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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8 views

Time-domain NMR or: When is the Fourier-Transformation not appropriate?

My question has two parts: One is general and has to do with the Fourier-Transformation, one has to do with Time-Domain NMR. Both parts are interlinked, of course. I tried to find out, why people do ...
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1answer
36 views

Relation of the cross product of the functions to the cross product of their Fourier spectra

I know that according to the Convolution theorem the Fourier transform of the convolution of two functions $f$ and $g$ is equal to the product of their Fourier spectra: $\mathcal{F}\{f*g\} = ...
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1answer
36 views

Decoupling of double discrete Fourier transform

I have a problem with a double Fourier transform I encountered: $$\sum_{j=1}^L \sum_{l=1}^L e^{-i\pi \frac{n_1}{L} (j+l)}e^{-i\pi \frac{n_2}{L} (j-l)}V(j-l)$$ where $n_1,n_2$ are integer. If the ...
1
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0answers
39 views

Edge states of Kitaev chain [closed]

I am reading paper about Kitaev chain of electrons, which can exhibit famous Majorana fermions at ends of wire. The Hamiltonian (his Eq. (6)) reads $H = \frac{i}{2} \sum_j - \mu c_{2j-1}c_{2j} ...
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0answers
32 views

How do calculate <p|x>? [duplicate]

In my quantum mechanics lectures it says that the relation between the basis $|x\rangle$ and $|p\rangle$ is given by: $\langle p | x \rangle = \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \, .$ ...
2
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1answer
35 views

Why does the 4f lenses configuration decrease aberration? [closed]

In many publications/lectures it is said that the 4f lenses configuration is the preferred configuration for imaging. My question is why does this configuration in an imaging actually minimizes the ...
3
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1answer
77 views

Numerically solving a simple Schrodinger equation with fast Fourier transforms

While trying to solve a stochastic Gross-Piaevskii equation I have found a problem that can be tracked down to something buggy occurring in the simplest Schrodinger equation possible: $$\partial_t ...
2
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0answers
55 views

Is my expansion of the state $| x \rangle$ correct? [duplicate]

In my quantum mechanics textbook it says that the relation between the basis $|x\rangle$ and $|p\rangle$ is given by: $\langle p | x \rangle = \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \, .$ ...
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0answers
18 views

Eigenvalues for correlation matrix which have the form of an harmonic function

I am trying to understand the written in the picture below. I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ...
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0answers
21 views

Comoving and physical momentum in a Friedmann universe

It is most probably a very basic question, but I'm a bit stuck with it. Let us consider a spatially flat Friedmann universe with the usual metric ...
7
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3answers
234 views

Why is the Fourier transform more useful than the Hartley transform in physics?

The Hartley transform is defined as $$ H(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t) \, \mbox{cas}(\omega t) \mathrm{d}t, $$ with $\mbox{cas}(\omega t) = \cos(\omega t) + \sin(\omega ...
3
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3answers
113 views

How can $F_0\cos\omega t$ change to $F_0e^{i\omega t}$ in driven oscillator equation?

I have one thing that confuses me on deriving the solution for the Linear Forced Oscillator. Suppose we have the equation as $$ma + rv + kx = F_0 \cos \omega t$$ What confuses me is when the driving ...
0
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1answer
61 views

Derivation of group velocity using Fourier transform

The aim is to determine the group velocity of a wave packet with the general form $$\Psi\left(x,t\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \phi\left(x\right)e^{i\left(kx-\omega ...
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0answers
49 views

How can we fix the constant of the energy eigenstates of a quantum free particle such that they satisfy the orthonormality condition?

For a quantum free particle, the momentum and energy eigenstates are compatible. The constants of the momentum eigenstates are fixed by their orthonormality. Similarly, how can we fix the constant for ...
0
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2answers
47 views

$Ae^{\mathrm{i}\omega t}$ assumption for oscillating systems (formal & intuitive)

When we obtain a system of ODE's for $n$ masses connected with springs (or otherwise obtained by small amplitudes approximation), the next steps are usually assuming a solution in form $Ae^{i\omega ...
4
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2answers
99 views

Shifting momentum by a constant in the Schrodinger Equation

My book states that if we perturb a given Hamiltonian for the Schrödinger Equation $$ H = \frac{p^2}{2m} +V(x) $$ to $$ H' = \frac{p^2}{2m} + V(x) + \frac{\lambda p}{m} $$ then we can rewrite ...
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0answers
41 views

Doubt in Path integral equation

In Pokorski's "Gauge Field Theories" book, page 108 we find equation (2.87) ...
3
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2answers
98 views

Derivation of canonical position-momentum commutator relation

We know that the position-momentum commutator is fundamental in quantum mechanics, but would it be possible to derive it starting from a different set of first principles, more specifically starting ...
0
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2answers
62 views

How does one get the first few terms of the S-matrix expansion?

According to a set of notes I'm reading $$\langle p_f | S | p_i \rangle = \delta(p_f-p_i) + 2 \pi \delta(E_f-E_i) \bigg[\langle p_f | V | p_i \rangle + \cdots\bigg] \tag{1.29}$$ I don't understand ...
0
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1answer
28 views

Using the fourier series to analyze the motion of a finite string [closed]

Q: Find the Fourier series for the motion of a string of length L if (a) $y(x,0) = Ax(L-x); \frac{\partial y}{\partial t}_{t=0}=0.$ (b) $y(x,0) = 0; \frac{\partial y}{\partial ...
2
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1answer
204 views

Analogy to Fourier transform in spherical coordinates with boundary at a certain radius

Suppose, we have a wavefuction $\phi(\vec{x})$ which is restricted in a sphere, with the spherical boundary condtion $$\phi(\vec{x}=R)=\phi_0.$$ How can I do the 'Fourier transformation' as the case ...
0
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0answers
26 views

Units of Fourier Transform [duplicate]

I am a bit confused about the units of continuous time Fourier transform. Let's say that $x(t)$ is an input signal and has units of volts. Taking the Fourier transform of this yields $X(f)$. I would ...
0
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2answers
27 views

Can you use Fourier transformations (or other) to read multiple superimposed barcodes?

If you printed bar codes on tracing paper/acetate etc. and then positioned several in front of one another, could you extract the individual codes from the aggregate overlaid image? I feel intuitively ...
0
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1answer
73 views

How Quantum Fourier Transform equal to Hadamard Transform on 4-by-4 matrix?

I just don't understand why $QFT_4$ become the same as Hadamard Transform $H_4$ The Hadamard matrix is as follwoing, $$ H_2 = \frac12 \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & ...
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0answers
62 views

How to Fourier transform creation/annihilation operators?

Zee's QFT in a Nutshell pages 65-66. For a complex scalar QFT $$ \varphi(\vec{x},t) = \int\frac{d^Dk}{\sqrt{(2\pi)^D2\omega_k}}\left[a(\vec{k})\mathrm{e}^{-i(\omega_kt-\vec{k}\cdot\vec{x})} + ...
0
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1answer
37 views

locator equation of motion

I strugle with folowing problem. I do start with the locator equation of motion: $$G_{i j} = g_i \delta_{i j} + g_i \sum\limits_{k \ne i} W_{i k} G_{k j}$$ where $G_{i j}$ are matrix elements of ...
0
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0answers
95 views

How do you fourier transform a tight binding hamiltonian numerically?

The task is to do a fourier transformation of a tight binding hamiltonian of a 1D-chain with unit cell size 2, but even after many tries and googling I still don't have a idea how to do it correctly. ...
1
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1answer
63 views

Why do physicists use a positive sign for the Fourier kernel / outward propagating waves? [closed]

I am not a physicist but rather an engineer / mathematician, so I've always wondered why is it that physicists use the positive sign convention in the forward Fourier transform. That is, in all of my ...
2
votes
1answer
86 views

Is the wavelet transform utilized at all in QM?

Excuse any ignorance, but something was on my mind today and my professor didn't give me a very clear answer... Obviously the Fourier Transform is used pretty constantly in QM. What about the wavelet ...
2
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2answers
67 views

Physical interpretation of Fourier $[x(t)]$ where $x(t)$ is the position of mass $m$ as a function of time?

If a macroscopic body of mass $m$ moves according to a certain law of motion like, for example, $$x(t)=A\cos(2\pi ft)$$ then what physical interpretation can be attributed to the Fourier transform of ...
1
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0answers
99 views

Transforming the Schrodinger equation from momentum-space to position-space [closed]

I am working on a problem right now that states the following: A particle of mass $m$ is subjected to a force $\mathbf{F}(\mathbf{r}) = - \nabla V(\mathbf{r})$ such that the wave function ...
1
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1answer
84 views

Does “sum over all paths” in the path integral imply “sum over all paths” in momentum space when one Fourier-transforms?

How is the Fourier-transformed-field path integral interpreted? Is it still a "sum of all paths" in momentum space? Just that with another action? Consider for instance the (Euclidean) partition ...
0
votes
1answer
44 views

Fourier transform for $W(J)$ in a free QFT

In chapter I.3 and I.4 of A. Zee's QFT in a Nutshell, he starts with the theory $$ \mathcal{L}(\varphi) = \frac 12[(\partial\phi)^2-m^2\varphi^2]+J\varphi $$ Using the path integral approach, he ...
0
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1answer
74 views

A question on using Fourier decomposition to solve the Klein Gordon equation

Given the Klein Gordon equation $$\left(\Box +m^{2}\right)\phi(t,\mathbf{x})=0$$ it is possible to find a solution $\phi(t,\mathbf{x})$ by carrying out a Fourier decomposition of the scalar field ...
2
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0answers
55 views

Hints on alternative quantum Fourier transform [closed]

Nielsen and Chuang's "Quantum Computation and Quantum Information" text, and I have gotten stuck on Exercise 5.14 as shown below. I do not especially know how to construct V using $O(L^3)$ gates. ...
0
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1answer
74 views

Motion of string fixed at both ends

I was reading about the Fourier analysis from Waves by Frank S Crawford Jr. But I got trapped at the very beginning; this is the excerpt that troubled me: Motion of string fixed at both ends. ...
1
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1answer
100 views

Switch from the position representation to the momentum representation

If we use Fourier Transform, we can switch from the position representation to the momentum representation, like the following formula here comes the problem, if we use dirac notation we can see it ...
2
votes
1answer
60 views

How to detect “noisiness” of sound wave?

Some phonemes like "ssss" are basically white noise. How would you determine which parts of a wave are white noise? From frequency analysis the white noise will have no tones so just using this would ...
1
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0answers
41 views

Coulomb potential of a periodic crystal in reciprocal space

Usually the Coulomb potential (electron-electron interaction) can be Fourier transformed (aside from prefactors) like that: $$ \frac{1}{|\vec r_1 -\vec r_2|} = \int \frac{\text d ^3 k}{(2\pi)^3} ...
1
vote
0answers
58 views

$i\epsilon$ versus $2i \epsilon E_k$ in the propagator

The Fourier Transform of the propagator can be written as $$\tilde{\Delta}(k) = \frac{i}{k^2-m^2+i\epsilon} \tag{1} $$ which is then "factored" into $$ = \frac{i}{\left( k^0-E_k ...
0
votes
1answer
32 views

In quantum Fourier transform, why can any controlled $R_{k}$ gate be formed by two controlled-Not gate?

Controlled $R_{k}$ gate is implemented in quantum Fourier transform like this: Each of the $R_{k}$ on a qubit is in this matrix form: My question is: Why each of these controlled $R_{k}$ gates, no ...
3
votes
0answers
64 views

Linear KDV eq. asymptotics

The question arises from the book Solitons by P. G. Drazin about the linearized KDV eq. $$ u_t+u_{xxx}=0 $$ My first step was to take a Fourier transform of the equation, find that the dispersion ...
1
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2answers
264 views

Normalized wave functions in position and momentum space

Using the following expression for the Dirac delta function: $$\delta(k-k')=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(k-k')x}\mathrm{d}x$$ show that if $\Psi(x,t)$ is normalized at time $t=0$, ...
2
votes
0answers
26 views

How to mathematically model a realistic aperture illumination?

I want to know a mathematical expression that I can use to model a realistic aperture illumination to produce the primary beam of an antenna so that the radial distribution of this aperture ...
-1
votes
1answer
28 views

Having trouble understanding the proof for Fourier Transform Scaling Property [closed]

Starting from Plancherel's Theorem: $$ f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F(k)e^{ikx}dk ...(1) $$ $$ F(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-ikx}dx ...(2) $$ I need to ...
1
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2answers
54 views

Characteristics of an Optical System in the Fourier Domain

An imaging system can be characterized by its point spread function (PSF), which in most cases is space-variant. The final image is the result of the convolution of the PSF with the object (2d ...
1
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0answers
40 views

Discrete Fourier transform for periodic signal?

From the Signal and System textbook, by Oppenheim, I learned that the discrete-time Fourier transform can be written as $$ x[n]=\frac{1}{2\pi}\int_{2\pi}X(e^{j\omega})e^{j\omega n}d\omega $$ $$ ...
1
vote
1answer
59 views

Finding $\phi(k)$

If you have $\Psi(x,0) = c(\psi_1 + \psi_2)$ where $\psi_n$ is an Energy eigenfunction for a quantum number $n$. I'm supposed to find $\phi(k,t)$ at $t$ = 0. This is for an infinite square well from ...
-1
votes
2answers
100 views

Fourier Transform with Branch Cuts [closed]

I want to compute the Inverse Fourier transform of the following function (it appears as a certain correlation function in a physical model I am interested it): $$ \widetilde{f}(\omega) = ...
3
votes
1answer
86 views

A question about Fourier Transformation [closed]

Recently I try to evaluate a integral in a paper: $$ \Gamma(x,y)=\int_{-\infty}^{\infty} \frac{dk}{2\pi} \sqrt{k^2+m^2} e^{ikx} $$ This is the Fourier Transform of: $$ f(k)=\sqrt{k^2+m^2} $$ The ...