Questions tagged [fourier-transform]
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 calculating the response of a time shift invariant linear system as the sum of its response to time harmonic excitation or for transforming a quantum state in position co-ordinates into one in momentum co-ordinates and contrawise. There is also a discrete, fast Fourier transform for discretised data.
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Vibrating string at many frequencies at once [closed]
I understand standing waves . When it vibrates faster it pushes air faster higher frequency . What about a plucked string? Does different segments have their own standing wave as the string as a whole ...
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About the vector potential
I have a question that is probably trivial concerning the vector potential used in electromagnetism. When solving the wave equations for the vector potential $\mathbf{A}$, we are essentially ...
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Beating the uncertainty principle for musicians
https://www.phys.unsw.edu.au/jw/uncertainty.html
The musician's uncertainty principle as above states that tuning can be less precise in short notes. But when we have a string with knowing its ...
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In what sense is "momentum" flowing in Feynman diagrams?
Given a Lagrangian $\mathcal{L} = \mathcal{L}_0 + \mathcal{L}_I$, we can construct the Feynman diagrams for some process by writing out the Taylor series for our interaction term and judiciously ...
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Fourier Optics Relation with Signal Processing Pulse
In signal processing, a very short pulse in the time domain can be understood as a superposition of many frequency components in the frequency domain.
In imaging or Fourier optics, can we find an ...
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Solutions for massless scalar field vs massive scalar field
My question relates to the difference between the solutions for massless scalar field vs massive scalar field, as it appears in the book: Quantum Field Theory for the Gifted Amateur from Lancaster &...
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How to derive Fourier-Laplace transform of random walk master equation on 2D lattice (Part 2)?
This question is linked with this question and is related to this paper.
The Fourier-Laplace transform is given by:
$$P(q,r,s)=\sum_{t=0}^{\infty}\sum_{m,n=-\infty}^{\infty}\frac{e^{iqm+irn}}{(1 + s)^{...
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How to derive Fourier-Laplace transform of random walk master equation on 2D lattice (Part 1)?
I have certain gaps in clearly understanding the derivation given in this paper. Suppose a particle moves on a 2D lattice randomly. The probability of going in any one direction outb of four available ...
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Density matrix of particle on a circle
Consider a particle constrained to a ring of circumference $L$. Following this paper, the position eigenstate on a circle can be expressed in terms of the position eigenstates on the real line as
$$
\...
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”Heisenberg-like” uncertainty principle(s) for antennas?
In harmonic analysis, we have the (one-dimensional) uncertainty principle:
$$\left(\displaystyle\int\limits_{-\infty }^{\infty }x^{2}|f(x)|^{2} \, \mathrm dx\right)\left(\displaystyle\int\limits _{-\...
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Diffraction Modeling Question
I'm trying to implement a model for diffraction-limited imaging, following "Microlithography" by Sheats and Smith. You can skip to the bottom for my question, but I'll explain the setup ...
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Lorentz transformation of momentum space and position space simultaneously
When we consider Lorentz transformations, do we consider the transformations of the momentum space and the position space simultaneously? Or do we do it depending on the problem i.e. if we work in ...
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On the Separability of the Coulomb Interaction into Ionic and Electronic Components within the Ewald Summation
Background and Context:
In calculations for periodic systems, such as ab initio MD, the Ewald method is employed to compute the Coulomb interaction. A known issue with the Ewald method is the ...
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Scalar fields satisfying Klein-Gordon equation
If a scalar field satisfies the following equation known as Klein-Gordon equation
$$
\phi_{;\mu\nu}g^{\mu \nu} + m^2 \phi=(\Box+m^2) \phi=0
$$
Let’s apply seperation of variables as
$$\phi = T(t)X(x)Y(...
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Convergence of $E$ for a system of free fermions using Green's functions
I am following "Quantum Theory of Many-Particle Systems" by Fetter and Walecka. The expression for the total ground-state energy of a homogeneous system of fermions in a box of volume $V$ (...
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Typo in "The Quantum Theory of Optical Coherence"?
I was reading through Glauber's "The Quantum Theory of Optical Coherence" Phys. Rev. 130, 2529, (1963) and found the following expressions for the positive and negative frequency components:
...
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Additional, non-integer harmonics in Fourier transform of a wave
Suppose we have a periodic wave, for example a note played on an instrument.
We assume ideal conditions, in the sense that the wave is periodic with frequency $\omega_0$.
We want to know the relative ...
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LSZ formula in momentum space
I'm reviewing the derivation of the LSZ formalism and I'm comfortable with the position-space version, which has the following form:
$$\left<{p_i};{\rm out}|{q_j};{\rm in}\right> = i^{n+m}\int \...
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Paradox regarding propagation of a pulse wave
In optics, we often use the complex amplitude to specify the electric and magnetic fields at any instant.
A rightward-propagating plane wave can be expressed as
$$
\exp(ikz)
$$
and
$$
\exp(-ikz)
$$
is ...
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Another question on Wick Rotation and field expansion
I’m sorry to be yet another person confused about Wick rotation, but it’s been all day and I’m still not sure I’ve got it right.
Let’s start by considering the field expansion of a scalar field:
$$
\...
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Diffraction through thick media
I want to calculate the diffraction pattern of light using (e.g.) the Rayleigh-Sommerfeld diffraction model.
For this, I am using the Python library Torchoptics.
However, I am struggeling on how to ...
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Why can't a particle with definite momentum be localised in space?
I'm watching this video by Float Head Physics, which presents a narrative where the YouTuber "Mahesh" is learning about the Heisenberg uncertainty principle from Feynman (starting around 13:...
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Intuition on the delta function as a basis [closed]
I'm following Littlejohn's notes on quantum mechanics, where it gave some examples of what can happen for operators acting on an infinite-dimensional Hilbert space such as 1d wavefunctions.
He first ...
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How do the different momentum definitions for the real scalar field differ among each other?
This is a question that I've had for a while now. This question of course is valid even when one considers some other type of theory e.g complex scalari field, spinor field etc. I am considering the ...
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Should a scalar field transform under Lorentz or Poincaré transformations?
I tried to check whether the usual plane-wave expansion for a Klein-Gordon Hermitian field is a scalar, meaning it transforms correctly under Poincaré transformations:
$$\hat{U}(\Lambda,a)\hat{\phi}(x)...
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How should mass-clusters after the shell-crossing look like?
I was trying to make a simulation of particles in the early universe and I was wondering how they should behave.
In order to do so, I placed them uniformly with the zeldovich initial condition, e.g ...
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Does applying a higher-order phase like TOD modulate a pulse's spectral intensity?
The spectral intensity of a pulse is usually described as being independent of its phase, $\phi(\omega)$:
$$E(\omega) = A(\omega)e^{i\phi(\omega)} \quad \rightarrow \quad I(\omega) = |E(\omega)|^2 = |...
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Non-equilibrium Green's functions for quantum transport - time vs energy domain
I am following Cuevas (Molecular Electronics, 2nd Edition) to learn about the use of non-equilibrium Green's functions for quantum transport and I have an issue with one particular step in his "...
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Kronecker's Delta definition in Ginzburg-Landau Ising Model
Introduction
I'm studying the Ising Model described by the partition function:
$$
Z = \sum_{\{s_{i}\}} \exp \bigg\{ \dfrac{1}{2} \, \vec{s} \, \mathbf{J} \, \vec{s}^{\,\scriptscriptstyle T} \bigg\} ...
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Fourier Transform for sinusoidal number-density to obtain the structure function
I was reading chapter 2 of Chaikin and Lubensky, where I got stuck at this derivation of structure function. While talking about Smectics-A liquid crystal, it was mentioned that the molecules are ...
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Momentum space Feyman rules $\phi^3$ theory [closed]
Although in my QFT we used mostly Srednicki's book, we did a derivation that is a little different from the one of the book regarding $\phi^3$ theory momentum Feynman rules.
Omitting the counterterms ...
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Question about an expansion of a scalar field
In my Quantum Field Theory (QFT) class I was asked to prove that the scalar field of the free theory can be written as a mode expansion of plane waves namely:
$$
\phi(x)=\int\frac{d^3p}{2\omega_p (2\...
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Gaussian integration of path integrals
I have a question regarding the Gaussian integration of path integrals with quadratic action.
Lets say the path integral has the standard form
$$ \int D[\Phi^\dagger,\Phi]e^{iS}$$
with a quadratic ...
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Derivation of Form Factor to Determine Nuclear Charge Distribution
Background
I am following Introductory Nuclear Physics by Kenneth S Krane. The derivation of form factor to determine nuclear charge distribution is given as follows:
Consider an electron with intial ...
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Deriving vertex factors in Feynman rules from the Lagrangian
In the book "Quantum Field Theory" by Srednicki on page 372, the author has written the interaction terms in the scalar elevctrodynamics as follows: $$\mathcal{L}_1=ieA^{\mu}\Big[ (\partial_{...
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Question about Srednicki's derivation of the Lehmann-Kallen form of the exact propagator
On page $108$, equation ($13.17$) of Srednicki's QFT textbook (draft version available online), he gives the Lehmann-Kallen form of the exact momentum-space propagator as
$$\begin{equation}
\tilde{\...
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What is so special about harmonics that they have a single frequency?
To the extent I understand, pure sinusoidal functions, $$\sin(\omega t), \quad \cos(\omega t) \quad \text{or} \quad e^{i\omega t},$$ that exists for an infinitely long time, $-\infty\leq t\leq \infty$,...
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QFT ground state of Klein-Gordon in basis
Against all warnings and guidelines I've tried to express some of the ideas in solving the Klein-Gordon equation in QFT using the wavefunctional in basis, using the Schrodinger picture directly.
We ...
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How is this quantity related to momentum? [closed]
I am studying the solutions of the Schrodinger equation where $V = 0$. That is, where the particle is free. This quantity is given:
$$\phi(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty}\Psi(x,0)e^{...
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Positive Frequency Modes in QFT and Lorentz Transformations in Minkowski Spacetime
For a real scalar field $\Phi$ in Minkowski spacetime, in the process of quantising this field, one has mode solutions that are obtained for the dynamical equation for $\Phi$. In particular, we define ...
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Confused about noise power spectral density measurement
My question was started from this paper. The topic of the paper is related to linewidth measurements, but my question is even before that, related to the way the noise power spectral density (PSD) ...
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Singing glass frequency - why is it audible?
I'm thinking about the typical singing glass demonstration, where you rub a finger around the edge of a wine glass and can hear a loud sound at the natural frequency of the glass. The fact that it is ...
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Units of wave functions in real and reciprocal space
I'm confused about the units of wave functions in reciprocal space and their Fourier transform in real space. On one hand, I believe the Fourier transform of a reciprocal space wave function in 2D is ...
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Analysis to periodic diffusion at long times (Sethna Q2.7)
This is problem 2.7 from Entropy, Order Parameters, and Complexity by James P. Sethna.
Consider a one-dimensional diffusion equation $\frac{\partial \rho}{\partial t} = D\frac{\partial^{2}\rho}{\...
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Diagrammatic interpretation of the four-point correlation function in momentum space
In Peskin & Schroeder's book (chapter 7.2,p227), they claim that the exact 4-point function, expressed by
$$
\left( \prod_{i=1}^{2} \int d^4 x_i \, e^{i p_i \cdot x_i} \right)
\left( \prod_{i=1}^{...
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Why can we extract the value of $S$-matrix element by simply analyzing the pole of correlation function?
In Peskin & Schroeder's book (Chapter 7.2, p.223-226), they analyze the pole of the correlation function by discarding the exponential term.
They first calculate the accurate value of the ...
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Computing position two-point function and Fourier transform of $p^4 \ln p$
I am computing a two-point correlator in 4D Euclidean space and I am struggling with one particular term. I have found that in momentum space my correlator goes as
$$\langle \mathcal{O}(p)\mathcal{O}(...
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Incompleteness of positive-energy solutions to Klein-Gordon equation
In non-relativistic Quantum Mechanics, plane waves (didn't attempt to normalized below) or eigenstates of momentum operator form a complete set of basis of the state space $L^2(\mathbb{R}^3)$ (...
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A particle confined without forming a standing wave
Can a particle remain confined given that the wavefunction $ψ$ at the boundaries is 0, even if it doesn't form a standing wave? What exactly is confinement? I think it's a condition where $Δx$ (the ...
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How do we express the quantum state of a particle in momentum eigenfunctions basis given that eigenfunctions of momentum are not square-integrable?
As far as I know, for a particle without spin the quantum state of the particle is described by a "point" in the Hilbert space of the equivalence classes of $L^2$ square-integrable functions ...
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