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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Calculation of fluctuation corrections to the saddle point approximation

From Statistical physics of fields by Mehran Kardar page 45 section 3.6 [...] the partition function including small fluctuations is $$\begin{align} Z \approx e^{-V\left(t/2 \cdot \overline m^2 +...
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Fourier Transform from lattice site into $k$-space in Hubbard-Holstein model

Say I have a one dimensional lattice with lattice constant $a$. With next nearest neighbor hopping (NNN) included, the hopping term that describe such system would be $$H_{hop} = -t\sum_j(\hat c_{j+1}...
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How to solve this problem involving the “longest interval”?

The problem is shown as follows: If one wants to make a digital record of sound such that no audible information is lost, what is the longest interval, $\Delta t$, between samples that could be ...
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The validity of some “applications” of the uncertainty principle

Given a $L^2$ function $f$ with $\int_\mathbb{R}xf(x)dx=0$, define its variance to be $\sigma_f^2=\int_{\mathbb R}x^2f(x)dx$. The uncertainty principle states that $\sigma_f\sigma_\hat f\geq 1/4\pi$,...
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Is there an explanation for this unexpected similarity between binomial coefficients and waves?

Background Binomial coefficients appeal mostly in probability, combinatorics number theory etc so were were surprised when we observed something that appeared to belong more to physics than pure ...
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Solving free particles with Fourier series

Here's a silly idea : take the action of a free particle, $$S = \int_{t_1}^{t_2} \dot{x}^2 dt$$ Our configuration space is the space of $C^1$ functions over $[t_1, t_2]$, which is spanned by the ...
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Triple infinite summation of a 3D Fourier series for Madelung Potential

I'm trying to evaluate the equation below excluding the case when $n_x=n_y=n_z=0$. I know this equation converges everywhere except where x,y, and z are all multiples of $2\pi$. I've attempted ...
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Projection into Lowest Landau Level and Fourier transform

I am studying Quantum Hall and therefore Laughlin wave functions and the Lowest Landau Level. States in the Lowest Landau Level have the form: $\phi_m(z,\bar{z}) \propto z^m exp( - z \bar{z} / 4 l^...
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Quantizing Klein Gordon Field: Sign Problem

I'm trying to re-derive the Quantization of the Klein Gordon Field but I'm running into sign problems. My starting point is: $$ \phi(x,t) = \frac{1}{(\sqrt{2 \pi})^3} \int \tilde{\phi}(k,t) e^{i kx}...
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Why is the optimum window length for a discrete fourier transform of a signal less than 100%?

I'm trying to determine the best settings for a discrete Fourier transform on a signal with noise. Now I've stumbled on something that I can't seem to explain, I'm hoping someone can give me some ...
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Momentum Wave Function gives strange expectation values

Suppose there's a particle with the wave function $\psi(x)=\frac{1}{\sqrt{L}}$ for $0<x <L$ and 0 everywhere else. One way to get the associated Momentum Wave function is direct integration on ...
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How many linear combinations of harmonics or normal modes can describe the same periodic function as a Fourier series?

Please note that I am not asking how many terms in a linear combination can describe a specific periodic function but if given that there exist a set or linear combination of normal modes that ...
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Impossibility of Monochromatic Light [duplicate]

Pages 24-25 of my textbook, Optics by Hecht, says the following: Using the above definitions we can write a number of equivalent expressions for the traveling harmonic wave: $$\psi = A\sin k(x \...
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Why does the resonant frequency disappear for a ball in a potential well being jiggled by multiple frequencies?

Here's what I'm doing: I'm using MATLAB to model a ball placed within a potential energy well. I'm then driving this ball with an external driving force. The function for the external driving force ...
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Can momenta eigenstate written in term of $x$ be an eigenfunction of position?

Being non-commuatable operators, momentum and position cannot have simultaneous eigenfunctions. But in "Theoretical Minimum: QM" by Lenny Susskind and Artsy Friedman, in explaining Heisenberg's ...
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Shifting identical but offset pulses in frequency domain

Supposes I have 2 identical pulses, but one is offset by some phase. I want to take fourier transform of the signal in time domain to frequency domain. In freq domain, their spectrums are identical, ...
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Quantum mechanics, Fourier transformation

Why do we use $p=-i\hbar\frac{\partial}{\partial x}$ in quantum physics? (I know the reason for $i\hbar$, quantization). Is this right to say we can't measure velocity and position of electrons at the ...
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Application in medical field

I have heard about many new developments in radiotherapy for treating cancer/tumors such as hadron therapy but why can't we use wave interference for it? Incoming waves could interfere destructively ...
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Wave equation calculation [closed]

when I have a 1 dimensional, linear wave equation: $\Box\ \psi(t,z)=(\partial_z^2-\frac{1}{v^2}\partial_t^2)\ \psi(t,z)=0$. $\Box$: d´Alembert operator So linear combinations of plane waves $e^{ik(...
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Heuristic for large $x$ behavior from small $q$ behavior of Fourier Transform

If I have a function $h(\mathbf x)$ which may be written $$h(\mathbf x)= \int \frac{\text{d}^d\mathbf q}{(2\pi)^d} \, h(\mathbf q) e^{-i \mathbf q \cdot \mathbf x}$$ and assume spherical ...
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Why does a non-linear system lead to interaction and frequency mixing between input's?

When we have a system that is nonlinear and we apply a sum of two different frequency sine waves as an input, we see the output of this system has components that are at the sum frequency of the two ...
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Fourier optics - Transfer function of free space

In all consulted literature, the transfer function of the free space is given as follows: $$\exp(-i k_z d) = \exp(-i2 \pi d \sqrt{1/\lambda^2 -\nu_x^2-\nu_y^2})$$ When referring to this source, they ...
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Fractional Fourier Transform and Fresnel Propagation

I am currently trying to wrap my head around Fresnel propagation, and I understand it is mathematically linked to the Fractional Fourier Transform, but I'm having a hard time with the units and the ...
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Exponential decay and growth oscillations at beginning and end of Fourier transform of data

I have some data from an photo-detector, which I think should correspond to white noise, but when I plot the Fourier transform of the data, I get a plot with exponential decay at the beginning and ...
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Why does jiggling a ball in a nonlinear potential well lead to multiple harmonics?

Here is my problem: In MATLAB, I am placing a ball inside of a potential energy well, I have control over what this potential energy well looks like (a parabola for example). I then drive the ball ...
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Contour for integration in 1D scattering problem

A plane wave scattered by a 1D potential can be described by, $$\psi(x) = \begin{cases} e^{ikx} + R e^{-ikx}, & x<0\\ T e^{ikx}, & x>0 \end{cases}$$ where $R$ is the reflection ...
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When taking the Fourier Transform of a sum of two different frequency sine waves, why doesn't the beating frequency show up as a peak?

If the Fourier Transform is "comparing how similar" each frequency is to the signal, then why doesn't the beating frequency show up? It's clear to see that the signal has an oscillatory term at the ...
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What quantities are conserved through for light passing through a converging lens?

I am currently learning Fourier Optics. I have a beam with a Gaussian profile which passes through a converging lens. The resultant cross section at the focal point is being calculated. I am using an ...
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Deriving Jefimenko's equations in Fourier space

From the Fourier-transformed Maxwell equations we have, with some algebraic manipulation, $$\mathbf{E}=\frac{1}{|k|^2}\left[\mathbf{k}\frac{\rho}{\epsilon_0}-\mathbf{k}\times k_0\mathbf{B}\right]$$ $$\...
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Transition from phi basis to occupation number in quantum field theory

We can construct the unitary transformation for change of basis from $x$ to number operator $n$ in harmonic oscillator by using $a|0\rangle=0$ and then multiply $\langle x|$ to the both side and ...
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Fourier transform of matrix element of evolution operator

I learn ''Path integrals in quantum mechanics'' by Jean Zinn-Justin now. There is a chapter about calculating the path integral for particle on a ring (rigid rotator). So, after some calculations we ...
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Finding normalization constant of a wave function with definite momentum

I try to read Sakurai's Modern Quantum Mechanics but I stuck at this point, $$\delta(x^{'}-x^{''})=|N|^{2}\int dp^{'}\exp\Biggl({ip^{'}(x^{'}-x^{''})2\pi\over h}\Biggr)$$ This is an expression for ...
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Physical Interpretation of Energy-wavenumber Graphs

Consider an energy-wavenumber graph, typical in solid state physics, like the one below. I can follow the mathematics in the derivations with a KP model. But I don't understand the physical ...
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Fourier transform of some discrete, finite, non-uniform signal

Suppose I have some finite signal $x(t)$ of $N$ data points. This signal is produced by some program or some experiment and so is discrete and the difference in time between each data point $\delta t$ ...
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Examples of non-sine waves? [closed]

What would be a non-sine wave? AFAIK, all sound is a sine wave, equally to waves on the sea. What would be a common example of something in nature that's a wave but not a sine wave? Or, would we have ...
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The action of a lens on a point dipole (Fourier optics)

I want to model the action of a lens on radiation from a point dipole. I have found examples but am struggling to interrupt them. My knowledge of Fourier optics is that from the Helmholtz equation, $...
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Properties of Dirac delta function in Integral

I was reading commutation relation of canonical momentum in KG Field from Lectures of Quantum Field Theory by Ashok Das. In page 179, He has used Integration to derive the result where he expressed ...
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Fourier transform of the wave equation

In string theory, as the one dimensional string propagates in time, it sweeps out a two-dimensional surface known as the string worldsheet. The spacetime coordinates are taken to be functions $X = X(x,...
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Path integral calculations $e^{i\omega 0^+}$

When computing correlation functions using the path integral formulation, I often need to compute integrals such as $$ \int_{-\infty}^\infty \frac{d\omega}{2\pi} \frac{1}{i\omega -\epsilon} $$ ...
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Can any solution to the three-dimensional wave equation be written as a superposition of plane waves?

Can any solution to the three-dimensional wave equation, $$\nabla^2f = \frac{1}{v^2}\frac{\partial^2 f}{\partial t^2},$$ be written as a superposition of sinusoidal plane waves? In "Introduction to ...
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Calculating the charge density from a form factor

For an atomic form factor $F(\textbf q)$, the corresponding charge density distribution is given by $$ \rho(\textbf r) = \frac{1}{(2\pi)^3}\int\text{d}^3 \textbf q \,F(\textbf q)\,\text{e}^{-\text{i}\...
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CFT in momentum space

Is there a way to see the conformal symmetry in momentum space in a CFT? I mean if I can recover the conformal group in some way in momentum space.
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Field operator commutation: If two operators commute, then their fourier transforms also commute?

Im doing this in the context of field operators $$\psi(x)=\sum_k a_k e^{ikx},$$ $$\psi^T(y)=\sum_k a_k^T e^{-iky},$$ and their being defined as the fourier transform of the creation/annihilation ...
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Fourier transform of the amplitude of a laser evolving along $x$-axis: how the cross-section appear?

Consider the vector potential describing the field of a laser defined for $x>0$. It has a cross-section $L^2$. The field propagates along the $x$-axis. In the book "Quantum measurement theory and ...
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What this quantum field operator represents $ b_{in}(t) = \frac{1}{\sqrt{2 \pi}} \int e^{-i \omega t} b(\omega)$?

In Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation Physics Department, Uniuersity of Waikato, Hamilton, ¹tuZealand (Received 29 ...
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Why EHT have to use Fourier transform to process the image?

I read that Fourier transform is a mathematical tool to deconstruct a wave taken from a source into basic sine and cosine waves, since visible light coming from the M87 accretion disk will be obscured ...
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Is there any nonlinear equations depending on Fourier coefficients?

A nonlinear partial differential equation is an expression depending on derivatives of $u$ $$f(x,t,u,u_x,u_t,\cdots)=0,$$ where the derivatives of $u$ can be obtained from the Taylor series of $u$. ...
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Structure factor in a homogenous system

I want to calculate the structure factor for a homogenous system. The system that I am dealing with is the results of a Vicsek type model simulation. The structure factor is defined as : $$S(q) = \...
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Error in FFT (Sectrum) analyzer Instrument

Recently I performed an experiment on finding the natural frequency of vertical cantilever. I used accelerometer and FFT analyzer instrument for measuring it. First I kept the accelerometer near the ...
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Why do people study plane wave in wave physics?

I have recently been studying a structure for high sound absorption. There are a lot of literature on similar design, where all of them are using the plane wave as an input to the structure. I have ...