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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Dimensional analysis of quantized Klein-Gordon Field

For the free Klein-Gordon Lagrangian density: $$\mathcal{L}=\frac{1}{2}\partial^{\mu}\phi\partial_{\mu} \phi-m^2\phi^2 .$$ Since we need the dimension of Lagrangian density equal to 4 (in this case ...
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Is momentum space "less physical" than position space?

In quantum mechanics and quantum field theory it is specially common to work in both position and momentum space. Passing the theory to momentum space is sometimes crucial, as one usually finds that ...
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How come a Fourier transform approaches zero as the oscillation frequency increases?

I'm somewhat confused as to why the Fourier transform goes to zero. The only mathematical proof I've found which might be applicable is the Riemann-Lebesgue lemma, but I'm not sure that applies here. ...
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Correlation function in polar coordinates

I know that the Wightman correlation function is defined by $\langle 0|\hat{\Phi}(x)\hat{\Phi}(x')|0\rangle$ and it's expansion leads to an explicit expression depending from the commutator $[\hat{a}...
4 votes
2 answers
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Applying measurement postulate to a continuous sum of eigenvectors (by analogy)

Measurement postulate: If we measure the Hermitian operator $\hat Q$ in the state $Ψ$, the possible outcomes for the measurement are the eigenvalues $q_1$, $q_2$, . . .. The probability $p_i$ to ...
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Property of reciprocal lattice

I came across the following property of the reciprocal lattices. Let be $\Lambda$ a Bravais lattice and $\Lambda^*$ its reciprocal lattice; let be $\vec{G} \in \Lambda^*$ and $\vec{G}_0$ the shortest ...
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Time dependent operators in QFT

In Quantum field theory, how does one define time-dependent operators? For example, let me generalize the operator fermion $\psi$: $\psi(x) = \int \frac{d^3p}{2 (\pi)^3} \frac{1}{\sqrt{2E_p}} \sum_s \...
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Wavelength and frequency associated with a wave pulse

What are the definitions of wave length and frequency of a wave pulse?
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To what extent can we use the informal version of the Dirac delta function in Physics?

Apparently expressions such as $$ \int \delta (x) f(x)dx = f(0)\tag{1}$$ are widely used in Physics. After a little discussion in the Math SE, I realized that these expression are absolutely wrong ...
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How to prove this equation?

In J.J. Sakurai's Modern QM(Third edition), section 6.4.1, the author says that the equation 6.104 $$ \frac{e^{i\bf{k\cdot x}}}{(2π)^{\frac{3}{2}}}=\frac{1}{(2π)^{\frac{3}{2}}}\sum_l(2l+1)i^lj_l(kr)...
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Peculiar calculation of the Klein-Gordon Propagator

I am reading Peskin & Schroeder's QFT textbook (page 29~30). Here, to calculate Klein-Gordon Propagator, author computes following integral. $$\left< 0 | [\phi(x), \phi(y)]|0\right> = \int \...
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Wave function Fourier transform with time

I found the Fourier transform at $t=0$ for the wave function of a wave packet (and it's inverse Fourier transform) : $$\Psi(x,0)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\Phi(k)e^{ikx}dk$$ $$\Phi(k)...
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Factor of $(2\pi)^4$ in momentum space Feynman rule

I'm trying to figure out the momentum space Feynman rules using Peskin and Schroeder. For simplicity I'll ask about section 4.6 for case of the $\phi^4$ theory. In section 4.5, we have $$\tag{4.72}S=1+...
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Fourier transformation for Neumann boundary condition above 1D

I want to apply integral transform method to a Poisson's equation in a sphere $$ \nabla^2 u=f(\boldsymbol r) $$ with Neumann boundary condition $$ \boldsymbol n\cdot \nabla u|_{r=R}=0. $$ Physically ...
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Converting Feynman Rules from in-out formalism to in-in formalism

For a standard set of Feynman rules (following in-out formalism) in momentum space, extracted from a generally given Lagrangian, is there a generic algorithm for converting them into the Feynman rules ...
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Construction of the Klein-Gordon field theory - what is missing?

Many references I know on QFT start the discussion of the Klein-Gordon field theory with some discussion about harmonic oscillators. One such reference is Folland's Quantum Field Theory book. The idea ...
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How can i derive conserved charge of $SO(3)$ internal symmetry?

I'm studying on QFT for gifted amateur written by Tom Lancaster chapter 13.1 and i'm not fully understanding about derivation of conserved charge of $SO(3)$ internal symmetry. So if there's three ...
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Why is $<x|p>$ a plane wave? [duplicate]

Starting from the reasoning that $\langle x|p \rangle=e^{\frac{ipx}{\hbar}}$, I understand why the momentum operator in position space is $-i\hbar \partial_{x}$. What I'm looking for is some sort of ...
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Question on the bounds for finding Fourier coefficients

In Griffit's E&M, when solving Laplace's equation for the potential, he uses the "Fourier trick" on Legendre polynomials, where my question is, why are the bounds from -1 to 1? because ...
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Inverse Fourier Transform of $e^{i\mathbf{k}\cdot\mathbf{R}}$ in Brillouin Zone in Proving Orthogonality of Wannier Functions [closed]

The relationship between wannier function and Bloch function like this: $$ |\mathbf{R}_n\rangle = \dfrac{V}{(2\pi)^3} \int_{\mathrm{BZ}} |\psi_{n\mathbf{k}}\rangle e^{-i\mathbf{k}\cdot \mathbf{R}}d\...
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Why is $e^{i\hat{p}L/\hbar}$ only an operator when it is outside an integral?

Looking at the screenshot provided below, which is an excerpt from this textbook, really nothing more than a derivation of momentum as the generator of translations, would someone be kind enough to ...
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How to decompose Dirac spinor into plane wave solutions?

Suppose that we know Dirac spinor $\psi$ (as complex numbers) in every point in 3d space (4th dimension is time) How do we decompose it into plane wave solutions $u^s(p) e^{ipx}$ and $v^s(p) e^{ipx}$? ...
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Doubts about Fourier transform of IR spectroscopy

I was studying a Michelson interferometer for infrared absorption in Fourier transform and I've found these two images (taken from https://pages.mtu.edu/~scarn/teaching/GE4250/ftir_lecture_slides.pdf )...
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What's the rationale of replacing the Fourier coefficients in a field expansion by operators?

Let's take a look on the particular case of the Fourier expansion of the Klein-Gordon field: $$\psi (x,t) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_0(p)}[a(p)e^{i(E_0(p)t-px)}+a^\star (p)e^ {-i(E_0(p)t-...
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How can I find an operator originally expressed in terms of raising and lowering operators in terms of the field operators?

I'm following this book on QFT called "Quantum Field Theory of Point Particles and Strings" by Brian Hatfield. After the end of the scalar field theory section on Exercise 3.6, it asks us to ...
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Is there a way of determining information content of a phase hologram?

For a normal image, it is possible to compute the two-dimensional Shannon entropy to determine the information content/complexity within the image. For example, an image of a natural landscape will ...
5 votes
3 answers
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Question About Momentum and Position Operators and the Postulates of Quantum Mechanics

I've surmised that there are four big facts about the relationships between the position and momentum operators, their Hilbert spaces, and their eigenstates in QM. I think I am just about at a point ...
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How to get the 'momentum space' equation from the Proca equation via $p_\mu \leftrightarrow i \hbar \partial_\mu$?

In Introduction to Elementary Particles (page 370, Second Edition) Griffiths writes that $$\left[ \left(-p^2 + (mc)^2 \right) g_{\mu \nu} + p_\mu p_\nu \right] A^\nu=0$$ derives from $$\partial_\mu (\...
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Energy minimization of complex periodic scalar field with modulus constraint

I'm looking for an elegant method to (in general numerically) minimize an energy functional $E(\psi)$ for a complex field $\psi(x,y,z)$ which I know will be periodic in $z$ direction (due to self ...
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A specific question about sakurai modern quantum mechanics

In the kronocker deltas for spin, $\lambda_4$ and $\lambda_3$ places changes. How can the author change their places? Also, I see both case can appear by calculation so there are two different ...
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How to calculate decay in plasmonic wave amplitude travelling inside a metal-insulator-metal waveguide?

I'm working on a system of plasmonic resonators along a waveguide, to calculate the transmitance of the system I'm using a system of coupled differential equations (CMT), the system consists of 8 ...
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1 answer
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If we perform fourier transform to a pure sine wave, why it gives a bunch of frequencies?

If you perform a fourier transform to a pure wave say 5 cycle per second you would get the dominant frequency of the wave it is made of. But not only it gives up the only frequency the wave has been ...
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Is there any physical advantage in placing the sample between the recombined beam and the detector in Fourier Transform spectrometry?

Usually, in all the implementations of FT spectroscopy the sample is always placed between the recombined beam and the detector like in this figure from Wikipedia: Theoretically, the interferometer ...
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On the Integral Representation of Greens Functions

In Birrell & Davies book "QFT in Curved Spacetime" the authors discuss in chapter 2.7 that all Greens Functions (which are the vacuum expectation values of products of fields) can be ...
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Showing that flux fluctuation is $q^2 P_n(q)/2\pi$

In the context of spectral analysis of a sufficiently small section of the sky, 1 states that the flux fluctuation is given by $q^2 P_n(q)/2\pi$ on the angular scale of wavelength $2\pi/q$, where $P_n(...
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How is differential momentum assigned in multiparticle system of QFT?

I've been following Schwartz's book on quantum field theory, and got stuck at page 59 on Section 5.1 'cross section' of the book which argues that the region of final state momenta is the product of ...
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Fourier transform of an equation of Matrices

I am confronted by an equation of the form \begin{equation} A[\omega] = (-i \textbf{1} \omega - i C[\omega])^{-1} \left( -i D[\omega] -i E[\omega] F[\omega] \right) \end{equation} where capital ...
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Can anyone refer me to a textbook or paper that explains the $k$-space diagrams used in Fourier optics to describe diffraction?

I am taking a course in Fourier Optics. Often, we work with these $k$-space diagrams, similar to the one shown in the image below. But we use them without understanding the mathematical justification ...
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1 answer
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Equal-time Canonical Commutation Relation for a scalar field

In chapter 2 of Quantum Field Theory and the Standard Model, Schwartz derives the equal-time commutation relations of the second-quantised field. Using $$ \phi(\vec{x}) = \int \frac{d^3p}{(2\pi)^3} \...
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Expression of Klein-Gordon field in Heisenberg picture

In Schrodinger picture, the scalar field is $$ \phi(\vec{x}) = \int \frac{d^3 p}{2E(\vec{p})} \left( a(\vec{p}) e^{i\vec{p}\cdot\vec{x}} + a(\vec{p})^{\dagger} e^{-i\vec{p}\cdot\vec{x}} \right). \...
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What would happen when two wave functions intersect in a Fourier series representation of periodic signals? [closed]

I saw a piece of code on github which transforms the planetary movement into the fourier wave function. These circles are given by the x and y ordinates: x=cos(ωt) y=sin(ωt), which are periodic. ...
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How to understand the term $\frac{1}{-2E(\vec{p})} e^{-ip(x-y)}$ of Klein-Gordon propagator in Peskin & Schroeder's book?

I am reading Peskin & Schroeder's book on Chapter 2. I have a question about how to get the term $\frac{1}{-2E(\vec{p})} e^{-ip(x-y)}$. The original equation for propagator is $$ \langle 0 | [\phi(...
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Can a point $\vec R$ in direct lattice be uniquely mapped to a point in the reciprocal space?

Given a point in the direct lattice $\vec R=\vec a_1+\vec a_2+\vec a_3$ (say), what is the reciprocal lattice vector $\vec G$ corresponding to $\vec R=\vec a_1+\vec a_2+\vec a_3$? The reciprocal ...
1 vote
1 answer
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What is the connection between reciprocal lattice vectors $\vec G$ and the Miller indices?

We know that a family of crystal planes with Miller indices $(hk\ell)$ is orthogonal to the reciprocal lattice vector $\vec G = h \vec b_1 + k \vec b_2 + \ell\vec b_3$. My question is the converse of ...
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Why are power spectrum plots $l^2 P(l)$ instead of just $P(l)$?

Why is it typically plotted $l^2 P(l)$, or $l(l+1) P(l)$, vs $l$ instead of just $P(l)$ in power spectrum plots? For example, we can see it in this plot found in Introduction to Gravitational Lensing ...
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Why isn't the time dependent Schrödinger equation in the momentum basis simply the time independent one? [duplicate]

If $-i \frac{ \partial}{\partial x}$ becomes $p$ in the momentum basis, I would expect the energy to be the same: $$i\frac{\partial}{\partial t} \to E$$ So the time dependent Schrödinger equation ...
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Fourier component of electric field

The paper deals with Terahertz beam production from ZnTe emitters. We've eqn. for generated THz beam from an area element of emitter crystal giveb by:- $E_{THz}(t)=K_{eff}W_{opt}T(t)$ - eqn(1) where, $...
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Hamiltonian Open String

On page 38 of Becker Becker Schwarz, we're given equation 2.69 which is the Hamiltonian for a string given as $$H=\frac{T}{2}\int_{0}^{\pi}(\dot{X}^{2}+X^{'2}).\tag{2.69}$$ Considering the open string ...
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Peskin and Schroeder confusion on promoting Classical Klein-Gordon equation to quantum field equation

I am reading "An introduction to quantum field theory" by Peskin and Schroeder and I am confused. I appreciate your help. Here's the context to my question: In chapter 2, the book introduces ...
4 votes
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Is a normalized wave function in the position basis automatically normalized in the momentum basis?

If the wave function is normalized in the position basis: $$\int \Psi^*(x)\Psi(x) dx =1.$$ This means the probability of finding the particle somewhere is one. Similarly, in the momentum basis: $$\int ...
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