The fourier-transform tag has no wiki summary.
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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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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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139 views
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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57 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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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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47 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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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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52 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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1answer
38 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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53 views
How to solve following equation (Yukawa field)?
By using Lagrangian for Yukawa interaction,
$$
L = -\frac{1}{c}A_{\alpha}j^{\alpha} + \frac{1}{8 \pi c}(\partial_{\alpha}A_{\beta})(\partial^{\alpha}A^{\beta}) + ...
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171 views
Fourier Transform on a Riemannian Manifold
The question is quite simple: What would be the definition of Fourier Transform (and it's inverse) on a Riemannian Manifold?
I've found that a similar question has been asked at Mathematics.SE but ...
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Weird integration of gaussian wave packet
I have been learning Fourier transformation of a gaussian wave packet and i don't know how to calculate this integral:
In the above integral we try to calculate $\varphi(\alpha)$ where $\alpha$ is ...
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114 views
Fourier transform between $x$ and $p$
On this page right at the top they mention two sets of fourier transform. First set is connection between $x$ (position) and $k$ (wave vector) space:
$$
\begin{split}
f(x) &= ...
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353 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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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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237 views
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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468 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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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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142 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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1answer
346 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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3answers
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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 ...
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1answer
193 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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254 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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200 views
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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682 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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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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553 views
Canonical Commutation Relations
Is it logically sound to accept the canonical commutation relation (CCR)
$$[x,p]~=~i\hbar$$
as a postulate of quantum mechanics? Or is it more correct to derive it given some form for $p$ in the ...
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Splitting light into colors, mathematical expression (fourier transforms)
I am trying to solve a problem that includes a function of the light hitting a certain area. My question is, how would I change a function $G(x)$ of photons hitting a certain area to include just ...
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232 views
Evaluating propagator without the epsilon trick
Consider the Klein–Gordon equation and its propagator:
$$G(x,y) = \frac{1}{(2\pi)^4}\int d^4 p \frac{e^{-i p.(x-y)}}{p^2 - m^2} \; .$$
I'd like to see a method of evaluating explicit form of $G$ ...
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332 views
What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?
I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following:
$|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$.
$|p\rangle$ is an eigenvector of ...
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150 views
Energy stored in space/frequency electric field
I've come across a problem with finding the energy stored in time/frequency electric field. In space/time we have (taking $\epsilon = 1$)
$$ Energy = \frac{1}{2} \int_V |\mathbf{E}(\mathbf{x},t)|^2 ...
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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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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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357 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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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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355 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
220 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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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:
...
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The appearance of volume $V$ in the Fourier series representation of a periodic cubic system
In the textbook Understanding Molecular Simulation by Frenkel and Smit (Second Edition), the authors represent a function $f(\textbf{r})$ (which depends on the coordinates of a periodic system) as a ...
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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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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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3answers
177 views
A question from Srednicki's QFT textbook
I have a question in Srednicki's QFT textbook.
In order to compute the vacuum to vacuum transition amplitude given by :
$$\left \langle 0|0 \right \rangle_{J}~=~\int \left [ d\varphi \right ]e^{i\int ...
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3answers
311 views
Does the Fundamental Frequency in a Vibrating String NOT Necessarily Have the Strongest Amplitude?
I am doing some experiments on musical strings (guitar, piano, etc.). After performing a Fourier Transform on the sound recorded from those string vibrations, I find that the fundamental frequency is ...
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Physics of a guitar
I understand that when you pluck a guitar string, then a bunch of harmonic frequencies are produced rather than just the frequency of the desired note.
If this is true, why does C2 sound so different ...
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What should the amplitude be when plotting 1-sided Amplitude Spectrum?
I have a continuous signal $x(t)$ such that
$$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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2answers
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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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Optics of the eye - do we see Fourier transforms?
I've recently been learning about Fourier optics, specifically, that a thin lens can produce the Fourier transform of an object on a screen located in the focal plane.
With this in mind, does the ...
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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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Describing quantum intereference with only currents and densities
I know about and believe to understand the general wave equation based Kirchhoff diffraction formula, which in the Fraunhofer limit leads to a farfield complex wave function by Fourier transforming ...
