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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Marginal theorem in frequency-resolved optical gating (FROG)

In ultrashort laser physics, frequency-resolved optical gating (FROG) has been a standard method since a few decades to measure the temporal profile of an unknown pulse's electric field $E(t)$ via a ...
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Hamiltonian in real and reciprocal space

I found that sometimes people mentioned that Hamiltonian in real space or Hamiltonian in reciprocal/$k$-space. I wonder what difference of Hamiltonian in real and reciprocal spaces are? For example, ...
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Plucked string at $t=0$ and its Fourier decomposition

Decomposing a function into a Fourier series is possible for periodic functions. Fourier transform, on the other hand, is used for aperiodic functions. How can we use Fourier series to analyse the ...
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Nearly-free-electron model in 2D

I'm trying to compute the first five energy gaps at the point (1,0) for nearly free electron in 2D lattice. Where the potential is describe by V(r) = e$^{\frac{-\mid r \mid}{b}}$ (A, b constants and r ...
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Fourier transform of current density for a single particle

I would like to obtain the Fourier transform (frequency domain) for the current density generated by a single charged particle in uniform linear motion. In time domain the expression would be like ...
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Is every pair of conjugate variables associated with a Fourier transform?

For example, in quantum mechanics, the commutator of the position and momentum is $$[\hat{P_i} ;\hat{Q_j} ] =i\hbar\delta_{ij}\neq 0, i\neq j$$ I know that the position space representation of the ...
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Fourier transform of angular function

Following the notes on Emergence of AdS/CFT I am trying to follow through the bulk reconstruction section. I understand how the bulk fields are given by $$ \phi(r,t,\Omega)=\sum_{nl\vec{m}}\left(f_{...
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A Naive Question of Canonical Quantization A Real Scalar Field

When we use canonical quantization method to quantize a free real scalar field in Schrodinger Picture. The free real scalar Lagrangian is $$\mathcal{L} = \frac{1}{2}\partial_\mu\phi\partial^\mu\phi - \...
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Step on canonical quantization

So I've been trying to solve the expression for the Hamiltonian using the canonical quantization of a complex scalar field and I am not sure of how the following step comes by, from $$\mathcal{H} = \...
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Fourier Transform in the Path Integral of a Harmonic Oscillator

My question comes directly from Section 7 of Srednicki's QFT textbook. I'm not able to reproduce Equation (7.5): $$\begin{aligned} [\cdots]=\frac{1}{2} \int_{-\infty}^{+\infty} \frac{d E}{2 \pi} \...
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Dissatisfied with textbook explanations for why $\vec k$ in Bloch's theorem can be restricted to thefirst Brillouin Zone (BZ)

By Bloch's theorem, all the eigenfunctions of a Hamiltonian with a periodic potential $$U({\vec r}+{\vec R})=U({\vec r})$$ can be chosen to have the form $$\psi_{n{\vec k}}({\vec r})=e^{i{\vec k}\...
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Stuck with the derivation of correlation function from Huang's Statistical Mechanics

Context Section $16.2$ of Kerson Huang's Statistical Mechanics ($2$nd edition) deals with a derivation of two-point correlation function $\Gamma({\bf r})$, defined in terms of an order parameter ...
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Understanding electrons in a weak periodic potential fourier analysis

I have been trying to understand Ashcroft's take on electrons in a weak periodic potential, and his approach by Fourier analysis is proving to be extremely challenging. I understand how to get to the ...
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Action of the Liouvillian in Koopman-von Neumann Mechanics

I am trying to calculate the effect of the KvN Liouvillian operator but am getting a bit stuck. As it is the generator of time translations in KvN mechanics, and since in classical mechanics a point ...
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Approximation from discrete Kronecker Delta to continuum Dirac Delta

I am working on second quantization of the Dirac field with discrete momentum I was asked to compute the creation/annihilation anticommutator by imposing the anticommutators on $\psi$ i.e. $$ \{\...
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Fourier Transform of probability density in real space as a convolution of its frequency amplitudes

I'm trying to find an efficient method to calculate the Fourier transform of a probability density at a cylindrical boundary an infinite distance away given some arbitrarily placed point emitters ...
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Fourier transform of fermionic creation/annihilation operator

How should I picture the Fourier transform of a fermionic creation (annihilation) operator acting on a site of a periodic, say one-dimensional, lattice? I mean, in a real-space picture, what are the ...
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Separating Hamiltonians in spontaneous symmetry breaking investigation

I read a book about spontaneous symmetry breaking. In the book, the author says: Using a Fourier transformation, it's always possible to divide the Hamiltonian into two parts, one is collective part ...
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Why this sum of creation and annihilation operator?

In Schwartz (2.75) he defines a free quantum field as follows: $$ \phi_0(\vec{x}) = \int \frac{d^3 p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_p}} (a_p e^{i\vec{p}\vec{x}} + a_p^\dagger e^{-i\vec{p}\vec{x}})....
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Fourier Transform of a Superlattice Hamiltonian

In a paper by Gábor B. Halász and Leon Balents they derive the energy band structure for a Hamiltonian that models a time reversal invariant realization of the Weyl semimetal phase. The model is a ...
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On the uniqueness of Wannier functions

Was reading this text 'Electronic structure : basic theory and practical methods' by Richard.M.Martin and came across this following line: 'The most serious drawback of the Wannier representation is ...
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What is the direction of spin in the Fourier transformed space for the following situation?

I have an expression, like, $$\frac{1}{2}\mathbf{\hat{q}}\cdot\xi^{r\dagger}\boldsymbol{\sigma}\xi^s,$$ where, $\mathbf{\hat{q}}$ is the momentum unit vector and $\xi^s$ is a Dirac spinor. The above ...
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Contradicting power spectral densities and autocorrelation in shot noise

I have a white noise generator circuit which involves a Zener diode meant to undergo breakdown across a resistor $R1$, and am trying to compare power spectral densities and autocorrelation of the ...
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Fourier Analysis for Physicists

My professor wanted me to master these topics from Fourier Analysis. I need a resource where these topics are discussed in brief. Although i know many of the topics in the list, i prefer a good ...
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Reciprocal lattice vectors

Given a lattice with primitive vectors $\{\mathbf{g_i}\}$, one can write the position of any lattice point as $\mathbf{R}_n = n^i \mathbf{g_i}$ with $n^i \in Z$. Quantites, such as electron density, ...
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Fourier transform of Fermi function

As an alternative approach to the Sommerfeld-expansion, my lecturer tries to motivate properties of free fermions, such as temperature dependencies of the chemical potential $\mu(T)$, electron number $...
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Fourier Optics - application of the 4f correlator experiment

I am in high-school and planning on basing a research essay on the topic of Fourier Optics. I was looking at the derivations behind the Fourier transform and the fact that the Fourier transform of the ...
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Fourier transform of $f(x)=1/x^n$ at physicist level of rigour

Functions such as $f(x)=1/x^n$ where $n$ is a positive integer and $x$ is a real variable in $-\infty\leq x\leq \infty$, strictly do not have Fourier transforms. But when applied to physics, can we ...
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On the quantum computer implementation of FFT

In Fig. 1 of the paper Exact Ising model simulation on a quantum computer, it appears a circuit to implement a Bogoliubov and (discrete) fast Fourier transform (FFT) over 4 qubits in order to ...
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The solution to the differential equation, equation 1.7.28 in Sakurai (2nd Edition) [duplicate]

While I know this is an extremely well known result, I was trying to derive the expression for $ \langle{x}|{p}\rangle$, as done in Sakurai. It is basically a two line equation, so our starting point ...
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Fourier Optics - Impulse Response of Free Space from Fresnel Transfer Function

I am currently reading the chapter "Fourier Optics" in the book "Fundamentals of Photonics" by Saleh and Teich. However I am not able to follow one specific mathematical derivation. On page 111 the ...
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Is it possible to specify the position of a photon to arbitrary amounts? [closed]

For a photon, $$E = pc$$ from the Einstein Energy Equation. Or $$E = hf$$ with $$p = hf/c$$ From the Heisenberg uncertainty principle $$(\Delta x)*(\Delta p) = h/2$$ The maximum value for $\Delta ...
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How to interpret the cross-spectral density of an incoherent field

I have been given a definition of the cross spectral density of a completely incoherent field: $$W(x_1,x_2)=S_0\delta(x_1-x_2)$$ How do I interpret this? As I understand it, this means that there ...
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Can you extend the range of a simulated (discrete) momentum wavefunction by interpolating in position space?

$\newcommand{\ket}[1]{|#1\rangle}$ I did all the calculations in Matlab, however this question is not exclusive to this software. I have two 1000x1 arrays, $\vec a$ and $\vec b$, containing complex ...
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Refocusing light field images via Fourier Slice Photograph theorem

I am trying to refocus images from a microlens array light field using Ren Ng's Fourier Slice photograph theorem found in his thesis chapter 5, equation 5.7, which is available at https://stanford.edu/...
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Density operator in momenta representation using Fourier transform

I'm determining the density operator in momentum space and by working out the Fourier transform from the coordinate representation I get to: $$\hat n_q=\sum_{kk'ss'}\left<k,s|e^{-iq\hat r}|k's'\...
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Momentum space wave equation of free particle: constant factors

I'm trying to solve problem 3.12 in D.J. Griffiths's "Introduction to Quantum Mechanics 3rd ed."; it is as follows: Find [the momentum space wave equation] $\Phi(p,t)$ for the free particle in ...
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What paths are allowed in the Fourier form of the Dirac Delta distribution?

In this form of the Dirac Delta distribution $$\delta(x) = \frac{1}{2 \pi i}\int_{- i \infty}^{i \infty}e^{-\omega x} d\omega$$ can $\omega(t)$ be evaluated over any path (that starts at $\omega(-\...
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Why does the S matrix always contain a factor of $(2\pi)^4?$

In quantum field theory, one usually defines the scattering amplitude as $$S-1=(2\pi)^4\delta(p_{out}-p_{in})M_{Scattering Amplitude}$$ Where S is the S matrix element for any scattering process. It's ...
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Input field conditions for beam gaussian propagation

I am trying to calculate a beam propagating through a lens with focal length $f_0$ and a gaussian shape perpendicular to the beam propagation axis. For that I use the equation $$\partial_zE=\frac{i}{...
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Power spectrum on wavelength

For a peak in hydrogen spectrum, we fit a peak function of Lorentzian. I seeked for an explanation and I found the power spectrum $|f(\omega)|^2$ is in fact the Lorentzian. $f(\omega)$ is the Fourier ...
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Dyson-Maleev transformation on an AFM model and Fourier transformation

In both Dyson-Maleev transformation (DMT) and Holstein-Primakoff transformation (HPT), spin operators are transformed into Boson operators. I am trying to transform an antiferromagnetic (AFM) model ...
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Fourier Transform in special relativistic scenarios

I am currently reading this paper https://arxiv.org/abs/nucl-th/9505032. I want to prove equation 2.11 using 2.3 and 2.1. The result 2.11 seems quite obvious as $S(x,k)$ looks like a Gaussian in 4D ...
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Action of quantum Fourier transform on two-fermion states

In section 2.2 of the paper https://arxiv.org/abs/1807.07112, there appears a Fourier transformation named $F_k^n$ that comes out of a matrix called $F_2$, $$ F_2 = \begin{pmatrix} 1 & 0 & 0 &...
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Fourier transform of the product

While reading the book “Modern Consensed Matter Theory”, I came across the following calculation. $f(k) g(k) = \int d^d r e^{-i kr} f(-i \nabla) g(r)$ I know the convolution theorem for Fourier ...
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Studying Chaos in RLD circuit

We are currently working on non-linear dynamics (chaos theory) by analysing a series circuit including a diode (the 1N4004), a 100 ohm resistor and a 20 mH inductance. It is driven by an alternative ...
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How to expand a wave function in terms of momentum eigenbasis without invoking the energy eigenbasis?

So, I am trying to learn Quantum Mechanics on my own and I am using MIT OCW's 8.04 lecture series and I was learning about expansion of wavefunctions in terms of energy eigenfunctions. For some reason,...
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Canonically conjugate variable of rapidity [closed]

Does the rapidity $\theta \in \mathbb{R}$ have a canonically conjugate variable? More specifically, for some smooth function $f \in \mathcal{S}(\mathbb{R})$, by Plancherel's theorem we have (up to ...
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Power Spectrum Density of real valued time series data

There are real valued time-series data X(t) and corresponding auto-correlation function ACF(t)=$\left<X(0)X(t)\right>$. As written in wikipedia, Power Spectrum Density (PSD) can be calculated ...
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Orthogonality of exponential functions in the infinite potential well of 1-dimension

I'm struggling to write the Fourier expansion of the wave-function associated to a particle confined in an infinite potential well. Suppose that $V(x) = \left\{\begin{array}{c c} \...

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