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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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Back Focal Plane After A Convergent Lens

Acutually this is related to computer simulation, especially MATLAB. I am very confused, assume a plane wave passes through a convengent lens, where the propagation direction is parallel to paraxial ...
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Translating between momentum eigenstate and position eigenstate

Say a particle is in an eigenstate of momentum: $$\phi_{\mathbf p} = N e^{i \mathbf p x}$$ Then, according to my lecture notes, apparently a position measurement renders $$\psi = \delta ^{(3)} (\...
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Feynman prescription in path integral by adding an imaginary part to time

It is known that the well-definiteness of the path integral leads to the Feynman's $i \varepsilon$ prescription for the field propagator. I've found many ways of showing this in the literature, but it ...
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Fraunhofer diffraction problem in Python: How to interpret discrete Fourier transform (DFT) spectrum?

I have a periodic phase grating consisting of lenslets along the x-direction, invariant in y. I want to use python to calculate the far-field (Fraunhofer) diffraction pattern that one gets when ...
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What is a “Fourier transform limited pulse”?

I have some doubts about the definiton of a Fourier transform limited pulse. For example if I consider a generic pulse: $$E(z,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}A(\omega)e^{i(-\beta(\...
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How do you expand a wavefunction in the basis of eigenfunctions of the free particle?

If we have an initial state given by $ \Psi(x,0) $ and we want to find $ \Psi(x,t) $, we would expand the function in the basis of eigenstates of the Hamiltonian, $\{\psi_n\}$: $ \Psi(x,t)=\sum _nC_n ...
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What is the relationship between directions in reciprocal and real space of a photonic crystal?

I am reading "Photonic crystals - molding the flow of light" by Joannopoulos et al. (available on-line). The figures below are reproduced from there. This is a diagram of a triangular lattice of air ...
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Question about Mode expansion of free compact boson

$(1+1)$-Dim free compact boson, Lagrangian is $$\mathcal{L}= \frac{1}{2}(\partial_\mu\phi(\sigma,t))^2$$ with $\phi(x,t)\sim\phi(x,t)+2\pi r$ and periodic boundary condition along $x$, i.e. $\phi(\...
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On the integration for the electromagnetic density

Upon integrating the electromagnetic hamiltonian density \begin{equation} \mathcal{H}=\left|\textbf{E}\right|^{2}+\left|\textbf{B}\right|^{2}, \end{equation} on a cube of lengths L and volume V we ...
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Why does the integration over momentum has normalization constant of volume?

If I Fourier transform a wave function in position space, integration carries no normalization constant: $$\displaystyle{\phi(k) \equiv \langle k|\psi\rangle = \sum\limits_x \langle k|x\rangle\langle ...
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What are the allowed wavenumbers in the finite size system?

Usually, we introduce wavenumber $\textbf{q}$ by Fourier transform, for example, an operator $A_{\textbf{q}}=1/\sqrt{N}*\sum_{i}e^{i \textbf{q}\cdot \textbf{r}_{i}}A_{i}$, where $N$ is number of sites,...
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Sign ambiguity when going from position to momentum space evaluating Feynman diagrams

When calculating a simple diagram I came across an ambiguity in the conservation of momentum, i.e. it seems to me that the particle could come out of the process with opposite momentum with respect to ...
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Why is a circular aperture defined by the circle function rather than a heaviside step function?

In the chapter 4 of the text "Introduction To Fourier Optics" by Goodman, the diffraction pattern of light passing through a circular aperture was studied. The diffraction pattern on the image plane $(...
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How is the wave function Lebesgue integrable?

Let's assume we have a plane wave $\psi(x,t)= A_{0}e^{i(kx-wt)}$ in position space. To find the momentum representation of this wave we'd apply the Fourier transform. However, I don't see how this is ...
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Fourier transforming properties of a converging lens

Following Goodman's Introduction to fourier Optics argument I understand that a converging lens will introduce the following difference in optical path (see pag.158, 4th edition) $$r^2/2f$$ where $r$...
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How to understand the two-point correlation function in momentum space?

Let's take the Ising model as an example and study the two point spin spin correlation function: $$\langle s_0 s_r\rangle = \frac{\sum_{\{s_i\}}e^{K\sum_{\langle i ,j\rangle}s_i s_j} s_0 s_r}{\sum_{\{...
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What does the width of the 2D Fourier transform of a speckle pattern mean?

Speckle patterns can be studied by capturing an image of the speckle pattern produced on a rough surface when light from a laser passes through an aperture stop. When analysing this speckle pattern, ...
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How to derive the retarded Green's function matrix for a quadratic Hamiltonian?

Start with the quadratic Hamiltonian for fermion: $$\hat{H}=\sum_{ij}H_{ij}\hat{c}_i^\dagger \hat{c}_j$$ and the definition of retarded Green's functon in time domain: $$G_{i,j}^r(t_1,t_2)=-i\theta(...
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Help me make sense of the spectrum for the quantum wave function of an infinitely hard equilateral triangle

I'm trying to solve the spectrum for a equilateral Tetrahedron with infinitely hard walls. My first guess is to sum up a infinite amount of separable solutions to match the boundary conditions on the ...
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Action of momentum operator on wavefunction in momentum space

In a previous question How to get the position operator in the momentum representation from knowing the momentum operator in the position representation? it was mentioned that $$\begin{align} \...
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Can someone Tong got this equation in his QFT notes

Can someone explain how D.Tong got equation 2.18 in his QFT notes in chapter 2? I am lost from equation 2.5, can someone explain? Link to notes: http://www.damtp.cam.ac.uk/user/tong/qft.html Can ...
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At what frequency does a string vibrate?

When a string with fixed ends vibrates (e.g. plucking a guitar string) Fourier Theorem says that the vibration can be expressed as a sum of its normal modes, which are sinusoidal vibrations with ...
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Can the four-f correlator be solved geometrically?

For an optics lab my class created a four-f correlator and saw how a lens can create a Fourier transform of an image, and we can filter out spatial frequencies of an image by blocking parts of the ...
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Mode Expansion in Klein-Gordon QFT

I have a confusion regarding the mode expansion of the Klein-Gordon field theory. I am following Peskin and Schroeder. My questions are about how we formally get to the expansion of the KG QFT in ...
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Why do the matrix elements of an operator correspond to the Fourier components of the observable in Heisenberg's Matrix Mechanics?

It is well-known that Heisenberg $a$ began developing his Matrix Mechanics by creating matrix components $$A(n,n-a,t)=A(n,n-a)e^{i\omega(n,n-a)t}$$ or $$A_{nm}(t)=A_{nm}e^{\frac{i}{\hbar}\omega(nm)t}$$...
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Relationship between aperture function and Fraunhofer diffraction patern

I have read that the Fraunhofer diffraction pattern of a source is the Fourier transform of the aperture, however I have not been able to follow the proofs. All of them seem to demonstrate that the ...
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What is the error propagation in an FFT (Fast Fourier Transform)?

I use an insert FFT graph feature on a program called logger pro. If I have the uncertainty of my input data, can I know what the uncertainty of the FFT computation will be?
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Reason for $(2 \pi \hbar)^{-\frac{3}{2}}$ prefactor for quantum mechanical wavepacket

My textbook states that the prefactor $(2 \pi \hbar)^{-\frac{3}{2}}$ is not required for the following superpositioned wave function, but should be included for practical reasons without stating what ...
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Why do the matrices in k-space and the final MRI image have to match?

Naturally, there is no correspondence pixel-to-pixel between Fourier space (k-space) and the final 2D image of an MRI - k-space stores the Fourier coefficients, hence each pixel in it affects the ...
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Solution to Klein-Gordon equation: real field condition and other questions

Sorry for the lengthy question, pretty much the whole text is the standard derivation of the solution of the KG equation which I included to illustrate my doubts, and some questions are at the end. ...
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Is it possible to make full-colour Holography using three lasers and three separate holograms?

Holograms need monochromatic coherent light sources like lasers so that the reference light can interfere with the light scattered by the object. Illuminating the holograms, the stored interference ...
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Does $f(x) = 1 $ belong to the Hilbert space of the infinitely-deep square well?

Let the square well be on the interval $(0,\pi )$. It is generally postulated that the wave function of this system should vanish at the end points, i.e., $g(0)= g(\pi) = 0$. The function $f(x) \...
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Fourier transform of power

In the notes I am reading they use the following. Let $U$ be the voltage (depends on time) and $I$ the current in a circuit with some resistor with resistance $R$. Then the power is given by: $$P(t)=...
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MRI k-space: why does the echo produced by different spins correspond to lines in k-space?

It is unclear to me why altering the phase and frequency of all spins making up a substance and then measuring the signal that this produces (the echo), would correspond to lines in k-space (assuming ...
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Using fourier analysis of the Klein Gordon equation

This question is more about a mathematical detail, and I am undoubtedly missing something very obvious. And note, I have sifted through the numerous questions on Fourier transform (FT) and the Klein-...
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How to transform the irradiance image on the screen in the double-slit experiment back to electric field in order to perform Fourier Transform?

Say I perform the double slit experiment while meeting the requirements of the far-field. I get some interference image on the screen (where R is great, as mentioned). This image is correlated with ...
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If $\sum_n \ c_n \ \psi_n(x,t)$ represents an arbitrary state for a given solution to the TISE, what are the bases for a free particle?

If $\sum_n c_n \psi_n(x,t)$ represents an arbitrary vector in the Hilbert space of solutions to Schrodinger's equation with a given potential function, this makes makes sense to me. Each $\psi_n$ can ...
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Analytic derivation of honeycomb lattice Brillouin zone

My goal is to analytically derive the first Brillouin zone of the honeycomb lattice. Geometrically it's clear how to do this by just finding the space on the lattice nearest to a particular point of ...
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Considering an arbitrary wavefunction for a free particle, are all normalizable functions valid?

One can show that a possible solution to a wavefunction with constantly zero potential is equal to, only considering the spacial piece: $$\psi(x) = \int_{-\infty}^{\infty} A(k) \ e^{ikx} dk$$ This ...
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Spacing of resonant modes in a (laser) cavity

I don't see why the frequency spacing between resonant cavity modes should decrease with higher frequency. Here's what I have: For a laser cavity of length $L$, from the condition of constructive ...
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Question regarding position and momentum representations

According to Cohen-Tannoudji's Quantum Mechanics book we can pick the following two bases composed by functions that doesn't belong to $\mathscr{F}\in L^2(\mathbb{R^3})$: $$ \xi_{\mathbf{r}_{0}}(\...
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Solution to the Klein-Gordon Equation

Most textbooks solve the Klein-Gordon equation with the ansatz $$\varphi(\mathbf{x},t) = \int \frac{\mathrm{d}^3\mathbf{k}}{f(k)}\left(a(\mathbf{k})\, \mathrm{e}^{i k x} + b(\mathbf{k})\,\mathrm{e}^{-...
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Why is the Laplace Transform essentially never used when dealing with problems involving resonance?

Both the Laplace Transform and the Fourier Transform can be applied to a PDE, for example the wave equation, and used to derive a solution to the equation. But I never see the Laplace Transform used ...
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Fourier tranform of Coulomb-like potential $1/|r-r'|$

I've found that Fourier transform of Coulomb potential $V= q/r$ is $F[V]= 4\pi q/k^2$. Now I need to calculate fourier transform of function $1/|r-r'|$. And I exactly don't know how to operate with ...
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Fourier Inverse Relation to Diffraction

My professor today in class stated that taking a measurement by light diffraction, such as X-Ray Diffraction, results in the same image as taking a 2-D fourier transform of an image in the lattice ...
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Points of symmetry in $k$-space

Can you relate a point in the reciprocal space with a vector in real space? How do I find the family of planes that represent a point of symmetry in the Brillouin zone? For example, germanium has ...
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Why does a narrower Gaussian wavepacket correspond to an increased $\Delta k$?

Please answer this question without too much reference to the Uncertainty Principle. I understand it in that respect. I'm just wondering in terms of Fourier analysis. Why does a more localized wave ...
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Coordinates in 1970 Labeyrie article

I'm reading the 1970 article by Labeyrie on speckles in astronomy and I encountered a small problem. The author uses two sets of coordinates, $(\alpha, \beta)$ and $(x, y)$, connected by Fourier ...
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Spectral full width at half max of a Gaussian light pulse [closed]

I need to calculate the spectral full width at half max (FWHM) of a Gaussian light pulse. The frequency spectrum of a Gaussian light pulse is $$ \tilde{E}(\omega)\propto \exp{-\frac{(\omega-\omega_0)...
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Obtaining real-space correlations from reciprocal space correlations

Consider a system of Ising variable $s = \pm 1$ on a rectangular lattice which has open boundary conditions on the top and bottom and periodic boundary conditions to the left and right. In other words,...