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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3
votes
1answer
186 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. $$ ...
3
votes
1answer
122 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 ...
1
vote
1answer
446 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
332 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
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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 ...
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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 ...
1
vote
0answers
189 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
392 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
120 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)
4
votes
4answers
481 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
101 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
129 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
61 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
120 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
66 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
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, ...
0
votes
1answer
351 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
159 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
68 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: $\Delta x = 1$ ...
4
votes
0answers
118 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
135 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
137 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
54 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
646 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 ...
5
votes
1answer
605 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
41 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
98 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 ...
1
vote
0answers
61 views

Fraunhofer Diffraction [closed]

A 1-dimensional aperture is illuminated by a parallel beam of light of wavelength $\lambda$ and the diffraction pattern is viewed on a distance screen. Show that the amplitude of the diffraction ...
0
votes
1answer
156 views

Fourier transform with periodicity at the harmonic frequency

Let's suppose I have a signal $F(t)$ that is periodic, with two periodicities $P_1$ and $P_2$, with $P_1 > P_2$. Suppose that I know the values of the two periodicities. Using the Fast Fourier ...
2
votes
2answers
178 views

Can someone please explain the “infrared catastrophe”?

In my readings I've run into this idea of an "infrared catastrophe" associated with 1/f noise. As far as I can tell it is because when you graph the periodogram of the 1/f signal you see the PSD goes ...
0
votes
1answer
86 views

How to find different operator representations in QM?

I read that any observable operator may be represented as: $$\Omega = \sum_n \omega _n | \omega _n \rangle \langle \omega_n |$$ Where the little omegas are the eigenvectors/eigenvalues of the ...
1
vote
1answer
94 views

Deriving commutation relations in second quantisation

I am trying to start from: \begin{align*} [\phi(x),\pi(x')] = i\hbar\delta(x-x') \\ [\phi(x),\phi(x')] = [\pi(x),\pi(x')]=0 \end{align*} to derive: \begin{align*} [a(k),a(k')^\dagger]=\delta_{kk'}\\ ...
3
votes
1answer
517 views

Derivation of Green's Function for Wave Equation

In the textbook Modern Methods in Analytical Acoustics (Crighton-1992, Amazon link to 2013 edition) the following relates the 3D Green's function in the time-domain to the frequency domain $g(x-y)$: ...
2
votes
2answers
114 views

Far Field Diffraction of EM waves: what does the zero frequency signify?

If you have a system of independently radiating electrons/point-charges, the far field distribution of the EM waves can be approximated by the fraunhoffer diffraction integral, or simply by the ...
3
votes
1answer
206 views

confusion in discrete transform to solve kronig penney matrix equation in fourier space

I have a periodic potential $$V(x) =\sum_{K}e^{iKx}V_{K} =\sum_{n}e^{\iota2\pi nx/a}V_{n} $$ where $K =\frac{2\pi n}a$ is the reciprocal lattice vector and $a$ is the lattice constant and $n =\pm ...
9
votes
4answers
1k views

Fourier Transforming the Klein Gordon Equation

Starting with the Klein Gordon in position space, \begin{align*} \left(\frac{\partial^2}{\partial t^2} - \nabla^2+m^2\right)\phi(\mathbf{x},t) = 0 \end{align*} And using the Fourier Transform: ...
2
votes
1answer
151 views

Three dimensional wave packets in momentum space

I am given the 3D wave packet: $$\psi(x,y,z)=N\,\exp\left(\frac{-(x^2+y^2+2z^2)}{2a^2}\right).$$ I was asked to find N (easy enough). Then I was asked the probability that we measure $z$ greater than ...
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
1answer
76 views

Transition from coordinate space to momentum space for SHO

I am given that the ground state of the SHO in position space is given as $$\langle q|\psi_0\rangle=\frac{1}{a^{\frac12}\pi^{\frac14}}e^{-q/4a^2}$$ Where a is a constant with units of length. I am ...
6
votes
1answer
257 views

Integral in $n$−dimensional euclidean space

I've asked this question in Mathematics Stack Exchange, but unfortunately there is no answer yet. I repost it because this integral comes from QFT and maybe someone here did it before or could help ...
4
votes
0answers
249 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( ...
0
votes
0answers
136 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
99 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, ...
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 $$ ...
4
votes
2answers
230 views

$2\pi$ and Feynman Rules

I notice a $2\pi$ term in the $\delta$-function when trying to construct an amplitude using the Feynman Rules. The $2\pi$ also appears as an integration measure to enforce normalisation in the phase ...
1
vote
0answers
79 views

Complex Fourier Particular Solution [closed]

I have found the complex Fourier series for my desired force. I now need to find the steady-state forced vibration of my oscillator as a Fourier Series. (The particular solution to the inhomogeneous ...
1
vote
0answers
327 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 ...
0
votes
1answer
840 views

Phonon dispersion calculation based on velocity autocorrelation function in reciprocal space?

I would like to ask about the method of calculating phonon dispersion relation. Up to my knowledge, there are 2 methods to calculate the phonon dispersion: By diagonalizing the dynamics matrix. ...
0
votes
2answers
201 views

Simplest derivation of Fourier transform for periodic functions (in crystal lattice)?

What is the simplest derivation of the following two well-known formulas that work for crystal lattice [1]: $$ F[f(\mathbf{x})] \equiv \tilde f(\mathbf{G}) = {1\over\Omega_\mathrm{cell}} ...
1
vote
2answers
195 views

How to derive inverse Fourier transform for periodic functions (in crystal lattice)?

I would like to derive the following two well-known formulas that work for crystal lattice [1]: $$ F[f(\mathbf{x})] \equiv \tilde f(\mathbf{G}) = {1\over\Omega_\mathrm{cell}} ...