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 ...

learn more… | top users | synonyms

1
vote
0answers
70 views

Fraunhofer Diffraction [closed]

A 1-dimensional aperture is illuminated by a parallel beam of light of wavelength $\lambda$ and the diffraction pattern is viewed on a distance screen. Show that the amplitude of the diffraction ...
0
votes
1answer
169 views

Fourier transform with periodicity at the harmonic frequency

Let's suppose I have a signal $F(t)$ that is periodic, with two periodicities $P_1$ and $P_2$, with $P_1 > P_2$. Suppose that I know the values of the two periodicities. Using the Fast Fourier ...
2
votes
2answers
214 views

Can someone please explain the “infrared catastrophe”?

In my readings I've run into this idea of an "infrared catastrophe" associated with 1/f noise. As far as I can tell it is because when you graph the periodogram of the 1/f signal you see the PSD goes ...
0
votes
1answer
88 views

How to find different operator representations in QM?

I read that any observable operator may be represented as: $$\Omega = \sum_n \omega _n | \omega _n \rangle \langle \omega_n |$$ Where the little omegas are the eigenvectors/eigenvalues of the ...
1
vote
1answer
100 views

Deriving commutation relations in second quantisation

I am trying to start from: \begin{align*} [\phi(x),\pi(x')] = i\hbar\delta(x-x') \\ [\phi(x),\phi(x')] = [\pi(x),\pi(x')]=0 \end{align*} to derive: \begin{align*} [a(k),a(k')^\dagger]=\delta_{kk'}\\ ...
3
votes
1answer
543 views

Derivation of Green's Function for Wave Equation

In the textbook Modern Methods in Analytical Acoustics (Crighton-1992, Amazon link to 2013 edition) the following relates the 3D Green's function in the time-domain to the frequency domain $g(x-y)$: ...
2
votes
2answers
129 views

Far Field Diffraction of EM waves: what does the zero frequency signify?

If you have a system of independently radiating electrons/point-charges, the far field distribution of the EM waves can be approximated by the fraunhoffer diffraction integral, or simply by the ...
3
votes
1answer
217 views

confusion in discrete transform to solve kronig penney matrix equation in fourier space

I have a periodic potential $$V(x) =\sum_{K}e^{iKx}V_{K} =\sum_{n}e^{\iota2\pi nx/a}V_{n} $$ where $K =\frac{2\pi n}a$ is the reciprocal lattice vector and $a$ is the lattice constant and $n =\pm ...
11
votes
4answers
1k 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: ...
2
votes
1answer
160 views

Three dimensional wave packets in momentum space

I am given the 3D wave packet: $$\psi(x,y,z)=N\,\exp\left(\frac{-(x^2+y^2+2z^2)}{2a^2}\right).$$ I was asked to find N (easy enough). Then I was asked the probability that we measure $z$ greater than ...
2
votes
0answers
100 views

Momentum representation of a state

I am trying to figure out the momentum representation of the state which has the properties $$\langle \psi |\hat q |\psi \rangle=-q_0,$$ $$\langle\psi|\hat p|\psi \rangle=p_0, $$$$\Delta q\Delta ...
2
votes
1answer
79 views

Transition from coordinate space to momentum space for SHO

I am given that the ground state of the SHO in position space is given as $$\langle q|\psi_0\rangle=\frac{1}{a^{\frac12}\pi^{\frac14}}e^{-q/4a^2}$$ Where a is a constant with units of length. I am ...
6
votes
1answer
262 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 ...
4
votes
0answers
263 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( ...
0
votes
0answers
155 views

Looking for raw interferogram data / raw FID data from FT-IR / FT-NIR / FT-NMR

I'm trying to get my hands on some raw interferogram data / raw FID data from an FT-IR / FT-NIR / FT-NMR so I can run some tests using FFT with it (it needs to be real data). Here's a picture below of ...
0
votes
0answers
105 views

Lenses and benefit of exact fourier transform

I have learned in an Optics class that a lens will "compute" the Fourier Transform of an electromagnetic wave passing through it at the focal point behind it (but with a quadratic phase). However, ...
1
vote
0answers
82 views

Interpretation in Fourier-Laplace domain

The Green's function describing the distribution of particles sent from $\def\v#1{\boldsymbol{#1}}\def\u#1{\hat{\v#1}}\v r=0$ at $t=0$ uniformly in every directions is, in two dimensions $$ ...
4
votes
2answers
236 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 ...
1
vote
0answers
88 views

Complex Fourier Particular Solution [closed]

I have found the complex Fourier series for my desired force. I now need to find the steady-state forced vibration of my oscillator as a Fourier Series. (The particular solution to the inhomogeneous ...
1
vote
0answers
350 views

Questions about Michelson interferometer

I have been doing experiment on Michelson experiment, but I don't quite understand why white light results in an interferogram with very few fringes, and why are they necessarily Gaussian? I know that ...
0
votes
1answer
949 views

Phonon dispersion calculation based on velocity autocorrelation function in reciprocal space?

I would like to ask about the method of calculating phonon dispersion relation. Up to my knowledge, there are 2 methods to calculate the phonon dispersion: By diagonalizing the dynamics matrix. ...
0
votes
2answers
216 views

Simplest derivation of Fourier transform for periodic functions (in crystal lattice)?

What is the simplest derivation of the following two well-known formulas that work for crystal lattice [1]: $$ F[f(\mathbf{x})] \equiv \tilde f(\mathbf{G}) = {1\over\Omega_\mathrm{cell}} ...
1
vote
2answers
207 views

How to derive inverse Fourier transform for periodic functions (in crystal lattice)?

I would like to derive the following two well-known formulas that work for crystal lattice [1]: $$ F[f(\mathbf{x})] \equiv \tilde f(\mathbf{G}) = {1\over\Omega_\mathrm{cell}} ...
0
votes
1answer
584 views

Wave packets and the derivation of Schrodinger's equation

I studied in my class, that a plane progressive wave cannot be used to represent the wave nature of a particle as it is not square integrable. Also, the phase velocity can get above the value of $c$, ...
0
votes
1answer
136 views

Partition functions in $\phi^{4}$ theory

The partition function in a $\phi^{4}$ theory is written \begin{equation}Z[J]=\int D\phi \, e^{-\int d^{4}x \left(\frac{1}{2}\left[(\nabla ...
0
votes
0answers
55 views

Frequency response of the wireless channel

We know that the signal attenuates out with distance and according to the channel transfer function or frequency response, the signal frequency components attenuate to different values based on ...
1
vote
1answer
171 views

Wave vector $\vec{k}$ vs position vector $\vec{x}$

My question is about the $k$-vectors in first Brillouin zone. If I am not misunderstood, the relation k = 2π/(Na) tells that when k goes to zero, we are very very far away from the reference atom and ...
1
vote
1answer
87 views

In what way do passive circuit elements change the functional form of the voltage?

I heard capacitors affect the valleys and mounds of voltage sine curves, so that you get DC from AC. It's related to Graetz bridge, flipping signs of sine waves and seemingly afterwards smoothing ...
0
votes
1answer
159 views

Help with the Heisenberg relation in Gaussian wave

In short laserpulses there is a minimal product of the frequency width and the pulselength for Gaussian pulses $\tau \cdot \Delta\omega \geq4\ln2$ this is the fourier boundary. So I know it origins ...
0
votes
1answer
141 views

Image K-Space and SNR

I am imaging a sample using CMOS camera. The pixel ratio is 6um x 6um and a total resolution {480, 752}. I understand that each pixel on the camera sensor is 6um x 6um in size (Have I understood this ...
3
votes
1answer
177 views

Getting an equivalent integral equation from a given one

I'm reading a paper and don't understand some of the calculations. We are given an integral equation with asymptotic boundary conditions $\rho_+(u)=\frac{1}{2\pi} ...
0
votes
1answer
81 views

Fourier Transform of E-Field with Decay Constant

Given an atomic transition with associated E-field $E(t) = E_{0}\cos(\omega_{0}t)e^{-t/\tau}$ where $\omega_{0}$ is the natural line frequency and $\tau$ is the decay constant of the simple harmonic ...
18
votes
1answer
561 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 ...
0
votes
1answer
105 views

Momentum representation of a function with discontinuous derivative [closed]

Consider the following wave packet $$\psi = Ce^{2\pi i p_0x/h}e^{-|x|/(2\Delta x)}$$ where $h$ is the Planck's constant and $C$ is the normalization constant. The derivative of this function is ...
3
votes
1answer
301 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 ...
12
votes
4answers
474 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 ...
0
votes
1answer
202 views

Amplitude and phase in vector wave field

Is it possible to make some separation of amplitudes and phase for a general vector-wave field? For example, like a paraxial approximation of a complex scalar field of the form $$\Phi(x,y,z) = ...
0
votes
0answers
138 views

Discrete Fourier Transform: Why do we only consider a full cycle?

I am studying Fourier analysis and I am still new to this topic. If I understand that the maximum frequency that can be used in a DFT is given by $N/2$, where $N$ is the number of samples in our ...
3
votes
0answers
368 views

How should I think about reciprocal lattice and Miller indices?

When I hear someone talking about a (100) plane or a (111) plane or an (hkl) in general, my first thought is, is the system cubic. The reason I think this is because I tend NOT to think of the planes ...
2
votes
3answers
722 views

Jacobian, Lorentz and Fourier Transformation

Jacobian, Lorentz and Fourier Transformation. I am confused with the physical interpretation/meaning of all these transformations. As far as I understood, Jacobian transforms from one coordinate ...
2
votes
2answers
365 views

What restrictions on time boundary conditions does it have to use Fourier transform to solve wave equation?

The wave equation can be solved using Fourier transform, by assuming a solution of the form of $$\mathbf{E}(x,y,z,t)~=~\mathbf{E}(x,y,z)e^{j\omega t}$$ and then reducing the equation to the Helmholtz ...
1
vote
3answers
7k views

What is the meaning of “frequency of a human voice”?

The term frequency for a periodic wave can be defined as the number of times a repeating pattern occurs in a given time period (eg: no. of crest and trough cycles per second for EM wave). But what ...
2
votes
1answer
143 views

How to learn the wavelet transform?

Is there any good literature if I want to learn the wavelet transform? Especially my project is related with marine electromagnetism?
1
vote
1answer
123 views

How can I model a two dimensional and three dimensional equivalents of one dimensional delta dirac (impulse) function?

I just started to read the book 'A Brief History of Time' by Stephen Hawking. Actually When he was talking of the idea of infinite density 'thing' before big bang suddenly the mathematical function ...
0
votes
2answers
78 views

Brewster angle with diffraction propagation?

Diffraction theory is scalar. How you deal with beam propagation in fourier optics that is sensitive to the to the polarization? If I have linearly polarized gaussian beam incident on glass surface, ...
1
vote
1answer
227 views

Unit analysis of a Fourier transform

This is an extension of a Phys.SE question I asked earlier, Fourier transform of two pulses of light. I start with 2 Gaussians, represented by the equation $e^{-(t/a)^2}\times e^{2\pi i d ...
4
votes
3answers
240 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 ...
1
vote
1answer
452 views

Parseval's Theorem on a Random Signal

NB - I'm re-posting this question in physics because I haven't had any luck getting a response from the maths StackExchange site - it's a rather applied problem so is probably better suited here ...
3
votes
3answers
898 views

How do human ears distinguish the frequencies in sound?

If they do a Fourier transform, how can they know the formula to find coefficients?
3
votes
2answers
881 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 ...