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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5
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1answer
166 views

Which position and momentum distributions arise from some wave function?

Consider a particle in one dimension with wave function $\psi(x)$. The probability density function describing how likely it is to find it in a given position is given by ...
4
votes
1answer
185 views

Poles for a particle scattered in a delta potential

I am working on problem a professor gave me to get an idea for the research he does, and have hit a point where I'm having a difficult time seeing where I need to go from where I'm at. I would also ...
4
votes
1answer
150 views

Kolmogorov/Energy spectrum for turbulent boundary layer

Previously, I have calculated energy spectrum for 3D isotropic turbulent flow data which is equally spaced in all three directions and then to compute the energy spectrum, one performs Fourier ...
2
votes
1answer
122 views

Diffraction and $k$-space

Regarding diffraction I am a little bit lost reading about reciprocal space and the space of $k$'s. As I understand it the Fourier relationship between a wavepacket $\Psi(\vec r,t)$ and the complex ...
2
votes
1answer
161 views

Massless boson in 2D and its (retarded) propagator

I have the retarded propagator for a free scalar field in 1+1 dimensions. Inside the light cone, this looks like $J_0(m \sqrt(t^2-x^2))$, J being a Bessel function. When I take the massless limit, ...
4
votes
0answers
267 views

Action of Parity operator on Impulse representation

Is my derivation of the action of the parity operator $\mathbb{P}$ on the $|p\rangle$ representation correct? $$\left( \mathbb{P}\tilde\psi \right)(p)= - \tilde\psi (p).$$ Obtained from $$\left( ...
3
votes
0answers
69 views

Light, Fourier Transforms, Spherical Harmonics

Mathematically, I'm having trouble understanding where we can use what with light. I read somewhere on this site that Huygen's Principle is effectively just taking an expansion of a wave onto the ...
3
votes
0answers
373 views

How should I think about reciprocal lattice and Miller indices?

When I hear someone talking about a (100) plane or a (111) plane or an (hkl) in general, my first thought is, is the system cubic. The reason I think this is because I tend NOT to think of the planes ...
3
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0answers
86 views

Non-Hermiticity when Fourier transforming onto a finite lattice

I'm doing numerical simulations. I have the Haldane model in a honeycomb lattice where $$ H = \sum \limits_{<ij>}a^\dagger_i b_j + h.c $$ Where $i$ belongs to sublattice $A$, and $j$ to ...
2
votes
0answers
35 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 ...
2
votes
0answers
35 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 ...
2
votes
0answers
139 views

Feynman propagator with general $\xi$ parameter

Hey from my notes in my PS book it seems I have solved this some time in the past, but I cannot seem to get the indices straight this time around. So in deriving the Feynman photon-propagator which ...
2
votes
0answers
100 views

Momentum representation of a state

I am trying to figure out the momentum representation of the state which has the properties $$\langle \psi |\hat q |\psi \rangle=-q_0,$$ $$\langle\psi|\hat p|\psi \rangle=p_0, $$$$\Delta q\Delta ...
2
votes
0answers
193 views

Fourier Transform of ribbon's beam Electric Field

I have a monochromatic ribbon beam with $E(x)e^{i(kz-\omega t)}$ being the electric field's amplitude. I want to show that the lowest order approximation in terms of plane waves is ...
2
votes
0answers
216 views

Discrete sum over an exponential with imaginary argument, considering only every second lattice site?

Let's say I sum an exponential function $e^{\imath \left(k-k^{\prime}\right) x_{i}}$ over a chain system where every member of the chain is of the same type, e.g., A-A-A-...-A-A (total of N sites) ...
1
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0answers
94 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 ...
1
vote
0answers
59 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 ...
1
vote
0answers
28 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 ...
1
vote
0answers
39 views

Can I calculate the form of the aperture from the diffraction pattern?

As I understand, the Fraunhofer diffraction pattern of light is the Fourier transform of the aperture. More precisely, the amplitude of light would be the Fourier transform and the intensity its ...
1
vote
0answers
31 views

Why isn't there a different phase after fourier transformation in two lattices

I am trying to understand some solutions for graphenes energy dispersion. While most of it is clear, I don't get one step, when changing into k-space. Consindering two sublattices A and B with ...
1
vote
0answers
131 views

Schrodinger Wave Functional (quantum fields) - Solving Functional Gaussian Integrals

Okay, So i'm doing some research that involves the Schrodinger representation in quantum field theory. The ground state wave functional for the Klein Gordon field is a generalized gaussian in position ...
1
vote
0answers
190 views

Decoupling the Hamiltonian by a Discrete Fourier transform

For $N$ coupled oscillators(periodic BC) whose Hamiltonian is given as $H=\sum\limits_{i=1}^N (\frac{p_i}{2m} + \lambda(x_{i+1} - x_i)^2)$ decoupling can be achieved by change of variables by using ...
1
vote
0answers
74 views

What excactly is a “fourier component of a density fluctuation”?

Light scattering texts say depending on the scattering angle, you are seeing a certain fourier component of a density fluctuation. This density fluctuation varies sinusoidally due to Brownian motion ...
1
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0answers
219 views

Transition Between Position and Momentum Basis

I'm having some trouble following pages 55-56 of Sakurai's Modern Quantum Mechanics. We're trying to transfer from position space into momentum space. Here's a quote: Let us now establish the ...
1
vote
0answers
75 views

Quick question on convolution - Diffraction through a pair of slits

We know that the fourier transform of the amplitude function (in terms of $y$) gives you the amplitude function (in terms of $\theta$) Consider a pair of triangular slits: Fourier transform of ...
1
vote
0answers
42 views

Is having full information about the resonances of a rigid body equivalent to having full information about its material parameters?

Lets say I have a mechanical system whose mechanical resonances (mode shape and frequency) I can measure with perfect accuracy. Is this theoretically equivalent to knowing the materials parameters, ...
1
vote
0answers
82 views

Interpretation in Fourier-Laplace domain

The Green's function describing the distribution of particles sent from $\def\v#1{\boldsymbol{#1}}\def\u#1{\hat{\v#1}}\v r=0$ at $t=0$ uniformly in every directions is, in two dimensions $$ ...
1
vote
0answers
360 views

Questions about Michelson interferometer

I have been doing experiment on Michelson experiment, but I don't quite understand why white light results in an interferogram with very few fringes, and why are they necessarily Gaussian? I know that ...
1
vote
0answers
104 views

Periodic sequence with exponentially increasing period?

I have to develop a physical model for a certain type of biological oscillation that can be built upon periodic sequences. From earlier questions I know that any periodic sequence (containing $0$s ...
1
vote
0answers
188 views

Splitting light into colors, mathematical expression (fourier transforms)

I am trying to solve a problem that includes a function of the light hitting a certain area. My question is, how would I change a function $G(x)$ of photons hitting a certain area to include just ...
0
votes
0answers
22 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
0answers
15 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
0answers
36 views

Using a mask with a (2D) Fourier Transform

For a project I'm working on, I (1) take a (gaussian) random field of 256x256 pixels, (2) apply numpy's fast fourier transform (numpy.fft package), (3) filter the k-space image (using a Tukey ...
0
votes
0answers
14 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
50 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 ...
0
votes
0answers
28 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 ...
0
votes
0answers
42 views

Fourier Transforming a $n$-dimensional ket (QM)

I would like to evaluate the Fourier Transform of $n$ functions. I am aware from the derivation of the convolution how this is done for the case of $n=2$. How could this be generalised for $n=3$? ...
0
votes
0answers
22 views

What is the transfer function in fft beam propagation for unpolarized light?

What is the transfer function in fft beam propagation for unpolarized light ? How to construct the fft beam propagation ? This is for homework. For coherent light the beam propagation is E(x,z) ...
0
votes
0answers
47 views

Measuring typical distance between patches using 2D Fourier Transform

I need to extract information about the typical distance between the black patches in an image like the one I attached here. I tried to perform 2D FFT on it (using OpenCF fdt function in Python), but ...
0
votes
0answers
53 views

how to find the frequency in a auto correlation function?

I am running some molecular dynamics simulation with carbon nano tubes and calculating the velocity auto-correlation function (VACF). Each 10 time steps is writen in a file the VACF and in the final ...
0
votes
0answers
50 views

Lissajous plots - additive synthesis

I am using Fourier analysis to recreate my data. I have some test data to work on as a way of testing out my synthesis approach. I have spatial test data in the x and y directions. when I plot x ...
0
votes
0answers
14 views

Constructing Echocardiography

I am trying to search the Mathematics behind Echocardiography and constructing the image. There seems to be Wigner distribution applied to the pictures. There also seems to be many different ...
0
votes
0answers
63 views

Why do books write $X(f)$ when they mean actually mean $\lvert X(f)\rvert$?

All books write $X(f)$ in plots - the Fourier transform of $x(t)$ - when they actually mean $\lvert X(f)\rvert$, without even mentioning in passing that they are dropping the mod sign. And also they ...
0
votes
0answers
44 views

Is spectrum of Discrete-Time Fourier Transform (DTFT) periodic or not

I can't think of any better title. Here is the content that I got question http://cnx.org/content/m10247/2.31/ As it state the nature of DTFT's spectrum is periodic as it show in figure 1 However, ...
0
votes
0answers
159 views

Looking for raw interferogram data / raw FID data from FT-IR / FT-NIR / FT-NMR

I'm trying to get my hands on some raw interferogram data / raw FID data from an FT-IR / FT-NIR / FT-NMR so I can run some tests using FFT with it (it needs to be real data). Here's a picture below of ...
0
votes
0answers
107 views

Lenses and benefit of exact fourier transform

I have learned in an Optics class that a lens will "compute" the Fourier Transform of an electromagnetic wave passing through it at the focal point behind it (but with a quadratic phase). However, ...
0
votes
0answers
56 views

Frequency response of the wireless channel

We know that the signal attenuates out with distance and according to the channel transfer function or frequency response, the signal frequency components attenuate to different values based on ...
0
votes
0answers
144 views

Discrete Fourier Transform: Why do we only consider a full cycle?

I am studying Fourier analysis and I am still new to this topic. If I understand that the maximum frequency that can be used in a DFT is given by $N/2$, where $N$ is the number of samples in our ...
0
votes
0answers
75 views

Outflow for fluid simulation based on “Stable Fluids”

I've implemented a fluid simulation based on the paper Stable Fluids. It works quite well, except I would like to have the velocity at the "upper" edge just to outflow and not to re-enter on the ...
0
votes
0answers
175 views

How to solve following equation (Yukawa field)?

By using Lagrangian for Yukawa interaction, $$ L = -\frac{1}{c}A_{\alpha}j^{\alpha} + \frac{1}{8 \pi c}(\partial_{\alpha}A_{\beta})(\partial^{\alpha}A^{\beta}) + ...