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
votes
1answer
250 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 ...
2
votes
1answer
31 views

Detect original sound waves

I have just presented a project of mine regarding sound recognition using neural networks. I told during the presentation that I decided to only recognize one sound (musical notes coming from a guitar)...
4
votes
1answer
220 views

Diffraction and $k$-space

Regarding diffraction I am a little bit lost reading about reciprocal space and the space of $k$'s. As I understand it the Fourier relationship between a wavepacket $\Psi(\vec r,t)$ and the complex ...
3
votes
0answers
56 views

Commutation Relation between Annihilation & Creation Operators and Ascending & Descending Operators

I am currently working on a QD-Cavity system. After the point Heisenberg Equation of motion is obtained from corresponding Hamiltonian of the system, in order to find the expression for bosonic ...
-1
votes
1answer
41 views

Why the function has to be a periodic function, if we want to find its Fourier series?

Why the function has to be a periodic function, if we want to find its Fourier series? What happens if the function has non-zero values over a finite range only? e.g. $ f(x) = e^{-x}; 0 \le x \le ...
5
votes
3answers
1k views

Why is a wave pulse a superposition of sine waves?

I have learned that to construct a wave pulse, we need to superimpose multiple sine waves of different frequencies which all interfere to produce the pulse. What I don't understand is that a wave ...
2
votes
1answer
565 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 ...
0
votes
1answer
126 views

Is there any way to find difference between two same sounds of different people?

I am trying to understand and find a way to distinguish two same sounds of different people by some physics formula, so could you guys help me? OK I'll try to explain my question in this way that, ...
3
votes
4answers
90 views

Convolution Theorem in Physics

I'm getting ready for my classes to start next semester in Grad school, and I'm reading over Fourier Transforms and their applications. I came across the Convolution Theorem, namely, that if we have a ...
0
votes
1answer
23 views

Connection between reciprocal space and momentum

Given a lattice in real space with lattice vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$, we can do the following. We take any function $f(\vec{r})$ and take the Fourier transform to obtain $\tilde{f}(\...
0
votes
0answers
41 views

A problem in Fourier transformation of Coulomb potential

Firstly, we calculate the Fourier transformation of $1/|\mathbf{k}|^{2}$ as follows, \begin{eqnarray*} \int\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{|\mathbf{k}|^{2}}e^{i\mathbf{k}\cdot\mathbf{x}} & = &...
0
votes
1answer
30 views

Intrinsic permutation in nonlinear susceptibilities and pulses

The nonlinear susceptibilities have an intrinsic permutation symmetry. This symmetry treats two frequency components that are equal differently than it treats two frequency components that are ...
0
votes
1answer
120 views

Using the Fourier transform to find the natural frequencies of coupled oscillators

How can I find the natural frequencies of a system consisting of a pair of coupled oscillators using Fourier transforms? The System consists of two masses and three springs. One of the springs ...
3
votes
1answer
430 views

Fourier Transforms of position and momentum space in Quantum Mechanics

Fourier transformations: $$\phi(\vec{k}) = \left( \frac{1}{\sqrt{2 \pi}} \right)^3 \int_{r\text{ space}} \psi(\vec{r}) e^{-i \mathbf{k} \cdot \mathbf{r}} d^3r$$ for momentum space and $$\psi(\vec{r}...
0
votes
0answers
9 views

How do optical anti-aliasing filters work from a frequency domain perspective [migrated]

To prevent aliasing, caused by the finite number of pixels on a sensor, a blurring filter is commonly used. How does that work from a frequency domain perspective? What is the transfer function of ...
0
votes
0answers
28 views

Fourier transform of a fluids kinetic energy equation

I'm currently looking at the 2D equations for a fluid and want to examine the kinetic energy equation in Fourier space. I start with the equation for momentum $$ \frac{\partial\textbf{u}}{\partial t} ...
0
votes
1answer
48 views

Uncertainty and Classical waves

My professor, introducing Heisenberg uncertainty principle, started from the Fourier transform and the classical uncertainty for waves. He told about the localized impulsive wave $\delta(x)$ which ...
5
votes
2answers
88 views

Uncertainty Relations, Conjugate Quantities, and Fourier Transforms

I've heard from a lot of people that the reason momentum and position have an uncertainty relation is because of the Fourier Transform. But is this in any way the case? If it were I would expect all ...
1
vote
1answer
147 views

Effective masses for different direction

Assume we have an indirect semiconductor where the effective mass becomes anisotropic in different directions. Usually, one talks about a mass in parallel and perpendicular direction referring to ...
6
votes
3answers
195 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 ...
2
votes
0answers
83 views

Wilsonian Renormalisation — Peskin & Schroder Sect. 12.1

I'm working my way through Peskin & Schroeder, but some of the details of the calculations done in their introduction to the renormalisation group are slipping past me. For concreteness, the ...
2
votes
0answers
135 views

Particle density in k-space

Question: Given a many particle wave function $|\Psi\rangle$, how can I calculate the occupation numbers in k-space? Setup: I have a 1D chain of molecules which contains 4 sites per unit cell. Let'...
1
vote
3answers
66 views

Multiples of frequencies in Fourier transforms [closed]

Given a sinusoidal signal with a frequency $\omega$, the Fourier transform ought to simply be a delta function at $\omega$. However, what happens to multiples of these? MATLAB's FFT shows nothing at $\...
3
votes
1answer
122 views

Replacing fermionic operators with their Fourier transform and boundary conditions

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring. To compute the complexity of the algorithm ...
5
votes
3answers
137 views

Can a physical wavefunction be non-smooth (its first derivative is discontinuous)?

Here's an argument that might support the statement that such a non-smooth wavefunction is not physical: You cannot add a finite number of smooth functions to get a non-smooth function. By fourier ...
2
votes
2answers
64 views

A general complex electric field

When dealing with a plane wave solution to the electric field such as $$\vec{E}(r,t)=E_{0}\cos(kz-\omega t+\phi)$$ we usually introduce a complex electric field $\tilde{E}(r,t)$ such that $\vec{E}(r,t)...
2
votes
0answers
44 views

What is abbe's rule in optics?

I have read wikipedia but can't really understand what they mean to say. The usual explanations are given in terms of Fourier optics, which I don't yet have the background for. Can anyone explain it ...
0
votes
1answer
60 views

Hamiltonian - Fourier transform of order parameter [closed]

I have a rather simple task, but it seems I can't move forward with the solution. I have a Hamiltonian as seen in the picture. I have to use the Fourier transform of the order parameter $\phi(x)$ and ...
0
votes
0answers
63 views

Confused about the substitution of the fermionic operators with their Fourier transform in an adiabatic Hamiltonian

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring. To compute the complexity of the algorithm ...
1
vote
1answer
41 views

Fourier optics (diffraction from pinholes)

A plane wave of wavelength $\lambda$ and unit amplitude is normally incident on a mask placed in the xy-plane at $z=0$. The mask contains two infinitesimally small pinholes, located on the x-axis ($y=...
0
votes
0answers
58 views

Fourier transform in two dimensions, Green's function for Schrodinger equation

I want to calculate this Fourier transform: $$ \int\limits_0^{\infty} \mbox{d}k \int\limits_0^{2\pi} \mbox{d}\varphi~ k \frac{e^{i \vec{k} \cdot(\vec{x}-\vec{x}')}}{k^2+B} $$ which will be 2D Green's ...
0
votes
0answers
25 views

How can I calculate the time domain correlation function in the frequency domain?

I have an operator $a(t)$, its correlation function is: $$<a(t)a^{\dagger}(t^{\prime})>=\delta(t-t^{\prime})$$ Now I need to find the correlation function in the frequency domain. i.e. to ...
0
votes
0answers
12 views

Relation between phase of dark field and bright field images

I am trying to understand how the super-resolution technique based on Fourier Ptychography 1. In the paper, we run phase retrieval algorithms using images captured using illumination at different ...
0
votes
1answer
83 views

Eigenstates of position and momentum operators in QM

In Griffiths pages 103-105 "Introduction to Quantum Mechanics" 2nd editiion he states that the eigenfunctions of the position and momentum operators are $$g_y(x) = \delta(x-y)$$ where the eigenvalue ...
3
votes
1answer
70 views

Vacuum expectation value in presence of a source

If a vacuum is translationally invariant i.e., $P^\mu|0\rangle=0$ or $e^{(\pm ip\cdot x)}|0\rangle=0$, we can express the the vacuum expectation value of a field as $\langle 0|\phi(x)|0\rangle$ as $$\...
3
votes
1answer
82 views

Anomaly, Ward identity [closed]

While studying notes on anomaly by Adel Bilal (http://arxiv.org/abs/0802.0634), I stuck in a calculation. Here it goes as follows: The three-current correlator in perturbation theory as a one-loop ...
0
votes
2answers
192 views

How does one calculate the minimum spectral linewidth in cm$^{-1}$ of a pulsed laser with pulse duration of 10 fs?

I calculated the minimum spectral linewidth given here using the uncertainty principle that $\Delta E\,\Delta t =h/2\pi$. Will this be correct?
1
vote
0answers
18 views

Help with two dimensional polar axis Fourier transform

This is a problem that I met in real-life physics research. This question is related to Wick's theorem. The question is: 1. In two dimensional plane with polar axis, why do we have the following ...
2
votes
0answers
41 views

What is the meaning of the Fourier Transform of the electric field of a well guided mode in a dielectric waveguide?

I have been studying waveguiding in dielectrics for a while now; however, I cannot understand the meaning of the Fourier transformed electric field. I will first give some background information. ...
0
votes
1answer
57 views

Interpreting group velocity of free particle wave packet

I am trying to understand the concept of group velocity of a free particle wave packet: $$\Psi(x,t) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty} \phi(k)e^{ikx}e^{-\frac{i \hbar k^2 t}{2m}}dk.$$ ...
0
votes
1answer
28 views

Which information do we get from the phase spectrum about the wave?

Let a wave is represented by an equation $$y=f(t)=10\sin(\frac{2\pi f_1t}{T} + \pi/6)+5\cos(\frac{2\pi f_2t}{T} +\pi/3)$$. Here, let us take $f_1=10 ,f_2=5 ,T=100$ Then, from the Fourier transform ...
0
votes
1answer
78 views

Quantum Mechanics: how exactly does “delta function normalization” work for eigenfunctions in 1-d free space case?

The definition of "delta function normalization" says a basis of eigenfunctions of a particle in free space are orthonormal when $$\int_{-\infty}^{\infty}\phi_n^*(\vec{r})\phi_m(\vec{r})\mathrm{d}\vec{...
0
votes
0answers
26 views

Weinberg Cosmology book Ch 5.2

I am working on Weinberg Cosmology book, and have one question about what contained in Ch 5.2 (page 229). Basically, this chapter is dealing with stochastic initial conditions. What he wrote is that ...
2
votes
4answers
143 views

Meaning of a certain value at Fourier Transform

Define the Fourier Transform of a certain signal in the time domain FT[$x(t)$]=$X(j\omega)$ $X(j\omega)$ = $\int$ $x(t)$ $e$^($j\omega$$t$)$ $dt I'd like to ask what is the meaning of the value ...
1
vote
1answer
225 views

Expanding free scalar field in terms of ladder operators

I'm having some difficulty with the finer points of expanding a field in terms of ladder operators. Note that this is not identical to the other related question I asked. From Peskin / Schroeder; ...
1
vote
0answers
59 views

Radial Distribution Function - Structure Factor relation, deriviation help

I'm attempting to prove the relation between the structure factor and the RDF, following the deriviation here (pg 92-94). The solution this source comes too disagrees with this paper which I trust ...
4
votes
3answers
112 views

Same quantum states represented in different basis

In literature on an introduction to quantum mechanics which I am working through, there is a section which explains that a vector has different representations based on the basis you choose and then ...
3
votes
2answers
340 views

Schrödinger equation in momentum space

In literature on an introduction to quantum mechanics which I am working through, there is a section which explains that a vector has different representations based on the basis you choose. It then ...
1
vote
0answers
39 views

Lippmann-Schwinger equation and time dependence

Consider the Lippmann-Schwinger equation (LSE) $$ |\psi\rangle = |\phi\rangle + \hat{G}_0(\epsilon) \hat{V} |\psi\rangle \tag{1}$$ where $\hat{G}_0(\epsilon) = \frac{1}{\epsilon - \hat{H}_0 + i\eta}$...
4
votes
0answers
64 views

Interpreting the Fourier transform of a Gibbs measure

Recall that a Gibbs measure gives a probability distribution on states $x$ of the form $$ p_X(x) = \frac{1}{Z(\beta)}\exp(-\beta E(x)) $$ As I understand, the function $E$ is interpreted as the ...