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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8 votes
1 answer
1k views

What is the position-space form of the photon propagator in a general $R_{\xi}$ gauge?

I’m interested in the form of the photon propagator in position space when expressed in a general $R_{\xi}$ gauge. The integral representation of this propagator is usually written as the sum of two ...
0 votes
0 answers
24 views

How to solve Fresnel diffraction integral of vortex beam

Solving Fresnel integral for vortex beam diffraction is necessary and difficult. Without the necessary integral formula, this isn't easy to derive. Therefore, I need some means to calculate it ...
0 votes
0 answers
44 views

How to write a general state in Fock space in the form of a series of products of $b_k^\dagger$s on a vacuum state?

Consider the state in bosonic system with $N$ sites with periodic boundary condition $$|n_1,n_2,\cdots,n_{N/2}>=\left(\prod_{i\in\{0,\pm 1,\dots,\pm N/2\}}\frac{(b^\dagger_i)^{n_i}}{\sqrt{n_i!}}\...
0 votes
4 answers
18k views

Why frequency is inversely proportional to time-period?

Why frequency is inversely proportional to time-period? While studying about Fourier transform that shows frequency representation. A doubt that came to me was a set of signal with same wavelength but ...
6 votes
1 answer
451 views

Cauchy boundary conditions and Greens functions with Fourier transform

Consider the wave equation \begin{align} (\partial_t^2-\nabla^2)\phi(t,x)=0, \end{align} with Cauchy boundary conditions $\phi(0,x)=f(x)$ and $\dot{\phi}(0,x)=g(x)$. Suppose we perturb the system with ...
0 votes
1 answer
626 views

What is the point of a reciprocal space?

This is not my field, but I need to have some surface level knowledge about the topic, the main thing I need to understand is this part, which is from a paper about linear buckling in microstructures: ...
0 votes
1 answer
163 views

Inverse of an operator [closed]

I want to understand how to find the Inverse of an operator. I know it involves the use of Green's function but I can't seem to figure out how. Here is the actual problem: On page 302 of Peskin&...
1 vote
1 answer
65 views

The equation for linear combinations of sinusoidal waves

I was reading Introduction to Electrodynamics by D.J Griffith and in the chapter of Electromagnetic waves, it gives an equation for the representation of any wave in terms of sinusoidal waves. This is ...
1 vote
1 answer
452 views

Explicit expressions for the creation and annihilation operators

What are the explicit expressions for the creation and annihilation operators $\hat{a_\vec p}$ and $\hat{a}^{\dagger}_\vec p$ for bosons? I can't find them anywhere, as every source seems to introduce ...
0 votes
1 answer
73 views

Confusion on the signs in the complex scalar field [closed]

I saw there are different ways we can write down the complex scalar field. For example, in most textbooks I can find, this is defined as $$\phi(x) =\int \dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2E_p}}\...
0 votes
2 answers
281 views

Solving forced harmonic oscillator differential equation using fourier transform

I am trying to solve the equation of a forced harmonic oscillator using Fourier Transform. I know that if a function $f(t)$ is such that $\lim_{x->\pm \infty} f(t) = 0$, then $$\frac{1}{\sqrt{2\pi}}...
2 votes
1 answer
258 views

How to set mask size when apply inverse-FFT to the power spectrum of HAADF-STEM image?

Hello everyone, I want to characterize dislocation in my FIB sample using Titan electron transmission microscopy and I want to use FFT filter to process my HAADF-STEM image with DigitalMicrograph (...
0 votes
2 answers
79 views

Physical meaning of Fourier transform of electric field

I have recently been thinking about what is the physical meaning of the Fourier transform of an electric field, and in particular, why does it seem to have different units in frequency and time ...
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0 answers
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Physical meaning of the decoupling of the transverse velocity field in linearized Navier-Stokes equation

The Navier-Stokes equations: \begin{equation} \begin{split} \partial_t \rho &= -\vec \nabla \cdot (\rho \vec u) \\ \partial_t(\rho \vec u) + \vec \nabla \cdot (\rho \vec u \...
1 vote
0 answers
280 views

Help With Derivations [closed]

I am currently studying Quantum Field Theory and am using the textbook Overview of Quantum Field Theory by Peskin and Schroeder. Equation (2.31) derives the total Hamiltonian of the Klein Gordon field,...
0 votes
1 answer
206 views

How to compute the Feynman propagator for the Proca field?

I was repeating each step of the exercise 6.4 of the Greiner's book "Field quantization" when I discovered that there is a passage which I can't reproduce, the calculations are lengthy and ...
1 vote
3 answers
211 views

Intuitive and/or qualitative meaning of Heisenberg's uncertainty principle

I'm trying to figure out what the uncertainty principle really means, and I've arrived to construct this 'mental experiment': let's suppose to know with great precision the momentum of an electron (or ...
0 votes
0 answers
52 views

Scaling relation of potential energy per particle of Rashba Spin Orbit Coupled (SOC) 2D electron gas for short and long range interactions

I am trying to prove the scaling relations of potential energy per particle($V(\vec{r}) = \frac{V_0}{|\vec{r}|^\alpha}$) of Rashba Spin Orbit Coupled (SOC) 2D electron gas in uniform liquid phase for ...
0 votes
0 answers
92 views

Schwinger proper-time representation of Feynman propagator in Coordinate space

I'm working on a QCDSR paper which calculates mass of $B$ meson in the present of external magnetic field. the author works with Schwinger proper-time representation of Feynman propagator in momentum ...
2 votes
1 answer
1k views

Why are particles still a thing? [closed]

Couldn't we just assume that waves have mass and momentum and can become localized? Dirac Deltas can be given a rigorous mathematical foundation but physicist do not use the Gelfand triple. Why not ...
0 votes
0 answers
90 views

Total momentum operator for the KG field

This question pertains to Equation (2.33) in Peskin and Schroeder: $$ \hat{\vec P}=-\int d^3\!x\,\hat\pi(\vec x)\vec\nabla\hat\phi(\vec x)=\int d^3\!p\,\vec p\,\hat a_{\vec p}^\dagger\,\hat a_{\vec p} ...
1 vote
0 answers
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How did Cohen-Tannoudji (QM) manipulated this? [closed]

In the first chapter of Cohen-Tannoudji’s Quantum Mechanics Book, there's this part of the chapter called "Time evolution of a free wave-packet". So one starts with the wave packet given by $...
1 vote
2 answers
159 views

Diffraction limit PSF and angular spectrum method?

I know from the angular spectrum method that given a field $U$ with a wavelength $\lambda$, we can decompose it with Fourier transform. \begin{equation} U(x, y,0) = \int \int {\tilde{U}_0(f_x,f_y)} ...
0 votes
1 answer
63 views

Doubt on time invariant system

Now I am delaying the output of a system (which takes $x \left( t \right)$ as input and gives $t \cdot x \left( t \right)$ as output) by $T$ then final output is: Let's denote the output of the ...
1 vote
1 answer
294 views

How can infinite sine waves localize to a single pulse in space? [duplicate]

I have heard countless times (and not just when discussing the Heisenberg's Uncertainty Principle) that making a short pulse using sine waves requires more and more sine waves to localize the pulse ...
1 vote
2 answers
154 views

Why can $ϕ(p)$ be Fourier expanded to $ψ(x)$ in quantum mechanics? [closed]

I know the Fourier transform is $$ F(\omega)=\int_{-\infty}^{\infty} f(x) e^{-i \omega x} \,d x $$ $$ f(x)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) e^{i \omega x} \,d \omega, $$ but in ...
-1 votes
2 answers
75 views

Cohen Quantum Mechanics Derivation? [closed]

I dont understand the argument on page 38 eq. (C-6) of Cohen's quantum mechanics. Could someone break down for me what is $g(k)$? Is it the initial condition?
1 vote
0 answers
58 views

Intuition behind the math of Diffraction

Is there an intuitive way to understand why the diffraction patterns arising from an arbitrary aperture is Mathematically the Fourier Transform of this aperture function?
0 votes
2 answers
128 views

Does the momentum operator applied to a position state vanish?

In quantum mechanics we have \begin{equation*} \langle x|p\rangle=C\exp\left(\frac{ipx}{\hbar}\right) \end{equation*} where $C$ is a normalization constant. It follows that \begin{equation*} -i\hbar\...
0 votes
0 answers
47 views

Connection between the Modulation Transfer Function (MTF) and Line Spread Function (LSF) in "Hecht - Optics, 5th ed"

I try to understand the derivation of the connection of the MTF and the LSF and found a derivation in the Book "Optics" by Eugene Hecht (5th edition), chapter 11, section 11.3.6. However, ...
0 votes
0 answers
21 views

Fourier Transform of Damped Oscillations - Zero Frequency Peak and Shift [duplicate]

A damped oscillator has the time evolution: $$ y(t) = e^{-\Gamma t}\cos^2(\tilde{\omega}_0 t)$$ where $\Gamma$ is the damping rate, $\tilde{\omega}_0^2=\omega_0^2-\Gamma^2$ and $\omega_0$ is the ...
5 votes
1 answer
222 views

Fourier transformation of the inverse Klein-Gordon propagator

On Peskin & Schroeder's QFT, page 30, the scalar field propagator as the retarded Green function is defined as $$(\partial^2+m^2)D_R(x-y)=-i\delta^4(x-y) \tag{2.56}$$ The Fourier transformation is ...
4 votes
2 answers
215 views

Quantization of non-relativistic complex scalar field

I found that the taking the non-relativistic limit of the Lagrgangian for complex scalar fields gives $$\mathcal{L} = i\dot{\psi}\psi^* -\frac{1}{2m}\nabla\psi \nabla\psi^*.\tag{1}$$ Now, when we ...
2 votes
0 answers
46 views

Fourier transform of $1/r$ with plane-wave expansion formula

I know that up to constant factors ($\pi$'s and such), the Fourier transform of the Coulomb potential $$\frac{1}{4\pi}\frac{1}{r}$$ in 3-dimensions is proportional to $$\int d^3\vec r\frac{e^{i\vec k \...
1 vote
1 answer
85 views

Fourier transform of electric quadrupole potential

I'm struggling to find the scattering amplitude by the first Born approximation (Fourier transform) given by $$ f(\vec{k}_f,\vec{k}_i)= -\frac{1}{2\pi}\langle \vec{k}_f | {V}| \vec{k}_i\rangle = -\...
2 votes
1 answer
108 views

How is differential momentum assigned in multiparticle system of QFT?

I've been following Schwartz's book on quantum field theory, and got stuck at page 59 on Section 5.1 'cross section' of the book which argues that the region of final state momenta is the product of ...
3 votes
1 answer
223 views

Problem obtaining Klein-Gordon equation solutions

I am having some problems and also some questions regarding how can one get the general solution to the Klein-Gordon equation, which usually appears in the literature as $$ \phi(t,\mathbf{x})=\int\...
1 vote
0 answers
51 views

SLM pulse shaping to delay (and advance) ultrashort pulses in time

I am trying to replicate some of the experiments discussed in this excellent publication: SLM for pulse shaping In particular, I replicated the setup configuration in Figure 14 and I want to use the ...
2 votes
1 answer
119 views

Massless limit of Dirac fermion correlation functions

In the 2D massless Dirac fermion CFT we have correlation functions like $$\langle J(z,\bar{z})J(0)\rangle \sim \frac{1}{z^2},$$ where in terms of real Euclidean coordinates $x^0,x^1$, we have $z=x^0+...
0 votes
2 answers
314 views

Superimposed Waves

This question has been bothering me for a very long time. Imagine a wire carrying electric current. It carries two alternating current (AC) signals of different frequencies (say $50$ Hz and $60$ Hz). ...
1 vote
0 answers
137 views

Determination of Intermediate Scattering Function in numerical simulation

The Fourier transform of single particle density : $$\rho(\textbf{k},t)=\int\rho(\textbf r,t)\exp(-i\textbf k.\textbf r)d\textbf r=\sum_j \exp(-i\textbf k.\textbf r(t))$$ The Intermediate Scattering ...
1 vote
1 answer
64 views

A standing wave may be expressed as superposition traveling waves - is the converse, traveling as superposition of standing, also true?

So I consider the wave equation: \begin{align*} \frac{\partial^2 u}{\partial t^2} = c^2 \left(\frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \cdots + \frac{\partial^2 u}{\...
0 votes
0 answers
70 views

Fourier transform on 1D bipartite lattice

I have a 1-D lattice Hamiltonian like $$ H=-t\sum_{\langle ij \rangle} (c^\dagger_i c_j + h.c.), $$ I have to do its Fourier transform for 1-D bipartite lattice, I can think of two possible ways of ...
2 votes
0 answers
42 views

How are frequency combs obeying the energy conservation?

The operational principle of frequency combs is that you generate very short pulses (in time domain), and that in the frequency domain (due to Fourier's transform) the spectrum of such pulses is a ...
1 vote
0 answers
79 views

Fourier Transform of Dielectric Function in 2D

According to "Dielectric screening in two-dimensional insulators: Implications for excitonic and impurity states in graphane", https://doi.org/10.1103/PhysRevB.84.085406, the Fourier ...
0 votes
0 answers
80 views

Fourier Transform of temperature Green Function

I am doing a calculation involving a temperature Green function for some operator $\hat{A}$: $$G_{\hat{A}}(\tau)=-\Big\langle{T_\tau\big(\hat{A}(\tau)\hat{A}^\dagger \big)} \Big\rangle=-\theta(\tau)&...
0 votes
3 answers
314 views

Schrödinger equation of the double delta potential in momentum space

This may be a silly question but it has gotten me stumped. I am currently trying to see if I can get anywhere by putting a simple symmetric double delta potential \begin{align} V(x) = -\gamma (\...
0 votes
0 answers
18 views

Scattering from a dirac delta potential in momentum space [duplicate]

I was curious about solving the reflection and transmission amplitudes for the dirac delta potential barrier using fourier transform. I'm able to take the fourier transform but I'm unable to interpret ...
6 votes
1 answer
453 views

Poles for a particle scattered in a delta potential

I am working on problem a professor gave me to get an idea for the research he does, and have hit a point where I'm having a difficult time seeing where I need to go from where I'm at. I would also ...
1 vote
1 answer
113 views

Fourier transform of the top-hat filter [closed]

In the context of a cosmology text, I have found the following function, called a spatial top-hat filter: $$W_{TH,R}(r)=\dfrac{3\theta(R-r)}{4\pi R^3}$$ It is claimed that this leads to: $$W_{TH,R}(k)=...

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