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
141 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
162 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 ...
2
votes
1answer
99 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
118 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, ...
1
vote
1answer
113 views

A few questions on wave packets and uncertainty relations

According to Cohen-Tannoudji the wave-function for a one-dimensional free particle can be written as $$ \psi (x,0)=\frac{1}{\sqrt{2 \pi}} \int g(k) e^{ikx} dk.$$ While $g(k)$ is not specified, there ...
1
vote
1answer
74 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
117 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 ...
4
votes
0answers
115 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 ...
4
votes
0answers
248 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
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0answers
65 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
639 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
0answers
335 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
votes
0answers
83 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
108 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
98 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
174 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
194 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
vote
0answers
31 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
41 views

Convert angular power spectrum to spatial power spectrum

If we have a signal projected on a sphere, one routinely decomposes this in spherical harmonics, in analogy to a Fourier decomposition in flat space. One can then make the decomposition: ...
1
vote
0answers
28 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
113 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
109 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
64 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 ...
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0answers
183 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
59 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
35 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
77 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 $$ ...
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vote
0answers
317 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
99 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
182 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
20 views

A linear response system with a periodic input

I'm currently trying to solve the following exercise: A linear system is driven by a periodic input $f(t)$ such that $f(t+T)=f(t)$. The response $g(t)$ of the system is such that a sinusoidal ...
0
votes
0answers
38 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
16 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
33 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
29 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
48 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
12 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
29 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
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
46 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
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0answers
52 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 ...
0
votes
0answers
157 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
65 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: $\Delta x = 1$ ...
0
votes
0answers
39 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
134 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
97 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
44 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
122 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
73 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
167 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}) + ...