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 calcalating the response of a time shift invariant linear system as the sum of its response to time harmonic excitation or for transforming a ...

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1answer
30 views

Having trouble understanding the proof for Fourier Transform Scaling Property [closed]

Starting from Plancherel's Theorem: $$ f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F(k)e^{ikx}dk ...(1) $$ $$ F(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-ikx}dx ...(2) $$ I need to ...
1
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2answers
58 views

Characteristics of an Optical System in the Fourier Domain

An imaging system can be characterized by its point spread function (PSF), which in most cases is space-variant. The final image is the result of the convolution of the PSF with the object (2d ...
1
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0answers
54 views

Discrete Fourier transform for periodic signal?

From the Signal and System textbook, by Oppenheim, I learned that the discrete-time Fourier transform can be written as $$ x[n]=\frac{1}{2\pi}\int_{2\pi}X(e^{j\omega})e^{j\omega n}d\omega $$ $$ X(e^{...
1
vote
1answer
64 views

Finding $\phi(k)$

If you have $\Psi(x,0) = c(\psi_1 + \psi_2)$ where $\psi_n$ is an Energy eigenfunction for a quantum number $n$. I'm supposed to find $\phi(k,t)$ at $t$ = 0. This is for an infinite square well from ...
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2answers
134 views

Fourier Transform with Branch Cuts [closed]

I want to compute the Inverse Fourier transform of the following function (it appears as a certain correlation function in a physical model I am interested it): $$ \widetilde{f}(\omega) = \frac{2i}{\...
3
votes
1answer
98 views

A question about Fourier Transformation [closed]

Recently I try to evaluate a integral in a paper: $$ \Gamma(x,y)=\int_{-\infty}^{\infty} \frac{dk}{2\pi} \sqrt{k^2+m^2} e^{ikx} $$ This is the Fourier Transform of: $$ f(k)=\sqrt{k^2+m^2} $$ The ...
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1answer
138 views

Wavefunction interpretations in QM

From two-slit electron-interference experiment we can infer that there is a wave $\psi(x,t)$ that can be associated with electron. The amplitude at some point is the sum of amplitudes reaching that ...
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0answers
38 views

Estimation of the autocorrelation for data on finite size interval

Let's consider we have a continuous random signal ${ t \in ] - \infty \,;\, + \infty [ \mapsto b (t)}$. We assume this signal to be stationary, so that when ensemble-averaged, one may introduce the ...
0
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0answers
27 views

Units of a fourier transform end energy density of the time dependent force

I have a signal, which is time dependent force F(t) (obtained from the Atomic Force Microscope) I wanted to estimate the energy content in the signal for given frequency band. I have calculated Energy ...
0
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1answer
83 views

Fourier transform & asymptotic expansion of Klein-Gordon equation

I am looking for an approximate analytical solution to the generalized Klein-Gordon equation \begin{equation} \frac{\partial^2{\phi}}{\partial{t^2}}+\frac{\partial^2{\phi}}{\partial{x^2}}+\phi=0 \end{...
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2answers
245 views

Why can you only measure velocity or location in a particle?

I was talking to a family friend in the field of optics at a quantum scale (not sure the proper name for this) and he was explaining to me why you can only determine either the velocity or location of ...
0
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0answers
114 views

Band Structure from Fourier Transform Solution of Dirac Comb

I have use Fourier transforms to solve the Schrödinger equation for an attractive Dirac-$\delta$ comb potential of the form $$ V(x) = -\alpha \sum_{j = -\frac{N}{2}}^{\frac{N}{2}} \delta \left( x - \...
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0answers
80 views

Riemannian generalization/adaption of the Hubbard–Stratonovich transformation

I'd like to write the Hubbard–Stratonovich (HS) transformation of a scalar function on a Riemannian manifold. This transformation is quite simple in Euclidean space. One can consider it as a Fourier ...
0
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1answer
192 views

Superposition of waves with different initial phase in Quantum Mechanics [closed]

In Quantum Mechanics, if a particle's state is a superposition of many states, then we say that its position is well-defined (by the Heisenberg uncertainty principle, because here we have ill-defined ...
1
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1answer
69 views

Books on waves with Fourier Transforms [duplicate]

There are many waves and oscillations books out there that also include Fourier analysis but very few give the subject a thorough treatment, they just pass it in a few pages. If anybody has any ...
0
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1answer
48 views

Fourier expansion and transform - what about the phase of the waves that i am adding?

Say we have a wave on the surface of the water and we want to describe it as a sum of other waves. So we use Fourier expansion to add waves of different wavelengths. For simplicity, say we have to ...
3
votes
1answer
430 views

Fourier Transforms of position and momentum space in Quantum Mechanics

Fourier transformations: $$\phi(\vec{k}) = \left( \frac{1}{\sqrt{2 \pi}} \right)^3 \int_{r\text{ space}} \psi(\vec{r}) e^{-i \mathbf{k} \cdot \mathbf{r}} d^3r$$ for momentum space and $$\psi(\vec{r}...
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0answers
29 views

Dilations in momentum space

I don't quite understand what's going on here. Let's suppose I have a dilation in real space. The generator is $D=x^j \partial_j$, so an infinitesimal dilation is $\delta x^i = Dx^i = x^j \partial_j ...
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1answer
73 views

Whether there is any relationship between the frequency of an input signal and the frequency of it's fourier transform?

Whether there is any relationship between the frequency of an input signal and the frequency of it's fourier transform? For example, suppose I gave a 100Hz signal, whether my FFT frquency will also be ...
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0answers
40 views

How to find a single coefficient of quantum Fourier transform reliably

Quantum Fourier transform transform $X \in \mathbb{C}^{2^n}$ to $Y \in \mathbb{C}^{2^n}$. Suppose one wishes to find $y_0$, the first coefficient of "vector" $Y$. However, as this is quantum process, ...
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2answers
140 views

Is there a mathematical relationship between Legendre conjugates and Fourier conjugates?

In quantum mechanics, there is an uncertainty principle between conjugate variables, giving rise to complementary descriptions of a quantum system. But the variables are conjugates in two different ...
0
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0answers
57 views

How to apply contrast sensitivity function to an image?

I would like to apply contrast sensitivity function (CSF) to an image. My idea is to do the Fourier tranform of the image and then do the filtering in the frequency domain by applying the CSF. However ...
0
votes
1answer
99 views

What does it physically mean to take the Laplace transform of a non-periodic position function?

What I'm trying to get through my head here is how taking the Laplace transform of a system with a position function like $X(t)=t$ is possible. To my current (admittedly incomplete) understanding,...
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2answers
132 views

Why is a sine wave considered the fundamental building block of any signal? Why not some other function? [closed]

It is mathematically possible to express a given signal as a sum of functions other than sines and cosines. With that in mind, why does signal processing always revolve around breaking down the signal ...
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1answer
703 views

Solving the Klein-Gordon equation via Fourier transform

I have been writing a personal set of notes on QFT and I'm currently writing up a section on solving the Klein-Gordon (K-G) equation. I many texts that I've read, the author starts by expressing the ...
2
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0answers
51 views

How is translational symmetry related to Fourier decomposition?

The book (The Cosmic Microwave Background By Ruth Durrer) about cosmological perturbations says that because of translational symmetry of the background at a constant time, we can decompose our ...
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votes
1answer
49 views

Transforms in physics? [closed]

In my studies I have heard of two types of transformations in the physical science 1) the Fourier transform for diffraction and 2) the Legendre transform for thermodynamic potentials. While ...
1
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1answer
225 views

Expanding free scalar field in terms of ladder operators

I'm having some difficulty with the finer points of expanding a field in terms of ladder operators. Note that this is not identical to the other related question I asked. From Peskin / Schroeder; ...
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1answer
156 views

Field expansion in Peskin & Schroeder

Peskin and Schroeder state something which I'm not fully understanding. More specificially I think it's just phrased in a way I'm not understanding. In the Schrodinger picture we can expand the real ...
1
vote
2answers
122 views

How do phase carries structural information about the function? [closed]

Suppose you are on a railway platform and you hear the sound of train coming towards you. Now, Using Fourier transformation we can convert the time domain function (here take sound as a function) ...
5
votes
1answer
140 views

Physical implications of the Gibbs phenomenon for Quantum Mechanics

From Wikipedia: The Gibbs Phenomenon is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. The nth ...
2
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0answers
143 views

Autocorrelation function corresponding to density of states with significant rotational motion

Most statistical physics textbooks (at least the ones I've found) state simply that the density of states of a system can be found as the temporal Fourier transform of the velocity autocorrelation ...
0
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1answer
83 views

What is the difference between the momentum in the Fourier transform of a scalar field and the conjugate momentum of the field?

What is the difference between the momentum $p$ in $e^{i\mathbf{p}\cdot{\mathbf{x}}}$ in the Fourier transform of a scalar field and the corresponding conjugate momenta $\pi(x)$ of the scalar field?
0
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1answer
347 views

Probability density for momentum in Quantum Mechanics

In a book i found the following equations: $$ \phi(k)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty \Psi(x,0)e^{-ikx}dx $$ and $$ \Psi(x,t)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty \phi(k)e^{ikx-\...
0
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1answer
92 views

Problem with momentum values in a QM problem

I have the following equation of $Ψ$ around a ring (the particle is bound to move only on the ring): To visualize the state(it dies before L/2 if L=2πR): We can see from the first picture that ...
4
votes
1answer
373 views

How is Green function in many-body theory introduced?

Normally, for a (linear) operator $L$ and a DE $$ Lu(x) = f(x) $$ the Green function is defined as $$ LG(x,s) = \delta(x-s) $$ and it is found that $$ u(x) = \int G(x,s) f(s) ds $$ is the ...
0
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2answers
192 views

How does one calculate the minimum spectral linewidth in cm$^{-1}$ of a pulsed laser with pulse duration of 10 fs?

I calculated the minimum spectral linewidth given here using the uncertainty principle that $\Delta E\,\Delta t =h/2\pi$. Will this be correct?
0
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1answer
158 views

How to transform the Laplacian from momentum space to coordinate space

I'm working through some quantum mechanics problems with solution sets (attempting the problems then looking at the solutions to compare), and a little part of a solution has stumped me. I'm not sure ...
1
vote
1answer
100 views

The proof of a discrete Fourier identity in quantum field theory

On page 25, in the book Quantum Field Theory for the Gifted Amateur by Tom Lancaster and Stephen. J Blundell, it states the following: We impose periodic boundary conditions forcing $e^{ikja}=e^{...
2
votes
1answer
75 views

Fourier transform of a set of L fermions operators

I have a set of L fermion creation and annihilation operators: $\lbrace{\hat{C}^+_1,...,\hat{C}^+_L\rbrace}$ and $\lbrace{\hat{C}^-_1,...,\hat{C}^-_L\rbrace}$. Every $\hat{C}^+_l,\hat{C}^-_l$ ...
2
votes
2answers
97 views

Why are reciprocal lattice vector periodic, and time-frequency not?

k-space vectors are related to each other by $k=k'+G$, where $G$ is the reciprocal lattice vector $G=2\pi/a$. This means that the frequency of oscillation in real space of a plane wave $e^{ikx}$ is ...
0
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0answers
155 views

Definition of Fourier Transform on a Lattice

I am reading a book(EDIT: the book is Czyholls theoretical condensed matter physics, though i am not sure if there is an english version) where for periodic functions $f(x_l+L)=f(x_l)$ the Fourier ...
1
vote
1answer
102 views

Propagating higher order Hermite Gaussian modes. What are complex amplitude coefficients?

I've been tasked with writing a code (in MatLab, but I'm currently using Mathematica because I don't know MatLab %\ ...) to simulate the propagation of a Gaussian beam. I don't really know anything ...
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0answers
246 views

Fourier transform of Coulomb potential in 1D

The Fourier transform of the Coulomb potential $V(r)=\frac{k}{r}$ is typically evaluated by computing the Fourier transform of the Yukawa potential given by $V_{Yukawa}=\frac{ke^{-\epsilon r}}{r}$ and ...
0
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1answer
56 views

Details of the radial Fourier transform pertaining to certain quantum integrals

Consider the integral $$U(t)=\int\frac{d^3p}{(2\pi)^3}e^{-ip^2t/2m}e^{i\vec p\cdot\Delta\vec x}$$ for the free non-relativistic propagator. I'm not quite sure about the gritty details of radial ...
0
votes
1answer
53 views

How to compute phases of the signals?

Let us take 4 signals which are sinusoidal and periodic. Suppose you are given a phase spectrum or (/and) equation of the (main) signal only and you are said that the given (main) signal is formed of ...
0
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2answers
84 views

Normalization of the overlap $\langle x'|p'\rangle$

Let $$\langle x'|p'\rangle = N \exp(\frac{ip'x'}{\hbar})$$ be the overlap between position and momentum space, where $N$ is a normalization constant to be determined. We can then compute $N$ by $$ \...
3
votes
2answers
328 views

What is the significance of the Fourier coefficients?

Let us take an example, a white ray (which is composed of bunch of frequency components) is passed through a prism, the ray gets split (decomposed) into its elementary vibgyor colours (i.e.different ...
0
votes
1answer
158 views

Have some queries about Fourier transform [closed]

I have some queries about the Fourier transform In most of the cases, the Fourier transform of a signal is symmetric with respect to positive and negative frequency. I think the computational ...
1
vote
3answers
145 views

Whether the job of Fourier Transform is just to convert signals from time domain to frequency domain only or more than it?

I am a beginner . We convert a signal in time domain to frequency domain by applying Fourier transform on the signal to obtain frequency and phase spectrum. So,whether the job of Fourier transform ...