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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 ...
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