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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7
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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 ...
2
votes
3answers
271 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 ...
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 ...
2
votes
3answers
263 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 ...
2
votes
2answers
507 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 ...
3
votes
1answer
866 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 ...
7
votes
1answer
552 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 ...
2
votes
1answer
359 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 ...
0
votes
2answers
3k views

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: ...
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 ...
2
votes
2answers
194 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 ...
2
votes
0answers
150 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) ...
2
votes
3answers
227 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 ...
1
vote
3answers
718 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 ...
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
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 ...
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 ...
4
votes
2answers
310 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 ...
3
votes
1answer
76 views

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 ...
0
votes
2answers
904 views

What's the physical meaning of the Fourier transform of magnetic flux density?

I have here below the distribution of the magnetic flux density $B$ across a 1 pole pitch in the airgap of a synchronous machine. The horizontal axis represents the distance along the arc length ...
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
4answers
709 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 $p$, ...
7
votes
1answer
170 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 ...
1
vote
2answers
216 views

Expressing a particle's matter wave in terms of its momentum

I'm following Zettili's QM book and on p. 39 the following manipulation is done, Given a localized wave function (called a wave packet), it can be expressed as $$\psi(x,t) = \frac{1}{\sqrt{ 2 \pi}} ...
6
votes
2answers
906 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 ...
1
vote
1answer
560 views

Conjugate Variables and Fourier Transforms in Classical Physics

Let q be a generalized coordinate with a conjugate momentum p and a potential resulting in a periodic motion of q. What is the meaning of the Fourier transform of q(t) over its period? Can this be ...
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 ...
1
vote
2answers
459 views

Duality and Fourier Transforms [closed]

I read that $(FF(f))(x)=2\pi f(-x)$, where $F$ is the Fourier transform and $F(f(x-a))(k)=\exp(-ika) X(k)$ where $X(k)=F(f(x))$ implies $F(\exp(iax)f(x))(k)=X(k-a)$. But I don't see how that is ...
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 ...
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
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 ...
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 ...
2
votes
2answers
412 views

Simulating eye diagrams

I'm trying to figure out how to simulate eye diagrams for communications systems using Python. I'm not sure I have the theory down completely, though. From what I could figure out using some old ...
4
votes
2answers
804 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: ...
5
votes
1answer
674 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: ...
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 ...
2
votes
1answer
612 views

Using Fourier Transforms to Solve Systems with springs of high frequency

I'm trying to numerically solve the differential equations of motion in a system with multiple springs of very high frequency. Because the solution is often a combination of rapidly-oscillating sine ...
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 ...
1
vote
2answers
334 views

Finding $\psi(x)$ from Fourier modes [closed]

In quantum physics we've defined: $$ \psi (x) = \sqrt{ \dfrac{1}{2 \pi \hbar} } \int^{ \infty }_{-\infty } \phi (p) \exp \left( i \dfrac{px}{ \hbar} \right) dp $$ Now, $$a(k) \equiv \sqrt{ \hbar } ...
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 ...
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 ...
2
votes
5answers
470 views

Does this statement make any sense?

I am asking this question completely out of curiosity. The other day, my roommate, by mistake, used 'Light year' as a unit of time instead of distance. When I corrected him (pedantic, much), he said ...
3
votes
1answer
955 views

How do I compute the eigenfunctions of the Fourier Transform? [closed]

I read today (ref) that the Continuous Fourier Transform has four eigenvalues: +1, +i, -1, and -i. Associated with each eigenvalue is a space of eigenfunctions: functions which retain their form ...
22
votes
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 ...