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 ...

learn more… | top users | synonyms

0
votes
0answers
11 views

Fourier Transform derivation resource

I am trying real hard to understand the derivation of the Inverse and Forward Fourier Transform. In particular, I struggle with: Sinusoidal wave displacement equation. I'd like to know the ...
0
votes
0answers
37 views

Solving Laplace's Equation for a point charge at orgin with Fourier transform?

Ok So I have Laplaces equation: $\nabla^{2} \phi(\vec{r}) = \frac{q}{\epsilon_{o}} \delta(\vec{r})$ And I want to solve by using Fourier transforms. So I take the Fourier transform of the right ...
1
vote
1answer
38 views

Books on waves with Fourier Transforms

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
votes
1answer
34 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 ...
2
votes
1answer
65 views

Fourier Transforms of position and momentum space in Quantum Mechanics

Fourier transformations: for momentum space and for position space. How do we know that Ψ is not the Fourier transform of Φ but we suppose that its the other way around(Ψ would be ...
0
votes
0answers
16 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 ...
0
votes
1answer
14 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 ...
0
votes
0answers
25 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, ...
3
votes
2answers
72 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
votes
0answers
14 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
53 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) ...
0
votes
2answers
78 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 ...
2
votes
1answer
99 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
votes
0answers
40 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 ...
1
vote
0answers
31 views

Fourier transformation and mode expansions [duplicate]

Sorry as this is a rather trivial question, but I'm stuck with a certain implication. I'm working on exercise 1.7 from Polchinski where we are given an open string with boundary conditions ...
0
votes
1answer
43 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
vote
0answers
97 views

Fourier transformation and commutators

Sorry as this is a rather trivial question, but I'm stuck with a certain implication. I'm working on exercise 1.7 from Polchinski where we are given an open string with boundary conditions ...
0
votes
0answers
19 views

Modelling Fourier Transform Profilometry

Basically I want to simulate a surface profilometry technique through Matlab. For that I want to create a GUI with controls for generating a grating pattern of light at a particular angle with respect ...
0
votes
1answer
55 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; ...
1
vote
1answer
95 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
98 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
104 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
votes
0answers
39 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
votes
1answer
44 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 $\exp(ipx)$ in the Fourier transform of a scalar field and the corresponding conjugate momenta $\pi(x)$ of the scalar field?
0
votes
1answer
64 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 ...
0
votes
1answer
87 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 ...
2
votes
1answer
97 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
votes
1answer
56 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
78 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 ...
1
vote
1answer
30 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 $\hat{C}^+_l,\hat{C}^-_l$ ...
0
votes
0answers
15 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
votes
0answers
55 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
79 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 ...
1
vote
0answers
71 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
votes
1answer
54 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
40 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
votes
2answers
46 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
186 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
115 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
102 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 ...
1
vote
0answers
31 views

Sum in the reciprocal lattice

I have to use this property but I don't understand at all the deduction, so I was wondering if someone could help me. We have a crystal lattice with vectors to each atom from one of them $R_j$, and ...
8
votes
2answers
307 views

The poles of Feynman propagator in position space

This question maybe related to Feynman Propagator in Position Space through Schwinger Parameter. The Feynman propagator is defined as: $$ G_F(x,y) = \lim_{\epsilon \to 0} \frac{1}{(2 \pi)^4} \int d^4p ...
0
votes
1answer
49 views

Question about group velocity and travelling waves

I'm trying to learn some basic quantum mechanics and I have a question related to group velocity of a travelling wave. I know there are already a few questions related to group velocity, but I ...
3
votes
1answer
75 views

Considering $\langle \underline{q} \mid \underline{p} \rangle=\frac{1}{(2\pi\hbar)^{n/2}}e^{i\underline{q}\cdot\underline{p}/\hbar}$ [duplicate]

I have been given the following complete systems of eigenvectors $$\mathbf{Q}\mid\mathbf{q} \rangle=\mathbf{q}\mid\mathbf{q} \rangle, \quad \mathbf{P}\mid\mathbf{p} \rangle=\mathbf{p}\mid\mathbf{p} ...
1
vote
0answers
45 views

In Solution of the equation $\Box A_\sigma=0$. I find some mathematical difficulties [closed]

The solution of the equation $$\Box A^\mu(x)=0$$ has the form $$A^\mu(x)=\int\frac{d^4k}{(2\pi)^4} e^{ik^\mu x_\mu}A^\mu(k)\delta(k^2).$$ Where we take $A^\mu(k)$ of the form $$A^\mu(k)=\sum ...
2
votes
1answer
84 views

Path integral measure Fourier transformation for case of real field

Let's have $$ Z[J] = \int D \varphi e^{iS[\varphi , J]}, $$ where $\varphi$ denotes real scalar field. Let's make Fourier transform, $$ \varphi (x) = \int e^{iqx}\varphi (q), \quad \varphi^{*} (q) = ...
0
votes
2answers
76 views

Derivation of plane wave from inner product of position ket and momentum ket

In textbooks it seems to be taken for granted that $$\langle \mathbf{r}|\mathbf{k}\rangle ~=~ \frac{1}{\sqrt{\Omega}}\exp(i\mathbf{k}\cdot\mathbf{r}).$$ I'm sure it's obvious but is there a ...
0
votes
1answer
65 views

Fourier transform of random variables

My question is concerning Fourier transforms of random variables. So if the question itself is too heavy a task but you know of any good resources to learn this topic that would also be very much ...
0
votes
1answer
107 views

Why is the position space free particle wavefunction a function of momentum?

This is one of those little things that has always confused me. If someone said to you "in quantum mechanics, the eigenfunctions of a free particle are $\exp(ipx/\hbar)$" how would you know that ...
0
votes
0answers
33 views

Functions of the form $F(x-ct)$ written as superposition?

In this section of the Wikipedia article on the wave equation they do the following: $$\int^{\infty}_{-\infty}s_+(\omega)e^{-i(kx+\omega t)}d\omega +\int^{\infty}_{-\infty}s_-(\omega)e^{i(kx-\omega ...