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 calcalating the response of a time shift invariant linear system as the sum of its response to time harmonic excitation or for transforming a ...

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5
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296 views

Action of Parity operator on Impulse representation

Is my derivation of the action of the parity operator $\mathbb{P}$ on the $|p\rangle$ representation correct? $$\left( \mathbb{P}\tilde\psi \right)(p)= - \tilde\psi (p).$$ Obtained from $$\left( \...
4
votes
0answers
61 views

Interpreting the Fourier transform of a Gibbs measure

Recall that a Gibbs measure gives a probability distribution on states $x$ of the form $$ p_X(x) = \frac{1}{Z(\beta)}\exp(-\beta E(x)) $$ As I understand, the function $E$ is interpreted as the ...
4
votes
0answers
100 views

Non-Hermiticity when Fourier transforming onto a finite lattice

I'm doing numerical simulations. I have the Haldane model in a honeycomb lattice where $$ H = \sum \limits_{<ij>}a^\dagger_i b_j + h.c $$ Where $i$ belongs to sublattice $A$, and $j$ to ...
3
votes
0answers
68 views

Linear KDV eq. asymptotics

The question arises from the book Solitons by P. G. Drazin about the linearized KDV eq. $$ u_t+u_{xxx}=0 $$ My first step was to take a Fourier transform of the equation, find that the dispersion ...
3
votes
0answers
927 views

Questions about Michelson interferometer

I have been doing experiment on Michelson experiment, but I don't quite understand why white light results in an interferogram with very few fringes, and why are they necessarily Gaussian? I know that ...
2
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0answers
75 views

Wilsonian Renormalisation — Peskin & Schroder Sect. 12.1

I'm working my way through Peskin & Schroeder, but some of the details of the calculations done in their introduction to the renormalisation group are slipping past me. For concreteness, the ...
2
votes
0answers
133 views

Particle density in k-space

Question: Given a many particle wave function $|\Psi\rangle$, how can I calculate the occupation numbers in k-space? Setup: I have a 1D chain of molecules which contains 4 sites per unit cell. Let'...
2
votes
0answers
44 views

What is abbe's rule in optics?

I have read wikipedia but can't really understand what they mean to say. The usual explanations are given in terms of Fourier optics, which I don't yet have the background for. Can anyone explain it ...
2
votes
0answers
36 views

What is the meaning of the Fourier Transform of the electric field of a well guided mode in a dielectric waveguide?

I have been studying waveguiding in dielectrics for a while now; however, I cannot understand the meaning of the Fourier transformed electric field. I will first give some background information. ...
2
votes
0answers
35 views

What is the shape of the MTF curve in coherent imaging?

For incoherent imaging, the shape of the diffraction-limited MTF curve would look roughly like a triangle, with normalized contrast starting at 1 for zero spatial frequency and decreasing to 0 at the ...
2
votes
0answers
64 views

Intervalley scattering in graphene in presence of impurities

A long range impurity like coulomb impurity does not induce an inter valley scattering between the two Dirac points. Is there any mathematical explanation for the same although this is explained by ...
2
votes
0answers
69 views

$i\epsilon$ versus $2i \epsilon E_k$ in the propagator

The Fourier Transform of the propagator can be written as $$\tilde{\Delta}(k) = \frac{i}{k^2-m^2+i\epsilon} \tag{1} $$ which is then "factored" into $$ = \frac{i}{\left( k^0-E_k +i\...
2
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0answers
29 views

How to mathematically model a realistic aperture illumination?

I want to know a mathematical expression that I can use to model a realistic aperture illumination to produce the primary beam of an antenna so that the radial distribution of this aperture ...
2
votes
0answers
50 views

How is translational symmetry related to Fourier decomposition?

The book (The Cosmic Microwave Background By Ruth Durrer) about cosmological perturbations says that because of translational symmetry of the background at a constant time, we can decompose our ...
2
votes
0answers
135 views

Autocorrelation function corresponding to density of states with significant rotational motion

Most statistical physics textbooks (at least the ones I've found) state simply that the density of states of a system can be found as the temporal Fourier transform of the velocity autocorrelation ...
2
votes
0answers
218 views

Schrodinger Wave Functional (quantum fields) - Solving Functional Gaussian Integrals

Okay, So i'm doing some research that involves the Schrodinger representation in quantum field theory. The ground state wave functional for the Klein Gordon field is a generalized gaussian in position ...
2
votes
0answers
239 views

Feynman propagator with general $\xi$ parameter

Hey from my notes in my PS book it seems I have solved this some time in the past, but I cannot seem to get the indices straight this time around. So in deriving the Feynman photon-propagator which ...
2
votes
0answers
121 views

Momentum representation of a state

I am trying to figure out the momentum representation of the state which has the properties $$\langle \psi |\hat q |\psi \rangle=-q_0,$$ $$\langle\psi|\hat p|\psi \rangle=p_0, $$$$\Delta q\Delta p=\...
2
votes
0answers
232 views

Fourier Transform of ribbon's beam Electric Field

I have a monochromatic ribbon beam with $E(x)e^{i(kz-\omega t)}$ being the electric field's amplitude. I want to show that the lowest order approximation in terms of plane waves is $\mathbf{E}(x,z,t)=...
2
votes
0answers
244 views

Discrete sum over an exponential with imaginary argument, considering only every second lattice site?

Let's say I sum an exponential function $e^{\imath \left(k-k^{\prime}\right) x_{i}}$ over a chain system where every member of the chain is of the same type, e.g., A-A-A-...-A-A (total of N sites) ...
1
vote
0answers
18 views

Help with two dimensional polar axis Fourier transform

This is a problem that I met in real-life physics research. This question is related to Wick's theorem. The question is: 1. In two dimensional plane with polar axis, why do we have the following ...
1
vote
0answers
32 views

Radial Distribution Function - Structure Factor relation, deriviation help

I'm attempting to prove the relation between the structure factor and the RDF, following the deriviation here (pg 92-94). The solution this source comes too disagrees with this paper which I trust ...
1
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0answers
39 views

Lippmann-Schwinger equation and time dependence

Consider the Lippmann-Schwinger equation (LSE) $$ |\psi\rangle = |\phi\rangle + \hat{G}_0(\epsilon) \hat{V} |\psi\rangle \tag{1}$$ where $\hat{G}_0(\epsilon) = \frac{1}{\epsilon - \hat{H}_0 + i\eta}$...
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0answers
30 views

Time-domain NMR or: When is the Fourier-Transformation not appropriate?

My question has two parts: One is general and has to do with the Fourier-Transformation, one has to do with Time-Domain NMR. Both parts are interlinked, of course. I tried to find out, why people do ...
1
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0answers
39 views

Comoving and physical momentum in a Friedmann universe

It is most probably a very basic question, but I'm a bit stuck with it. Let us consider a spatially flat Friedmann universe with the usual metric $$ds^2=dt^2-a^2(t)\left(dr^2+r^2d\vartheta^2+r^2\sin^...
1
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0answers
67 views

Coulomb potential of a periodic crystal in reciprocal space

Usually the Coulomb potential (electron-electron interaction) can be Fourier transformed (aside from prefactors) like that: $$ \frac{1}{|\vec r_1 -\vec r_2|} = \int \frac{\text d ^3 k}{(2\pi)^3} \frac{...
1
vote
0answers
52 views

Discrete Fourier transform for periodic signal?

From the Signal and System textbook, by Oppenheim, I learned that the discrete-time Fourier transform can be written as $$ x[n]=\frac{1}{2\pi}\int_{2\pi}X(e^{j\omega})e^{j\omega n}d\omega $$ $$ X(e^{...
1
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0answers
78 views

Riemannian generalization/adaption of the Hubbard–Stratonovich transformation

I'd like to write the Hubbard–Stratonovich (HS) transformation of a scalar function on a Riemannian manifold. This transformation is quite simple in Euclidean space. One can consider it as a Fourier ...
1
vote
0answers
237 views

Fourier transform of Coulomb potential in 1D

The Fourier transform of the Coulomb potential $V(r)=\frac{k}{r}$ is typically evaluated by computing the Fourier transform of the Yukawa potential given by $V_{Yukawa}=\frac{ke^{-\epsilon r}}{r}$ and ...
1
vote
0answers
43 views

Sum in the reciprocal lattice

I have to use this property but I don't understand at all the deduction, so I was wondering if someone could help me. We have a crystal lattice with vectors to each atom from one of them $R_j$, and ...
1
vote
0answers
66 views

Can I calculate the form of the aperture from the diffraction pattern?

As I understand, the Fraunhofer diffraction pattern of light is the Fourier transform of the aperture. More precisely, the amplitude of light would be the Fourier transform and the intensity its ...
1
vote
0answers
34 views

Why isn't there a different phase after fourier transformation in two lattices

I am trying to understand some solutions for graphenes energy dispersion. While most of it is clear, I don't get one step, when changing into k-space. Consindering two sublattices A and B with ...
1
vote
0answers
436 views

Decoupling the Hamiltonian by a Discrete Fourier transform

For $N$ coupled oscillators(periodic BC) whose Hamiltonian is given as $H=\sum\limits_{i=1}^N (\frac{p_i}{2m} + \lambda(x_{i+1} - x_i)^2)$ decoupling can be achieved by change of variables by using ...
1
vote
0answers
100 views

What excactly is a “fourier component of a density fluctuation”?

Light scattering texts say depending on the scattering angle, you are seeing a certain fourier component of a density fluctuation. This density fluctuation varies sinusoidally due to Brownian motion ...
1
vote
0answers
402 views

Transition Between Position and Momentum Basis

I'm having some trouble following pages 55-56 of Sakurai's Modern Quantum Mechanics. We're trying to transfer from position space into momentum space. Here's a quote: Let us now establish the ...
1
vote
0answers
117 views

Quick question on convolution - Diffraction through a pair of slits

We know that the fourier transform of the amplitude function (in terms of $y$) gives you the amplitude function (in terms of $\theta$) Consider a pair of triangular slits: Fourier transform of G(y)...
1
vote
0answers
65 views

Is having full information about the resonances of a rigid body equivalent to having full information about its material parameters?

Lets say I have a mechanical system whose mechanical resonances (mode shape and frequency) I can measure with perfect accuracy. Is this theoretically equivalent to knowing the materials parameters, ...
1
vote
0answers
94 views

Interpretation in Fourier-Laplace domain

The Green's function describing the distribution of particles sent from $\def\v#1{\boldsymbol{#1}}\def\u#1{\hat{\v#1}}\v r=0$ at $t=0$ uniformly in every directions is, in two dimensions $$ G(r,t)=\...
1
vote
0answers
111 views

Periodic sequence with exponentially increasing period?

I have to develop a physical model for a certain type of biological oscillation that can be built upon periodic sequences. From earlier questions I know that any periodic sequence (containing $0$s ...
1
vote
0answers
202 views

Splitting light into colors, mathematical expression (fourier transforms)

I am trying to solve a problem that includes a function of the light hitting a certain area. My question is, how would I change a function $G(x)$ of photons hitting a certain area to include just ...
0
votes
0answers
8 views

How do optical anti-aliasing filters work from a frequency domain perspective

To prevent aliasing, caused by the finite number of pixels on a sensor, a blurring filter is commonly used. How does that work from a frequency domain perspective? What is the transfer function of ...
0
votes
0answers
11 views

Fourier transform of a fluids kinetic energy equation

I'm currently looking at the 2D equations for a fluid and want to examine the kinetic energy equation in Fourier space. I start with the equation for momentum $$ \frac{\partial\textbf{u}}{\partial t} ...
0
votes
0answers
62 views

Confused about the substitution of the fermionic operators with their Fourier transform in an adiabatic Hamiltonian

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring. To compute the complexity of the algorithm ...
0
votes
0answers
56 views

Fourier transform in two dimensions, Green's function for Schrodinger equation

I want to calculate this Fourier transform: $$ \int\limits_0^{\infty} \mbox{d}k \int\limits_0^{2\pi} \mbox{d}\varphi~ k \frac{e^{i \vec{k} \cdot(\vec{x}-\vec{x}')}}{k^2+B} $$ which will be 2D Green's ...
0
votes
0answers
23 views

How can I calculate the time domain correlation function in the frequency domain?

I have an operator $a(t)$, its correlation function is: $$<a(t)a^{\dagger}(t^{\prime})>=\delta(t-t^{\prime})$$ Now I need to find the correlation function in the frequency domain. i.e. to ...
0
votes
0answers
11 views

Relation between phase of dark field and bright field images

I am trying to understand how the super-resolution technique based on Fourier Ptychography 1. In the paper, we run phase retrieval algorithms using images captured using illumination at different ...
0
votes
0answers
26 views

Weinberg Cosmology book Ch 5.2

I am working on Weinberg Cosmology book, and have one question about what contained in Ch 5.2 (page 229). Basically, this chapter is dealing with stochastic initial conditions. What he wrote is that ...
0
votes
0answers
30 views

How to get the vibratonal frequency of a bond using FFT of velocity autocorrelation function?

I guess there is some errors in the way I am calculating VAC since I ma ending up with a peak whose frequency is two times the actual frequency. I ran an MD simulation long enough with 60 molecules of ...
0
votes
0answers
48 views

Using Plancherel's theorem on delta function

Plancherel's Theorem states that for $f \in L^{2}(\mathbb{R})$ we have $$f(x) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty}F(k)e^{ikx}dk \Longleftrightarrow F(k) = \frac{1}{\sqrt{2 \pi}}\int^{\...
0
votes
0answers
20 views

Eigenvalues for correlation matrix which have the form of an harmonic function

I am trying to understand the written in the picture below. I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\...