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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22
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6answers
2k views

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 ...
16
votes
1answer
371 views

Why do quasicrystals have well-defined Fourier transforms?

I was recently reading about quasicrystals, and I was really surprised to learn that even though they do not have a periodic structure, and only have long range order in a very different sense to the ...
10
votes
3answers
724 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 $G$ ...
9
votes
3answers
257 views

Reconstruction of “wavefunction” phases from $|\psi(x)|$ and $|\tilde \psi(p)|$

Consider a "wavefunction" $\psi(x)$, which has a Fourier transform $\tilde \psi(p)$ Suppose that we know, for each $x$, $|\psi(x)|^2$, and that we know, for each $p$, $|\tilde \psi(p)|^2$. Have we ...
8
votes
3answers
408 views

Fourier Transforming the Klein Gordon Equation

Starting with the Klein Gordon in position space, \begin{align*} \left(\frac{\partial^2}{\partial t^2} - \nabla^2+m^2\right)\phi(\mathbf{x},t) = 0 \end{align*} And using the Fourier Transform: ...
7
votes
2answers
532 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 ...
7
votes
4answers
2k views

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 ...
7
votes
2answers
431 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?
7
votes
2answers
215 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 ...
7
votes
1answer
172 views

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 ...
7
votes
1answer
555 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 ...
6
votes
2answers
908 views

How to distinguish female and male voices via Fourier analysis?

What makes one, without looking, be able to identify the gender of the talker as male or female? I mean if we Fourier analysed the voice of males and females, how the 2 spectrums are different which ...
6
votes
1answer
191 views

Integral in $n$−dimensional euclidean space

I've asked this question in Mathematics Stack Exchange, but unfortunately there is no answer yet. I repost it because this integral comes from QFT and maybe someone here did it before or could help ...
6
votes
3answers
1k views

Why is the bispectrum not commonly used in experimental physics?

Power spectra, coherence spectra, and linear transfer functions are ubiquitous tools of experimental physics. However, our instruments often retain small nonlinear effects which can contaminate ...
6
votes
1answer
453 views

Can one canonical conjugate variable be considered to be the “frequency” of the other one? (which could be a “wavelength”)?

So, from http://en.wikipedia.org/wiki/Conjugate_variables#Derivatives_of_action, we have... The energy of a particle at a certain event is the negative of the derivative of the action along a ...
5
votes
3answers
1k 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 ...
5
votes
3answers
395 views

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 ...
5
votes
3answers
475 views

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 ...
5
votes
1answer
677 views

What is the meaning of the Fourier transform of Feynman propagator?

I know $K(a,b,t)$ is the probability amplitude of find a particle that starts at point a in b in a time t later. There is also an expression that sometimes is called green function: ...
5
votes
3answers
1k views

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 ...
5
votes
2answers
271 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 ...
5
votes
2answers
529 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 ...
5
votes
1answer
456 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 ...
5
votes
1answer
332 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 ...
4
votes
3answers
1k 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 ...
4
votes
2answers
186 views

Does a finite wave necessarily have to be non-monochromatic in reality?

Does a finite wave necessarily have to be non-monochromatic in reality, or is that implication just a result of the mathematical analysis? I always wonder at these sort of things that come out of a ...
4
votes
4answers
710 views

Uncertainty Principle for a Totally Localized Particle

If a particle is totally localized at $x=0$, its wave function $\Psi(x,t)$ should be a Dirac delta function $\delta(x)$. Accordingly, its Fourier transform $\Phi(p,t)$ would be a constant for all $p$, ...
4
votes
3answers
435 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 ...
4
votes
3answers
165 views

Fourier transform of two pulses of light

I have laser beam path that fires two pulses of light in a gaussian distribution, so the intensity graph over time is two identical gaussians separated by a distance $t_0$. In other words, a gaussian ...
4
votes
2answers
306 views

A four-dimensional integral in Peskin & Schroeder

The following identity is used in Peskin & Schroeder's book Eq.(19.43), page 660: ...
4
votes
3answers
155 views

No well-defined frequency for a wave packet?

There are similar questions to mine on this site, but not quite what I am asking (I think). The de Broglie relations for energy and momentum $$ \lambda = \frac{h}{p}, \\ \nu = E/h .$$ equate a ...
4
votes
3answers
2k 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.
4
votes
2answers
235 views

Inverse Fourier Transform Of K-space Image…what is the object space scale?

Checked around a buch and could not find any help. But I needed help with: Understanding that if I get the Inverse FT of K-space data, what is the scaling on the X-space (object space) resultant ...
4
votes
2answers
312 views

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 ...
4
votes
2answers
1k views

Physical interpretation of Parseval's theorem

I have read that Parseval's theorem, relating the norm of a function $f$ and the norm of its Fourier transform $g(k)$: \begin{equation} \int |f(x)|^2 dx=\int|g(k)|^2 dk \end{equation} has the ...
4
votes
1answer
273 views

Fourier transformation, electric field and magnetic field to have a shielding lattice against particles

With Fourier-Series Expansion, we can write a function as sum of many non-repating different frequncied different amplituded sine and cosine functions. Lets assume we know electric-field and ...
4
votes
2answers
805 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: ...
4
votes
0answers
106 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( ...
3
votes
2answers
291 views

Why higher frequencies in Fourier series are more suppressed than lower frequencies?

One can expand any periodic function in sines and cosines. When calculating the coefficients $a_0$, $a_n$, and $b_n$ one find that $a_1>a_2>...>a_n>...$, similarly for $b_n$. Is there an ...
3
votes
3answers
180 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 ...
3
votes
4answers
1k 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 ...
3
votes
2answers
186 views

$2\pi$ and Feynman Rules

I notice a $2\pi$ term in the $\delta$-function when trying to construct an amplitude using the Feynman Rules. The $2\pi$ also appears as an integration measure to enforce normalisation in the phase ...
3
votes
1answer
2k views

How does the Fourier Transform invert units?

I don't really understand how units work under operations like derivation and integration. In particular, I am interested in understanding how the Fourier transform gives inverse units (i.e. time ...
3
votes
2answers
404 views

“Optically performed” Fourier Transform

This article says that they are only able to achieve such extremely high fiberoptic data rates because the multiplex light and then use a Fourier Transform to split it up again. But they say that ...
3
votes
2answers
156 views

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 ...
3
votes
2answers
458 views

Particle in a 1D box (momentum representation)

I have this problem. I want to find the wave function in the momentum space for the particle in a 1D box. We know that the wave function in the position space is: $$Y_n(x) = A\sin{(n\pi x/L)}$$ ...
3
votes
1answer
849 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 ...
3
votes
1answer
110 views

Determining Fourier Coefficients by inspection

I'm beginning to learn about Fourier series/transforms. My teacher hopes that by now we should be able to examine a simple potential function and decompose it without having to actually do the ...
3
votes
2answers
371 views

>What is the physical interpretation of the Fourier transform $(\mathcal{F}Z)(t)$ an impedance?

If I compose a impedances out of smaller ones in series and parallel configurations, e.g. $$Z(\omega)=i\omega L_2+\tfrac{1}{\tfrac{1}{R_1}\ +\ i\omega C_1+\ \tfrac{1}{i\omega L_2}},$$ then I get a ...
3
votes
1answer
221 views

Coulomb potential

It is known that the Coulomb potential can be obtained by Fourier transform of the propagator from E&M. Is this because one of Maxwell's equations have the form $\nabla \cdot \mathbf{E}=\rho$?