# 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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### 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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### 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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### 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 ...
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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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### 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 ...
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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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