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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68 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 ...
2
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3answers
75 views

What's the difference between frequency domain and time domain spectra?

If I have a mechanical oscillator and want to observe the dynamical behavior of the oscillator, is there any additional information to observe it in time domain and frequency domain? Normally, we ...
0
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1answer
72 views

Massless boson in 2D and its (retarded) propagator

I have the retarded propagator for a free scalar field in 1+1 dimensions. Inside the light cone, this looks like $J_0(m \sqrt(t^2-x^2))$, J being a Bessel function. When I take the massless limit, ...
5
votes
3answers
107 views

How do we know that the Fourier transform of space is momentum?

How do we know that the Fourier transform of real space $x$ is the momentum $p$ space or for energy and time, receptively? What's the mathematical process and physical logic?
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1answer
434 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 $g(x-y)$: ...
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2answers
59 views

Proof that quantum Fourier transform is unitary

I'm trying to work through the proof that the quantum Fourier transform can be described by a unitary operator, i.e $F^{\dagger}F=\mathbb{1}$, where ...
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2answers
61 views

Interpretation of the four-vector $k$ in scalar QFT

I'm studying the canonical quantization of the Klein-Gordon real scalar quantum field theory, given by the classical Lagrangian density $$\mathscr L = {1\over ...
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1answer
95 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 ...
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2answers
86 views

Numerically solving 2D poisson equation by FFT, proper units

The 2D Poisson equation is: (1)$$\frac{d^2\varphi(x,y)}{dx^2}+\frac{d^2\varphi(x,y)}{dy^2}=-\frac{\varrho(x,y)}{\epsilon_0\epsilon}$$ And in $k$-space it is in form of: (2)$$(k_x^2+k_y^2) ...
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1answer
77 views

A few questions on wave packets and uncertainty relations

According to Cohen-Tannoudji the wave-function for a one-dimensional free particle can be written as $$ \psi (x,0)=\frac{1}{\sqrt{2 \pi}} \int g(k) e^{ikx} dk.$$ While $g(k)$ is not specified, there ...
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0answers
12 views

how to find the frequency in a auto correlation function?

I am running some molecular dynamics simulation with carbon nano tubes and calculating the velocity auto-correlation function (VACF). Each 10 time steps is writen in a file the VACF and in the final ...
5
votes
3answers
221 views

The ubiquitous Planewave Ansatz

In physics, the planewave ansatz (meaning: an educated solution guess) is very ubiquitously used, when solving differential equations, in different domains of physics. E.g. to solve the dispersion ...
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1answer
53 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 ...
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votes
1answer
26 views

Scattering from a potential, matrix elements of momentum eigenstates, and the Fourier transform [closed]

I am working on my last quantum homework and don't know where to begin with part (i) in this question 4. Do I need to use a product rule in the FT and use convolution? Not sure how to go about the ...
3
votes
1answer
120 views

Poles for a particle scattered in a delta potential

I am working on problem a professor gave me to get an idea for the research he does, and have hit a point where I'm having a difficult time seeing where I need to go from where I'm at. I would also ...
2
votes
1answer
137 views

Fourier and inverse fourier transform in QFT

According to my lecture notes, the inverse Fourier transform of an operator $\phi(p)$ is given by $$\phi(x)=\int \frac {d^4p}{(2\pi)^4}\phi(p)e^{-ip\cdot x}.$$ As @WenChern pointed out below, Peskin ...
1
vote
1answer
22 views

Why are unilateral Laplace transforms suitable for causal systems and bilateral aren't?

https://en.wikipedia.org/wiki/Two-sided_Laplace_transform#Causality The above section says that bilateral transforms will not necessarily make sense for causal systems. In the course of advanced ...
5
votes
1answer
116 views

Which position and momentum distributions arise from some wave function?

Consider a particle in one dimension with wave function $\psi(x)$. The probability density function describing how likely it is to find it in a given position is given by ...
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1answer
49 views

What does $σ$ equal to zero mean?

Consider the Laplace transform of an RC filter. For those who can't immediately summon it, refer equation (46) at this link: http://web.mit.edu/2.151/www/Handouts/FreqDomain.pdf for a refresher. In ...
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1answer
80 views

Autocorrelation function for deterministic nonlinear dynamical systems

I am quite puzzled with the problem that spectral analysis has been either applied to noisy dynamical systems or to chaotic ones. I was wondering why nobody makes analysis of non-linear dynamical ...
2
votes
1answer
90 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 ...
0
votes
1answer
11 views

Calculating a resonance fluorescence spectrum (Mollow Triplet)

I was working through a lecture on quantum optics, in which we calculate the spectrum of electric field correlations of fields produced by two level emitters. Now, the part where I got stuck was ...
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vote
1answer
31 views

Water wave packet variance

Consider the following quantity, $$I = \int x^2|\eta(x)|^2 \ dx,$$. For $\eta(x)$ a solution to some linear equation, we have $\eta(x) = \int a(k) e^{ikx} \ dk$ where, for $\eta$ to be real, we ...
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votes
1answer
42 views

Integral over scalar product of eigenfunction of momentum operator and harmonic oscillator one

Recently I've met following expression: $$ \tag 2 \sum_{n}f(n)\int dp~ |\langle p | n\rangle|^{2} = 2\pi \sum_{n}f(n). $$ Here $|n>$ is eigenfunction of harmonic oscillator with energy $$E_{n} = ...
6
votes
1answer
817 views

What is the meaning of the Fourier transform of Feynman propagator?

I know $K(a,b,t)$ is the probability amplitude that a particle that starts at point $a$ is found at point $b$ at a time $t$ later. There is also an expression that sometimes is called green function: ...
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3answers
149 views

Why frequency is inversely proportional to time-period?

Why frequency is inversely proportional to time-period? While studying about Fourier transform that shows frequency representation. A doubt that came to me was a set of signal with same wavelength ...
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0answers
25 views

Why isn't there a different phase after fourier transformation in two lattices

I am trying to understand some solutions for graphenes energy dispersion. While most of it is clear, I don't get one step, when changing into k-space. Consindering two sublattices A and B with ...
4
votes
2answers
134 views

Renormalization, integrating out high momenta Wilson way

In equation $(12.5)$ in Peskin and Schroeder, they write out the generating function but leave out all quadratic terms of the form $\phi\hat{\phi}$ arguing that they vanish since Fourier ...
3
votes
2answers
104 views

Is a wave packet physically realizable as a Fourier series?

In QM a wave packet is modeled as an infinite, or almost infinite, Fourier series, and the Fourier transform provides a transformation between momentum space and position space. To what extent is ...
2
votes
3answers
154 views

Why does $\nabla \to ik$ when you Fourier transform?

I am reading a text that describes the scattering of light by a particle with dielectric constant $\epsilon$ After a bit of maths starting from Maxwell's equations they obtain: $$\nabla (\nabla ...
4
votes
2answers
250 views

Position operator in QFT

My Professor in QFT did a move which I cannot follow: Given the state $$\hat\phi|0\rangle = \int \frac{d^3p}{(2\pi)^3 2 E_p} a^\dagger_p e^{- i p_\mu x^\mu}|0\rangle,$$ he wanted to show that this ...
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0answers
75 views

Schrodinger Wave Functional (quantum fields) - Solving Functional Gaussian Integrals

Okay, So i'm doing some research that involves the Schrodinger representation in quantum field theory. The ground state wave functional for the Klein Gordon field is a generalized gaussian in position ...
7
votes
1answer
187 views

Why is there $1/2\pi$ in $\int\frac{dp}{2\pi}|p\rangle\langle p|$?

I'm reading Richard MacKenzie's lectures on path integrals and on page 7 he derives the propagator for the free particle as follows: $$ \begin{align} K &= \langle q'|e^{-iHT}|q\rangle \\ &= ...
2
votes
0answers
65 views

Feynman propagator with general $\xi$ parameter

Hey from my notes in my PS book it seems I have solved this some time in the past, but I cannot seem to get the indices straight this time around. So in deriving the Feynman photon-propagator which ...
0
votes
0answers
40 views

Lissajous plots - additive synthesis

I am using Fourier analysis to recreate my data. I have some test data to work on as a way of testing out my synthesis approach. I have spatial test data in the x and y directions. when I plot x ...
1
vote
1answer
81 views

Momentum Representation vs Position Representation

We are given an operator $g$ from $\mathcal{l}^2(\mathbb{Z})$ to $\mathcal{l}^2(\mathbb{Z})$, i.e., the space of functions that are square summable over $\mathbb{Z}$ such that ...
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0answers
63 views

Decoupling the Hamiltonian by a Discrete Fourier transform

For $N$ coupled oscillators(periodic BC) whose Hamiltonian is given as $H=\sum\limits_{i=1}^N (\frac{p_i}{2m} + \lambda(x_{i+1} - x_i)^2)$ decoupling can be achieved by change of variables by using ...
4
votes
3answers
1k views

What is the specific meaning of “Fourier frequency” (as opposed to simply “frequency”)?

I've noticed that many journal articles (in optics) use the phrase "Fourier frequency" to describe, well, the frequency of something. Google scholar search for "Fourier frequency". Example: ...
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votes
0answers
11 views

Constructing Echocardiography

I am trying to search the Mathematics behind Echocardiography and constructing the image. There seems to be Wigner distribution applied to the pictures. There also seems to be many different ...
1
vote
1answer
363 views

What's the average position of oscillating particles in a box with periodic boundary conditions?

Imagine an open box repeating itself in a way that a if a particle crossing one of the box boundary is "teleported" on the opposite boundary (typical periodic boundary position in 3D). Now put a ...
12
votes
4answers
397 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 ...
3
votes
1answer
45 views

How does one plot the frequency and time domain of a distribution on a 3D plot?

I have seen this plot on Wikipedia's Fractional Fourier Transform, where it discusses the rotations between frequency and time domains of a distribution, however I do not understand how to plot a ...
9
votes
2answers
746 views

What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?

I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following: $|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$. $|p\rangle$ is an eigenvector of ...
2
votes
1answer
115 views

Fourier integral form of the delta function?

I'm learning basic maths for physicist and was wondering what do we use the Dirac delta function for? What is the difference between "the Fourier integral form" and the usual way of expressing the ...
2
votes
1answer
54 views

Physical interpretation of the relation $\dot{x}(t) \rightarrow i \omega \tilde{x}(\omega)$

If $x(t)$ is some time-dependent real quantity i can interpret its Fourier Transform $\tilde{x}(t)$ as representing, in a generic sense, the frequency components of $x(t)$. What about the FT of ...
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vote
1answer
46 views

How do you handle a functional input in a Dirac delta function and prove these types of relations?

I have a quadratic relation inside of a Dirac delta function with the following relation \begin{align} \delta((x-x_1)(x-x_2)) = \dfrac{ \delta(x-x_1) + \delta(x-x_2) }{|x_1-x_2|}. \end{align} How do ...
3
votes
1answer
141 views

Quantum field theory: field operators in terms of creation/annihilation operators

I am learning Quantum Field Theory and there is a step in my notes that I do not really understand. It starts with the classical definitions of position $q$ and momentum $p$: $$ q = ...
1
vote
2answers
76 views

How is the integrand concluded to be identically zero?

In expanding the classical Klein-Gordon field in Fourier space to write it in terms of $\phi(\mathbf{p})$ instead of $\phi(\mathbf{x})$, I reached the following result. $$\int ...
1
vote
1answer
358 views

Initial condition for Fourier transformed Schrödinger equation

I asked in this thread Time-dependet Schrödinger equation how to solve the Time-dependent Schrödinger equation. One of JamalS' recommendations was the Fourier transform, which is why I want to quote ...
2
votes
2answers
108 views

Modeling the free space propagation of laser beams using Fourier transforms

I am trying to model the propagation of a laser beam in free space. I have an initial field $E_{in}(x,z=0)$ (a Gaussian beam) and need to find the fields at other points on the optical axis $E(x,z=d)$ ...