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
45 views

Differential Equations [on hold]

$$ \nabla^2 B = - \mu_0^2 \epsilon_0^2 \frac{\partial B}{\partial t} $$ $$ \nabla^2 E = \mu_0 \epsilon_0 \frac{\partial B}{\partial t} $$ Find as many E and B field solutions as you can from this. ...
0
votes
0answers
18 views

Frequency response of a dynamical system under band and time limited input

Assume that I force the following dynamical system $$ \begin{align} \dot{\boldsymbol{x}} &= \boldsymbol{f}(\boldsymbol{x},u), \\ y &= g(\boldsymbol{x},u), \\ \boldsymbol{x}(t_0) &= ...
1
vote
2answers
64 views

How is the integrand concluded to be identically zero?

In expanding the classical Klein-Gordon field in Fourier space to write it in terms of $\phi(\mathbf{p})$ instead of $\phi(\mathbf{x})$, I reached the following result. $$\int ...
1
vote
1answer
32 views

Effective masses for different direction

Assume we have an indirect semiconductor where the effective mass becomes anisotropic in different directions. Usually, one talks about a mass in parallel and perpendicular direction referring to ...
1
vote
1answer
80 views

Why does $\nabla \to ik$ when you Fourier transform?

I am reading a text that describes the scattering of light by a particle with dielectric constant $\epsilon$ After a bit of maths starting from Maxwell's equations they obtain: $$\nabla (\nabla ...
3
votes
3answers
162 views

The Dirac-Delta function as an initial state for the quantum free particle

I want to ask if it is reasonable that I use the Dirac-Delta function as an intial state ($\Psi (x,0) $) for the free particle wavefunction and interpret it such that I say that the particle is ...
2
votes
2answers
66 views

Modeling the free space propagation of laser beams using Fourier transforms

I am trying to model the propagation of a laser beam in free space. I have an initial field $E_{in}(x,z=0)$ (a Gaussian beam) and need to find the fields at other points on the optical axis $E(x,z=d)$ ...
1
vote
0answers
41 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 ...
0
votes
0answers
4 views

Verification of Fourier transformationn of Io-sinh function [migrated]

I try to match. What could not match $I_o-\sinh$ pair developed by Ben Logan, transform pair also published in The Practical Application of the Fourier Integral Campbell, George A .Sir J.F. Kaiser ...
0
votes
0answers
18 views

Question on envelope-carrier description of traveling wave

I'm doing a research internship in attosecond physics right now, and one of the really important things in the field is the description of a propagating laser pulse as the combination of a slowly (or ...
0
votes
2answers
34 views

Peak at zero in one device and not the other

I was wondering if anyone could shed some light on this problem. I have placed two accelerometers on an animal one sampling at 50 Hz the other at 100 Hz. They were placed in the same position. I then ...
1
vote
0answers
247 views

Fourier Transform of Poisson Equation [migrated]

While trying to solve the Poisson Equation by using Green's Function I have to Fourier transform the equation i.e $$-\nabla^{2}\phi(r)=\rho(r).$$ In the book after Fourier transform, the solution ...
2
votes
1answer
78 views

Solution to Klein-Gordon equation

I have a sound grounding on ODE's, not that much on PDE's, i've read many books on QFT and most if not all come to the conclusion that the solution to the Klein-Gordon equation ...
0
votes
1answer
84 views

Fourier expansion of the Klein-Gordon field

Is there a reason(both physical and mathematical) why the Klein-Gordon field is represented as a fourier expansion in the second quantization as opposed to other mathematical expansions? Be gentle ...
1
vote
1answer
83 views

String Theory and Fourier Analysis [closed]

Me and my friend, both many years from learning string theory, had a recent debate about it anyway. He said he already partially discounts it because after learning waves, he believes any function, ...
0
votes
0answers
56 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 ...
3
votes
1answer
92 views

Plane wave complex notation

As far as I know, the function: $$ \vec{E}(\vec{r},t)=\vec{E_0}\cdot e^{i(\vec{k}\cdot \vec{r}-\omega t)} \hspace{2cm}(1) $$ is a mathematical solution of the wave equation: $$ \nabla^2 ...
6
votes
3answers
81 views

Units of a discrete Fourier transform

Normally a Fourier transform (FT) of a function of one variable is defined as $$f_k=\int^\infty_{-\infty}f(x)\exp\left(-2\pi i k x\right) dx.$$ This means that $f_k$ gets the units of $f$ times the ...
1
vote
1answer
91 views

Inverse Fourier transform of Yukawa potential (troubles with Mathematica)

It can be proved that the potential $\frac{e^{-u|r|}}{|r|}$ has Fourier transform $\frac{4\pi}{u^2+q^2}$. Now, I'm trying to go backwards and do the inverse Fourier transform but I'm running into ...
3
votes
1answer
83 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 ...
0
votes
1answer
50 views

Phase and amplitude information of an image

By applying Fourier Transform to an image we can get its magnitude as well as phase spectrum. A magnitude spectrum describes how various frequencies are attenuated and accentuated in that image but ...
3
votes
1answer
83 views

Image Reconstruction:Phase vs. Magnitude

Figure 1.(c) shows the Test image reconstructed from MAGNITUDE spectrum only. We can say that the intensity values of LOW frequency pixels are comparatively more than HIGH frequency pixels. $$ ...
2
votes
1answer
94 views

Divergent solution in time-dependent Schrödinger equation

if I transform the time-dependent Schrödinger equation without a potential I get: $$ - \hbar \omega \psi(\omega,x) = \frac{- \hbar^2}{2m} \frac{\partial^2 \psi(\omega,x)}{\partial x^2}$$ The ...
0
votes
1answer
119 views

Initial condition for Fourier transformed Schrödinger equation

I asked in this thread Time-dependet Schrödinger equation how to solve the Time-dependent Schrödinger equation. One of JamalS' recommendations was the Fourier transform, which is why I want to quote ...
1
vote
1answer
78 views

Inner product of position and momentum eigenkets

Let's define $\hat{q},\ \hat{p}$ the positon and momentum quantum operators, $\hat{a}$ the annihilation operator and $\hat{a}_1,\ \hat{a}_2$ with its real and imaginary part, such that $$ \hat{a} = ...
0
votes
0answers
45 views

Quantum Fourier Transform question regarding measurement

When we use the quantum fourier transform, for a function, the output is entangled, so if a measurement is made on the output, the result may not be that of the function that one wanted in the first ...
0
votes
0answers
33 views

Quantum Fourier Transform question

We can formulate a Quantum Fourier Transfrom which is derived from a DFT. This DFT performs a polynomial operation by interpolating over specific sample points, and then when we read the output from ...
1
vote
0answers
45 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 ...
0
votes
1answer
43 views

Frequency spectrum and histogram of white noise

I haven't been able to find any images with, so here goes: In the frequency/Fourier spectrum, how does white noise look like ? Is that just random dots all over the place, making it very hard to ...
0
votes
1answer
77 views

In quantum mechanics, why position and momentum are related by Fourier Transformation(only)? [duplicate]

We know that if we take Fourier transform of momentum we go to position space. But why Fourier transform only.(credit_ Abh Gupta)
3
votes
4answers
129 views

Continuous Fourier transform vs. Discrete Fourier transform

Continuous Fourier transform vs. Discrete Fourier transform. Can anyone tell me what the difference is physics-wise? I know the mathematical way to do both, but when do you use the other instead of ...
2
votes
1answer
44 views

Inverse of a series (solid state)

I am working with the expression involving the equilibrium displacement ($y_n$) for the $n$th particle in a 1D harmonic lattice in terms of the normal modes coordinates $A_k$. Let me show you the ...
0
votes
2answers
109 views

Representations in quantum mechanics [closed]

This might be a very simple question. I just want someone to point me the right direction to understand things like this: $$ \langle x|x'\rangle=\delta(x-x') \\ \psi(x)=\langle x|\psi\rangle \\ ...
1
vote
0answers
30 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
1answer
67 views

Resolution in a Fourier transform spectroscopy setup

I am a bachelor physics student and as an assignment we had to perform measurements on an FT spectroscopy setup. Context. Our setup consisted of a Michelson interferometer through which the light ...
3
votes
0answers
47 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 ...
1
vote
0answers
26 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, ...
0
votes
1answer
79 views

Quantum Mechanics - Finding momentum probability density [closed]

everyone. I got a bit stuck on 2(iii), this is supposed to be a easy question, but i don't know how you get the square term? I thought you just do the Fourier transform, but then I got some ...
0
votes
0answers
59 views

Corrections and Normalization for Power Spectrum Calculation

So I'm hoping I can get some help. I have a 2d image and need to get the 1d power spectrum. I know the basic steps: take fft, take fft^2 to get power, then take average power in radial bins to get 1d ...
0
votes
0answers
32 views

Drawing the wave function for a wave packet

I have the following infotmation: Amplitude-Function: $U(k) = Ae^{-a|k-k_0|}$ Wave Function: $u(x,t) = \frac{A}{\sqrt{2\pi}} \frac{2a}{(x-vt)^2+a^2}e^{ik_0(x-vt)}$ Uncertainty in x: $\Delta x = 1$ ...
3
votes
0answers
66 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 ...
1
vote
1answer
76 views

How to derive quantum Fourier transform from discrete Fourier transform (DFT)?

I am interested in Shor's algorithm, and I am reading several papers that related to the quantum Fourier transform (QFT). I know the there is a difference between the output of QFT and DFT (DFT). ...
1
vote
2answers
101 views

Fourier transformation [closed]

I have recently studied Fourier and Laplace transformation in maths. I wanted to understand the utility in physics with some examples that requires this change in dimension and the reason why.
0
votes
0answers
31 views

How do RGB colors work? [duplicate]

They say that all colors can be formed by mixing Red, Green, and Blue appropriately. Is it true? Isn't the Fourier basis infinite dimensional? Or does it turn out to be the case that only three ...
3
votes
0answers
165 views

“Derivation” of the Heisenberg Uncertainty Principle

Ok, so I posted this in the mathematics StackExchange, but got no response. The question I outline below is my textbook's "derivation" of the Heisenberg Uncertainty Principle. The "derivation" my ...
3
votes
1answer
144 views

Finding the creation/annihilation operators

Using Minkowski signature $(+,-,-,-)$, for the Lagrangian density $${\cal L}=\partial_{\mu}\phi\partial^{\mu}\phi^{\dagger}-m^2\phi \phi^{\dagger}$$ of the complex scalar field, we have the field ...
0
votes
0answers
26 views

Looking for Solutions to Symmetric Potential

I'm a little confused on the basic method of finding a separable solution to a give potential distribution. If we have a symmetric potential, say it hits zero and $-a$ and $a$, constituting two sides ...
0
votes
0answers
20 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
1answer
64 views

Physical meaning of taking twice the real part of a Fourier transform

In my previous question, Calculating the coherence length from a spectrum, it turned out that I can calculate the coherence length of my light source from the autocorrelation function, which can be ...
0
votes
0answers
26 views

dual variables for lattice fermions

I am quite familiar with duality transformations for lattice spin systems (i.e. systems with global $O(n)$ symmetry) and pure gauge systems (i.e. local $SU(n)$). However, after searching for a bit, I ...