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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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
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0answers
20 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} ...
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1answer
40 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
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2answers
83 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 ...
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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
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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
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0answers
77 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
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0answers
133 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'...
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3answers
65 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
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1answer
120 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
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3answers
135 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 ...
4
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1answer
211 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 ...
7
votes
1answer
244 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
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2answers
62 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
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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 ...
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1answer
58 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 ...
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0answers
62 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 ...
0
votes
1answer
116 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, ...
1
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1answer
40 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
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0answers
56 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 ...
3
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1answer
414 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}...
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0answers
23 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 ...
2
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1answer
552 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 ...
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0answers
15 views

Relation of rate of decay of a function with width of peaks of its Fourier transform

Consider a function $f(t)=θ(t)e^{−σ_0 t}sin(ω_0 t)$, where $θ(t)$ is $1$ for positive $t$ and $0$ for negative $t$. Its Fourier transform can be easily computed and has the form: $$ \hat{f}(\omega)=\...
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1answer
40 views

Energy Transferred to a Spring by a Time Dependent Force (using Fourier Transformations)

I found an excersice in Byron-Fuller's: "Mathematics of Classical and Quantum Physics", about the energy which is transferred to a spring by a time depended force of the form: $F(t)=\left\{ \begin{...
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0answers
11 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
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1answer
81 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 ...
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1answer
112 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
64 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
79 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
188 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?
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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 ...
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0answers
36 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
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1answer
56 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
27 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
72 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{...
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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
223 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
33 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
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3answers
110 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
294 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
61 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 ...
2
votes
3answers
51 views

Position and momentum measurement effects on wave functions

I have a few short questions about an interpretation of what happens with position and momentum wave functions described in literature I am using. Given momentum space wave function and position space ...
1
vote
1answer
111 views

A question on using Fourier decomposition to solve the Klein Gordon equation

Given the Klein Gordon equation $$\left(\Box +m^{2}\right)\phi(t,\mathbf{x})=0$$ it is possible to find a solution $\phi(t,\mathbf{x})$ by carrying out a Fourier decomposition of the scalar field $\...
2
votes
0answers
36 views

What is the shape of the MTF curve in coherent imaging?

For incoherent imaging, the shape of the diffraction-limited MTF curve would look roughly like a triangle, with normalized contrast starting at 1 for zero spatial frequency and decreasing to 0 at the ...
6
votes
1answer
6k 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 ...
4
votes
1answer
257 views

Significance of higher harmonics

I am analyzing a noise signal and have identified the fundamental frequency of a tone and it's higher harmonics. Say for example given the signal below, The fundamental frequency has a sound ...
2
votes
0answers
64 views

Intervalley scattering in graphene in presence of impurities

A long range impurity like coulomb impurity does not induce an inter valley scattering between the two Dirac points. Is there any mathematical explanation for the same although this is explained by ...