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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25 views

The proof of a discrete Fourier identity in quantum field theory

On page 25, in the book Quantum Field Theory for the Gifted Amateur by Tom Lancaster and Stephen. J Blundell, it states the following: We impose periodic boundary conditions forcing ...
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1answer
19 views

Fourier transform of a set of L fermions operators

I have a set of L fermion creation and annihilation operators: $\lbrace{\hat{C}^+_1,...,\hat{C}^+_L\rbrace}$ and $\lbrace{\hat{C}^-_1,...,\hat{C}^-_L\rbrace}$. Every $\hat{C}^+_l,\hat{C}^-_l$ ...
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0answers
11 views

Why are reciprocal lattice vector periodic, and time-frequency not?

k-space vectors are related to each other by $k=k'+G$, where $G$ is the reciprocal lattice vector $G=2\pi/a$. This means that the frequency of oscillation in real space of a plane wave $e^{ikx}$ is ...
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0answers
39 views

Compare phase spectrum and magnitude spectrum [on hold]

For a given signal,using Fourier transform, we compute it's magnitude and phase spectrum. Now,I want to compare these both spectrums with each other by below points Which information they give ...
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0answers
14 views

Separating sources in a three dimensional space using spherical harmonics?

Suppose I have sensors arranged along a surface that is roughly a hemisphere (half of a sphere). The sensors pick up signals originating from two different positions in a three dimensional space. The ...
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32 views

Definition of Fourier Transform on a Lattice

I am reading a book(EDIT: the book is Czyholls theoretical condensed matter physics, though i am not sure if there is an english version) where for periodic functions $f(x_l+L)=f(x_l)$ the Fourier ...
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1answer
63 views

Propagating higher order Hermite Gaussian modes. What are complex amplitude coefficients?

I've been tasked with writing a code (in MatLab, but I'm currently using Mathematica because I don't know MatLab %\ ...) to simulate the propagation of a Gaussian beam. I don't really know anything ...
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0answers
39 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 ...
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1answer
48 views

Details of the radial Fourier transform pertaining to certain quantum integrals

Consider the integral $$U(t)=\int\frac{d^3p}{(2\pi)^3}e^{-ip^2t/2m}e^{i\vec p\cdot\Delta\vec x}$$ for the free non-relativistic propagator. I'm not quite sure about the gritty details of radial ...
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1answer
34 views

How to compute phases of the signals?

Let us take 4 signals which are sinusoidal and periodic. Suppose you are given a phase spectrum or (/and) equation of the (main) signal only and you are said that the given (main) signal is formed of ...
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2answers
36 views

Normalization of the overlap $\langle x'|p'\rangle$

Let $$\langle x'|p'\rangle = N \exp(\frac{ip'x'}{\hbar})$$ be the overlap between position and momentum space, where $N$ is a normalization constant to be determined. We can then compute $N$ by $$ ...
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2answers
163 views

What is the significance of the Fourier coefficients?

Let us take an example, a white ray (which is composed of bunch of frequency components) is passed through a prism, the ray gets split (decomposed) into its elementary vibgyor colours (i.e.different ...
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1answer
89 views

Have some queries about Fourier transform [closed]

I have some queries about the Fourier transform In most of the cases, the Fourier transform of a signal is symmetric with respect to positive and negative frequency. I think the computational ...
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3answers
64 views

Whether the job of Fourier Transform is just to convert signals from time domain to frequency domain only or more than it?

I am a beginner . We convert a signal in time domain to frequency domain by applying Fourier transform on the signal to obtain frequency and phase spectrum. So,whether the job of Fourier transform ...
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0answers
23 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 ...
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0answers
13 views

Length and spacing autocorrelation with FFT [migrated]

I have calculated, with FFT and A-FFT, the autocorrelation of a discrete signal (coming from an integration) with spacing $\Delta T$ composed by $N$ samples padded to $2N$ with zeroes. What is the ...
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2answers
259 views

The poles of Feynman propagator in position space

This question maybe related to Feynman Propagator in Position Space through Schwinger Parameter. The Feynman propagator is defined as: $$ G_F(x,y) = \lim_{\epsilon \to 0} \frac{1}{(2 \pi)^4} \int d^4p ...
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1answer
34 views

Question about group velocity and travelling waves

I'm trying to learn some basic quantum mechanics and I have a question related to group velocity of a travelling wave. I know there are already a few questions related to group velocity, but I ...
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1answer
69 views

Considering $\langle \underline{q} \mid \underline{p} \rangle=\frac{1}{(2\pi\hbar)^{n/2}}e^{i\underline{q}\cdot\underline{p}/\hbar}$ [duplicate]

I have been given the following complete systems of eigenvectors $$\mathbf{Q}\mid\mathbf{q} \rangle=\mathbf{q}\mid\mathbf{q} \rangle, \quad \mathbf{P}\mid\mathbf{p} \rangle=\mathbf{p}\mid\mathbf{p} ...
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43 views

In Solution of the equation $\Box A_\sigma=0$. I find some mathematical difficulties [closed]

The solution of the equation $$\Box A^\mu(x)=0$$ has the form $$A^\mu(x)=\int\frac{d^4k}{(2\pi)^4} e^{ik^\mu x_\mu}A^\mu(k)\delta(k^2).$$ Where we take $A^\mu(k)$ of the form $$A^\mu(k)=\sum ...
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1answer
69 views

Path integral measure Fourier transformation for case of real field

Let's have $$ Z[J] = \int D \varphi e^{iS[\varphi , J]}, $$ where $\varphi$ denotes real scalar field. Let's make Fourier transform, $$ \varphi (x) = \int e^{iqx}\varphi (q), \quad \varphi^{*} (q) = ...
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2answers
63 views

Derivation of plane wave from inner product of position ket and momentum ket

In textbooks it seems to be taken for granted that $$\langle \mathbf{r}|\mathbf{k}\rangle ~=~ \frac{1}{\sqrt{\Omega}}\exp(i\mathbf{k}\cdot\mathbf{r}).$$ I'm sure it's obvious but is there a ...
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1answer
57 views

Fourier transform of random variables

My question is concerning Fourier transforms of random variables. So if the question itself is too heavy a task but you know of any good resources to learn this topic that would also be very much ...
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1answer
64 views

Why is the position space free particle wavefunction a function of momentum?

This is one of those little things that has always confused me. If someone said to you "in quantum mechanics, the eigenfunctions of a free particle are $\exp(ipx/\hbar)$" how would you know that ...
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23 views

Functions of the form $F(x-ct)$ written as superposition?

In this section of the Wikipedia article on the wave equation they do the following: $$\int^{\infty}_{-\infty}s_+(\omega)e^{-i(kx+\omega t)}d\omega +\int^{\infty}_{-\infty}s_-(\omega)e^{i(kx-\omega ...
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1answer
59 views

The general equation for a wave packet derivation? [closed]

On Wikipedia it gives the general equation for a wave packet (and therefore for a wave?) to be: $$u(x,t)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}A(k)e^{i(kx-\omega t)} dk$$ I have been trying to ...
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1answer
47 views

How do you derive the Dirac equation for momentum space?

$\require{cancel}$ \begin{align} 0 &= i \gamma^\mu \partial_\mu \psi(x) - m \psi(x) \\ &= \int \frac{d^4 k}{(2\pi)^4}e^{-i k x}\left( \gamma^\mu k_m \tilde{\psi}(k) - m \tilde{\psi}(k) ...
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3answers
55 views

Why is wave a function of volts?

I'm looking at a beginner's book on Fourier and waves, and the very first graph shows a periodic wave where the horizontal axis is time (msec) and the vertical axis is something noted as "MAG(V)" ...
2
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1answer
66 views

White noise and Fourier transform

I try to solve a Langevin equation in the Fourier space. My understanding of the white noise in the Fourier space seems to be wrong. Suppose I have a particle with its time evolution of the position ...
0
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1answer
37 views

Fourier series for a wave on an infinite string?

From "Vibrations and Waves" by A.P. French I know that any wave on a string length $L$ can be represented by: $$y(x,t)=\Sigma^\infty_0 A_n \sin(\frac{n\pi x}{L})\cos(\omega_nt-\delta_n)$$ But can we ...
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3answers
100 views

Fourier Transforms Related to Green's Functions

I'm reading a text on field theory where there are a number of assertions made about Fourier transforms that I'm finding confusing. For example, let $G^R = -i \theta(t - t')e^{-i \omega_0 (t - t')}$. ...
4
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1answer
172 views

Analogies between Fraunhofer diffraction and Josephson junctions

There are several analogies between diffraction patterns and Josephson junctions, especially between a double slit experiment and two Josephson junctions in a superconducting ring (like this): Both ...
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1answer
52 views

Using the Fourier transform to find the natural frequencies of coupled oscillators

How can I find the natural frequencies of a system consisting of a pair of coupled oscillators using Fourier transforms? The System consists of two masses and three springs. One of the springs ...
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0answers
61 views

Why do we use Fourier transforms in QFT? [duplicate]

I ask this question, as someone has recently asked me this and I'm not sure I gave them a satisfactory/correct answer. I explained that in QFT we describe particles (and there interactions) in terms ...
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3answers
148 views

Calculating $\langle x | \hat{x} | p \rangle$ in $p$ basis

I am trying to calculate $\langle x\ |\ \hat{x}\ |\ p\rangle$. I can work in the $x$-basis like so: $$\langle x\ |\ \hat{x}\ |\ p\rangle=\int dx'\langle x\ |\ \hat{x}\ |\ x'\rangle\langle x'\ |\ ...
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1answer
49 views

How do I take take the partial derivatives of the general solution to the TDSE for a free particle? [closed]

Consider the general solution to the time-dependent Schrödinger equation for a free particle \begin{align*} \Psi(x,t) &=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} \phi(k) e^{i\left(\hbar ...
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0answers
39 views

Fourier Transforming a $n$-dimensional ket (QM)

I would like to evaluate the Fourier Transform of $n$ functions. I am aware from the derivation of the convolution how this is done for the case of $n=2$. How could this be generalised for $n=3$? ...
0
votes
1answer
57 views

Is this a frequency domain plot for audio? [closed]

I have a program "spectrum" that draws an chart for an audio file (a short .wav with an human voice recorded on it). I believe it is a frequency domain chart. The ...
1
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4answers
136 views

What does “spread of momentum” actually mean?

I was reading Feynman's lecture in which Feynman invoked his own way of explaining the uncertainty principle using single-slit experiment. There I found: To get a rough idea of the spread of ...
3
votes
1answer
68 views

Quantization of a free field: Klein-Gordon case

I am a beginner and reading this course text on QFT. The author first introduces the KG equation: $$\partial_\mu\partial^{\mu}\phi+m^2\phi=0$$ [with Minkowski signature $(+,-,-,-)$]. Then the ...
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0answers
18 views

What is the transfer function in fft beam propagation for unpolarized light?

What is the transfer function in fft beam propagation for unpolarized light ? How to construct the fft beam propagation ? This is for homework. For coherent light the beam propagation is E(x,z) ...
1
vote
1answer
53 views

Why can't we define a unique wavelength for a short wave train? [duplicate]

Here we encounter a strange thing about waves; a very simple thing . . .namely, we cannot define a unique wavelength for a short wave train. Such a wave train does not have a definite wavelength; ...
1
vote
1answer
110 views

Second Quantisation, Fourier Transform, minus sign [closed]

I want to expand a field \begin{equation} \Phi (x) = \int \frac{d^3 p}{(2 \pi)^3} e^{ipx} \end{equation} in terms of the second quantisation \begin{equation} \Phi = \frac{1}{\sqrt{2 E}} (a + ...
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1answer
24 views

a question regarding Fourier transform in electron microscopy

I have recorded a micrograph of a 2-D array at a magnification of 43,000x on my DE-20 digital camera, which has a 6.4 μm pixel size and a frame size of 5120 × 3840 pixels. This magnification is ...
6
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2answers
194 views

Can the momentum eigenstates be non-orthogonal?

Consider the Hilbert space of a particle, whose position domain is confined to $q\in[0,1]$ (e.g. a particle in a box with unit width). Using $$ 1=\int_0 ^1 dq |q\rangle\langle q| $$ and the position ...
0
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1answer
29 views

Inverse Fourier Transfrom of a wavefunction

I was reading about how a Fourier transform yields the wave-function expressed in terms of the momenta which constitute it, i.e. the wave-function in momentum space. I'm not so good at calculus yet ...
0
votes
1answer
108 views

How to prove that the position operator in momentum is $i\hbar \partial/\partial p$ - One Missing Sign [duplicate]

I am trying to prove that the position operator in momentum space is $i\hbar \partial/\partial p$ but my derivation is missing one sign. Can someone spot the error? Start with $$<\hat x> ...
1
vote
1answer
135 views

What intermediate steps of the Dirac Delta Function and Fourier Series am I missing in finding a solution to the Kronig-Penney Model?

Intro We're looking at the Kronig-Penney model in class and one of the conundrums is related to the Kronig-Penney potential for a chain of $N$ atoms. I'm supposed to squeeze out some expression for ...
2
votes
1answer
77 views

What am I REALLY doing when I take the Fourier transform of the momentum operator

I was playing around with some equations and found a surprising relationship when I took the fourier transform of the momentum operator Define $\hat P = \frac{\hbar}{i} \partial_x$, then $F(\hat P) = ...
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0answers
39 views

Measuring typical distance between patches using 2D Fourier Transform

I need to extract information about the typical distance between the black patches in an image like the one I attached here. I tried to perform 2D FFT on it (using OpenCF fdt function in Python), but ...