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
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0answers
74 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
116 views
+50

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
62 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 $\...
0
votes
0answers
29 views

Fourier transform in gravitational lensing? [closed]

I am reading this text about weak gravitational lensing. Could someone please explain to me how the author got the Eqs(5.57-5.59) by fourier transformation Thanks.
5
votes
3answers
134 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
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)...
3
votes
1answer
117 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 ...
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
57 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
59 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
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
votes
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 ...
0
votes
0answers
22 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 ...
-1
votes
0answers
14 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)=\...
0
votes
0answers
10 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
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 ...
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 $$\...
0
votes
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{...
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
0answers
55 views

Asymptotic behaviours from Fourier transforms [migrated]

I have completely forgotten how one derives the asymptotic behavior in frequency space, given the asymptotic behavior of the function in real space (e.g. time). As an example example, it is often said ...
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
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
votes
1answer
54 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
70 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
25 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
0answers
29 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
109 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
277 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
60 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
50 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 ...
2
votes
0answers
35 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 ...
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 ...
1
vote
1answer
73 views

Deriving Schrodinger equation from QFT with the definition $\psi(\textbf{x},t)\equiv \langle 0|\phi_0(\textbf{x},t)|\psi\rangle$

In the book "Quantum Field theory and the Standard Model" by Matthew Schwartz, he uses the equation $$\partial_t^2\phi_0=(\nabla^2-m^2)\phi_0$$ (i.e., the Klein-Gordon equation for the free ...
0
votes
0answers
30 views

How to get the vibratonal frequency of a bond using FFT of velocity autocorrelation function?

I guess there is some errors in the way I am calculating VAC since I ma ending up with a peak whose frequency is two times the actual frequency. I ran an MD simulation long enough with 60 molecules of ...
1
vote
2answers
58 views

Fourier transform of Hamiltonian for scalar field

In the Srednicki notes (http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf) page 36 he goes from $$H = \int d^{3}x a^{\dagger}(x)\left( \frac{- \nabla^{2}}{2m}\right) a(x) $$ to $$H = \int d^{3}p\...
1
vote
3answers
65 views

Normal mode analysis

I'm reading lots of texts about normal modes and I've seen that normal modes are solutions of the wave function produced by separation of variables. However, when most of authors I've read perform the ...
3
votes
3answers
255 views

How to measure an image's contrast?

I'm studying Fourier optics and Interferometry and I intend to determine the contrast of an image using computer software. My teacher of Experimental Physics didn't tell me how to do it, and so, I'm ...
0
votes
1answer
27 views

Help understanding the wave number in light

For my final optics project I want to implement the beam propagation method using Fourier transforms. I came across the following document http://ecee.colorado.edu/~mcleod/pdfs/NMIP/lecturenotes/NMiP%...
3
votes
3answers
59 views

Why is response of system same frequency as driving force frequency

Super basic question: why does a system (to be definite, perhaps assume a collection of coupled harmonic oscillators) respond (in the steady-state, after transient effects have dissipated) with all ...
-1
votes
1answer
45 views

Expectation value of position operator $X$ in momentum space [closed]

I'm solving the following question: If $\psi(p)$ is the wavefunction of a particle in momentum space, write down the expression for the expectation value of the position operator $\langle x\rangle$? ...
1
vote
0answers
42 views

Help normalising and taking the inverse Fourier transform of this wavefunction [closed]

Normalising Consider the wavefunction $$\psi(x,0)=Ne^{-\frac{|x|}{\lambda}}.$$ In order to normalise this I take the integral, which due to the modulus on the $x$ I evaluate just from zero to ...
1
vote
0answers
65 views

Expanding a wavefunction [closed]

I have a wave function that I have already normalised: $$ \psi(x) = \sqrt{\frac{30}{a^{5}}}x(a-x) $$ but now I have been asked to expand it to get: $$ \psi(x) = \sqrt{\frac{960}{\pi^{6}}}\sum_{k} \...
0
votes
0answers
48 views

Using Plancherel's theorem on delta function

Plancherel's Theorem states that for $f \in L^{2}(\mathbb{R})$ we have $$f(x) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty}F(k)e^{ikx}dk \Longleftrightarrow F(k) = \frac{1}{\sqrt{2 \pi}}\int^{\...
-1
votes
1answer
79 views

Commutation relations in Quantum Field Theory [closed]

\begin{align} [a, a^\dagger] =& \left[\int d^3 x e^{-ikx} (\omega \phi(x) + i \Pi^\dagger(x)), \int d^3 x' e^{ikx'} (\omega \phi^\dagger(x') - i \Pi(x')) \right] \\ =& \int d^3x \, d^3x' \, e^{...
3
votes
1answer
48 views

Finding the noise spectral density of a quantity made from different noisy components

I'm looking for the expression of the noise spectral density of the magnetic flux $\Phi$ generated by a noisy voltage signal $V$ applied to a resistor with Johnson-Nyquist noise $R$ which produces a ...
0
votes
1answer
89 views

Momentum and position for free particle

In the section of 'The free particle' in 'Introduction to quantum mechanics, second edition' by Griffiths page 65. He has the wave equation as $$\Psi(x,t) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\...
1
vote
0answers
28 views

Time-domain NMR or: When is the Fourier-Transformation not appropriate?

My question has two parts: One is general and has to do with the Fourier-Transformation, one has to do with Time-Domain NMR. Both parts are interlinked, of course. I tried to find out, why people do ...