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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2
votes
1answer
165 views

Inverse of a series (solid state)

I am working with the expression involving the equilibrium displacement ($y_n$) for the $n$th particle in a 1D harmonic lattice in terms of the normal modes coordinates $A_k$. Let me show you the ...
0
votes
2answers
150 views

Representations in quantum mechanics [closed]

This might be a very simple question. I just want someone to point me the right direction to understand things like this: $$ \langle x|x'\rangle=\delta(x-x') \\ \psi(x)=\langle x|\psi\rangle \\ ...
1
vote
0answers
104 views

Quick question on convolution - Diffraction through a pair of slits

We know that the fourier transform of the amplitude function (in terms of $y$) gives you the amplitude function (in terms of $\theta$) Consider a pair of triangular slits: Fourier transform of ...
1
vote
1answer
154 views

Resolution in a Fourier transform spectroscopy setup

I am a bachelor physics student and as an assignment we had to perform measurements on an FT spectroscopy setup. Context. Our setup consisted of a Michelson interferometer through which the light ...
4
votes
1answer
100 views

Light, Fourier Transforms, Spherical Harmonics

Mathematically, I'm having trouble understanding where we can use what with light. I read somewhere on this site that Huygen's Principle is effectively just taking an expansion of a wave onto the ...
0
votes
0answers
150 views

If the discretized $x$ and $p$ are conjugate variables, why are there two $U$s in $U_{FFT}xU_{FFT}^{\dagger}=p $?

While studying some quantum mechanics from Nielsen's book on quantum computing, specifically the simulation of the single-particle Schrödinger equation using a quantum computer, I came across ...
1
vote
0answers
54 views

Is having full information about the resonances of a rigid body equivalent to having full information about its material parameters?

Lets say I have a mechanical system whose mechanical resonances (mode shape and frequency) I can measure with perfect accuracy. Is this theoretically equivalent to knowing the materials parameters, ...
0
votes
1answer
584 views

Quantum Mechanics - Finding momentum probability density [closed]

everyone. I got a bit stuck on 2(iii), this is supposed to be a easy question, but i don't know how you get the square term? I thought you just do the Fourier transform, but then I got some ...
6
votes
1answer
199 views

Kolmogorov/Energy spectrum for turbulent boundary layer

Previously, I have calculated energy spectrum for 3D isotropic turbulent flow data which is equally spaced in all three directions and then to compute the energy spectrum, one performs Fourier ...
1
vote
1answer
243 views

How to derive quantum Fourier transform from discrete Fourier transform (DFT)?

I am interested in Shor's algorithm, and I am reading several papers that related to the quantum Fourier transform (QFT). I know the there is a difference between the output of QFT and DFT (DFT). ...
1
vote
2answers
157 views

Fourier transformation [closed]

I have recently studied Fourier and Laplace transformation in maths. I wanted to understand the utility in physics with some examples that requires this change in dimension and the reason why.
0
votes
0answers
146 views

How do RGB colors work? [duplicate]

They say that all colors can be formed by mixing Red, Green, and Blue appropriately. Is it true? Isn't the Fourier basis infinite dimensional? Or does it turn out to be the case that only three ...
17
votes
8answers
3k views

Why are sine/cosine always used to describe oscillations?

What I am really asking is are there other functions that, like $\sin()$ and $\cos()$ are bounded from above and below, and periodic? If there are, why are they never used to describe oscillations in ...
3
votes
1answer
932 views

“Derivation” of the Heisenberg Uncertainty Principle

The question I outline below is my textbook's "derivation" of the Heisenberg Uncertainty Principle. The "derivation" my textbook uses involves wave packets. Suppose there are seven waves of ...
5
votes
1answer
1k views

Finding the creation/annihilation operators

Using Minkowski signature $(+,-,-,-)$, for the Lagrangian density $${\cal L}=\partial_{\mu}\phi\partial^{\mu}\phi^{\dagger}-m^2\phi \phi^{\dagger}$$ of the complex scalar field, we have the field ...
0
votes
0answers
70 views

Is spectrum of Discrete-Time Fourier Transform (DTFT) periodic or not

I can't think of any better title. Here is the content that I got question http://cnx.org/content/m10247/2.31/ As it state the nature of DTFT's spectrum is periodic as it show in figure 1 However, ...
0
votes
1answer
109 views

Physical meaning of taking twice the real part of a Fourier transform

In my previous question, Calculating the coherence length from a spectrum, it turned out that I can calculate the coherence length of my light source from the autocorrelation function, which can be ...
1
vote
0answers
88 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
207 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
308 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
94 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
126 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
652 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 $g(x-y)$: ...
4
votes
2answers
167 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
241 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
198 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
108 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
93 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 ...
7
votes
1answer
273 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 ...
5
votes
0answers
284 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
211 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
117 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
91 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
245 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
102 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 ...
2
votes
0answers
564 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
1k 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
302 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
246 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
676 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
161 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
67 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
255 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
99 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
178 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
190 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 ...
4
votes
1answer
224 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
85 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
631 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 ...