The fourier-transform tag has no wiki summary.
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What's the physical significance of using fourier transform for diffraction?
I am studying some basic idea of diffraction and there mention in far field, the diffraction pattern could be understood by Fourier transform. But I just don't understand what's the physical fact for ...
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6answers
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Fourier transformation in nature/natural physics?
I just came from a class on Fourier Transformations as applied to signal processing and sound. It all seems pretty abstract to me, so I was wondering if there were any physical systems that would ...
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70 views
Prove that the position operator is $\hat{x} = i\hbar \frac{d}{{dp}}$ in the momentum representation [closed]
Proof that: $x = i\hbar \frac{d}{{dp}}$
I did this, could you tell me if I am false or true
$\begin{array}{l}
x{e^{\frac{{ipx}}{\hbar }}} = - i\hbar \frac{{d{e^{\frac{{ipx}}{\hbar }}}}}{{dp}} = ...
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0answers
27 views
Problem with Discrete Parseval's Theorem [migrated]
I think I must be missing something obvious, but I can't for the life of me see what it is. The discrete version of Parseval's theorem can be written like this:
$\sum_{n=0}^{N-1} |x[n]|^2 = ...
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1answer
140 views
What's the average position of oscillating particles in a box with periodic boundary conditions?
Imagine an open box repeating itself in a way that a if a particle crossing one of the box boundary is "teleported" on the opposite boundary (typical periodic boundary position in 3D).
Now put a ...
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2answers
51 views
Convolution kernel of poisson equation by FFT
I'm trying to solve poisson equation using FFT. In genral it is a convolution of the charge density with potential well of point charge ( Green's function of laplace equation ) which is $1/r$
I'm ...
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1answer
36 views
Why pulse waves results in wave packets?
I was doing experiments of measuring sonic velocity and I generate pulse waves from sensor 1, but when they are received by sensor 2, I saw wave packets on the oscilloscope, can you explain why?
I was ...
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0answers
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Difficulty in obtaining the Lorentzian lineshape for natural broadening [migrated]
Not sure if this maybe belongs more in the maths section, but since it comes from a physics problem i'll post here.
when calculating the natural broadening lineshape for a laser we have to take the ...
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2answers
148 views
Energy stored in space/frequency electric field
I've come across a problem with finding the energy stored in time/frequency electric field. In space/time we have (taking $\epsilon = 1$)
$$ Energy = \frac{1}{2} \int_V |\mathbf{E}(\mathbf{x},t)|^2 ...
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0answers
46 views
Fourier Transform of ribbon's beam Electric Field
I have a monochromatic ribbon beam with $E(x)e^{i(kz-\omega t)}$ being the electric field's amplitude. I want to show that the lowest order approximation in terms of plane waves is
...
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2answers
140 views
A four-dimensional integral in Peskin & Schroeder
The following identity is used in Peskin & Schroeder's book Eq.(19.43), page 660:
...
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What should the amplitude be when plotting 1-sided Amplitude Spectrum?
I have a continuous signal $x(t)$ such that
$$x(t)=12\cos(6\pi t)+6\cos(24\pi t)+3\cos(30 \pi t)$$
and is asked to sketch a 1-sided Amplitude Spectrum of the signal $x(t)$ if sampled above the ...
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51 views
Definition of frequency domain coordinates
I am using the Fourier Transform in Optics to perform differentiation with a filter by making use of the relation
$\frac {\partial}{\partial x} f(x)=2\pi i \int^{\infty}_{-\infty} u F(u) \exp (2i\pi ...
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0answers
53 views
How to solve following equation (Yukawa field)?
By using Lagrangian for Yukawa interaction,
$$
L = -\frac{1}{c}A_{\alpha}j^{\alpha} + \frac{1}{8 \pi c}(\partial_{\alpha}A_{\beta})(\partial^{\alpha}A^{\beta}) + ...
5
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1answer
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Fourier Transform on a Riemannian Manifold
The question is quite simple: What would be the definition of Fourier Transform (and it's inverse) on a Riemannian Manifold?
I've found that a similar question has been asked at Mathematics.SE but ...
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1answer
148 views
Weird integration of gaussian wave packet
I have been learning Fourier transformation of a gaussian wave packet and i don't know how to calculate this integral:
In the above integral we try to calculate $\varphi(\alpha)$ where $\alpha$ is ...
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3answers
997 views
What is the relation between position and momentum wavefunctions in quantum physics?
I have read in a couple of places that $\psi(p)$ and $\psi(q)$ are Fourier transforms of one another (e.g. Penrose). But isn't a Fourier transform simply a decomposition of a function into a sum or ...
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1answer
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Fourier transform between $x$ and $p$
On this page right at the top they mention two sets of fourier transform. First set is connection between $x$ (position) and $k$ (wave vector) space:
$$
\begin{split}
f(x) &= ...
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3answers
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Can Laplace's equation be solved using Fourier transform instead of Fourier series?
Sorry for the long text, but I am unable to make my question more compact.
Any periodic function can be Fourier expanded. Usually, they say in mathematical physics books, if the function is not ...
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1answer
326 views
Gaussian wave packet
At our QM intro our professor said that we derive uncertainty principle using the integral of plane waves $\psi = \psi_0(k) e^{i(kx - \omega t)}$ over wave numbers $k$. We do it at $t=0$ hence $\psi = ...
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1answer
67 views
Dynamic structure factor
Dynamic structure factor is the spatial and temporal Fourier transform of Van Hoves time dependent pair correlation function. It is written as
$$ S(k,\omega)= \frac{1}{2\pi}\int F(k,t)\exp(i\omega ...
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2answers
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Why are AC quantities represented by sine waves always?
Usually we use a sinusoidal wave form to represent a alternating quantity. Why not a cosinusoidal wave or a ramp wave form?
In sine wave forms we can indicate the maximum and minimum amplitude and ...
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3answers
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How to design an experiment that shows that a rectangular pulse can be expressed as a series of infinite sinusoids?
Is it possible to design a physical experiment that shows that a time limited signal, such as a rectangular pulse is composed of infinite continuous sine/cosine waves?
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3answers
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Why use Fourier expansion in Quantum Field Theory?
I have just begun studying quantum field theory and am following the book by Peskin and Schroeder for that.
So while quantising the Klein Gordon field, we Fourier expand the field and then work only ...
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649 views
What is the significance of negative frequency in Fourier transform?
What is the significance of negative frequency in Fourier transform? Why we include the band widths of the negative frequency also while calculating band width of the signal.
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3answers
421 views
Magnitude of the Fourier Transform of White Noise
Say you have two white noise signals with different variation amplitudes A1 and A2 as shown in this beautiful Excel graph:
Ignoring the DC offset as it's been represented here, how do you relate ...
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1answer
326 views
Physical Significance of Fourier Transform and Uncertainty Relationships
What is the physical significance of a fourier transform?
I am interested in knowing exactly how it works when crossing over from momentum space to co ordinate space and also how we arrive at the ...
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4answers
589 views
Intuitive explanation of why momentum is the Fourier transform variable of position?
Does anyone have a (semi-)intuitive explanation of why momentum is the Fourier transform variable of position?
(By semi-intuitive I mean, I already have intuition on Fourier transform between ...
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1answer
217 views
Is there a relation between quantum theory and Fourier analysis?
These days I was studying the quantum theory.I found that some theories about that is similar to Fourier Transform theory.For instance, it says "A finite-time light's frequency can't be a certain ...
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3answers
114 views
What information is stored on gramaphones/tape recorders/CDs/DVDs
I'm a Software Developer by profession and my physics knowledge is limited what I had learned at high school level. Please excuse me if the question is trivial.
Question:
From what I know, a sound ...
7
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1answer
352 views
Calculating diffraction patterns using FFT
I'm trying to write a piece of code that calculates a diffraction pattern similar to an X-ray experiment using a FFT.
From my knowledge, the diffraction pattern for point particles can be calculated ...
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1answer
186 views
What would we never know in Physics if Fourier Transform were not discovered? [closed]
I am still unsure if Fourier Transform has any fundamental significance in Physics. Is it anything more than a calculation tool? For example sometimes people Fourier transform an equation to solve it ...
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1answer
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Fourier Methods in General Relativity
I am looking for some references which discuss Fourier transform methods in GR. Specifically supposing you have a metric $g_{\mu \nu}(x)$ and its Fourier transform $\tilde{g}_{\mu \nu}(k)$, what does ...
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2answers
249 views
Was uncertainty principle inferred by Fourier analysis?
I would like to know: did Heisenberg chance upon his Uncertainty Principle by performing Fourier analysis of wavepackets, after assuming that electrons can be treated as wavepackets?
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1answer
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Intuition behind Fourier transformed spaces
Intuitively I've been able to understand a Fourier transform a change-of-basis formula - you're basically moving from position to momentum basis or from time to frequency - but what does it mean that ...
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2answers
136 views
Does light have timbre?
Timbre is a property associated with the shape of a sound wave, that is, the coefficients of the discrete Fourier transform of the corresponding signal. This is why a violin and a piano can each play ...
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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 ...
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3answers
544 views
Canonical Commutation Relations
Is it logically sound to accept the canonical commutation relation (CCR)
$$[x,p]~=~i\hbar$$
as a postulate of quantum mechanics? Or is it more correct to derive it given some form for $p$ in the ...
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2answers
231 views
Evaluating propagator without the epsilon trick
Consider the Klein–Gordon equation and its propagator:
$$G(x,y) = \frac{1}{(2\pi)^4}\int d^4 p \frac{e^{-i p.(x-y)}}{p^2 - m^2} \; .$$
I'd like to see a method of evaluating explicit form of $G$ ...
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Effect of a wavefront deformation on the far-field diffraction pattern of a TEM00
By performing Matlab simulations on a TEM00 mode (approximated by a gaussian intensity profile with a flat wavefront), I got the impression that applying wavefront deformations (such as a single ...
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2answers
176 views
Measurement and uncertainty principle in QM
The Wikipedia says on the page for the uncertainty principle:
Mathematically, the uncertainty relation between position and momentum arises because the expressions of the wave function in the two ...
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2answers
474 views
What is the specific meaning of “Fourier frequency” (as opposed to simply “frequency”)?
I've noticed that many journal articles (in optics) use the phrase "Fourier frequency" to describe, well, the frequency of something.
Google scholar search for "Fourier frequency".
Example:
...
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394 views
Very simple example of the way the Fourier transform is used in quantum mechanics?
According to a book I'm reading, the Fourier transform is widely used in quantum mechanics (QM). That came as a huge surprise to me. (Unfortunately, the book doesn't go on to give any simple examples ...
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2answers
167 views
Modeling stochastic process with frequency-dependent power spectrum
I'm trying to model of Johnson-Nyquist noise propagation in a nonlinear circuit. An ideal (linear) resistor can be modeled very nicely by the Fokker-Planck equation (equivalently, the drift-diffusion ...
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0answers
97 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)
...
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2answers
351 views
Field theory:functional derivative involving Fourier Transform
I have to solve the following functional derivative
$$
\frac{\delta}{\delta \Lambda(\mathbf{x})}\log[A-\mathbf{k}^2\Lambda(\mathbf{k})]
$$
where $\Lambda(\mathbf{k})$ is the Fourier transform of ...
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2answers
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What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?
I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following:
$|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$.
$|p\rangle$ is an eigenvector of ...
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1answer
331 views
Finding $\psi(x,t)$ for a free particle starting from a Gaussian wave profile $\psi(x)$
Consider a free-particle with a Gaussian wavefunction,
$$\psi(x)~=~\left(\frac{a}{\pi}\right)^{1/4}e^{-\frac12a x^2},$$
find $\psi(x,t)$.
The wavefunction is already normalized, so the next thing to ...
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3answers
173 views
Acausality in solving time-domain inhomogeneous differential equations with Fourier transforms?
I was always wondering about the acausal nature of solutions obtained by Fourier transforms in the case of inhomogeneous equations. The solution usually revolves around the integration of the ...
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2answers
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Can the equation $v=\lambda f$ be made true even for non sinusoidal waves?
The known relation between the speed of a propagating wave, the wave length of the wave, and its frequency is
$$v=\lambda f$$
which is always true for any periodic sinusoidal waves.
Now consider:
...
