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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1answer
59 views

Eigenstates of position and momentum operators in QM

In Griffiths pages 103-105 "Introduction to Quantum Mechanics" 2nd editiion he states that the eigenfunctions of the position and momentum operators are $$g_y(x) = \delta(x-y)$$ where the eigenvalue ...
-1
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0answers
52 views

How to find the correlation function? [on hold]

I have an operator $\hat{a}(t)$, and the its correlation is given as: $$\langle\hat{a}(t)\hat{a}^{\dagger}(t^{\prime })\rangle=\delta(t-t^{\prime})$$ Then how can I find the correlation function in ...
3
votes
1answer
53 views

Vacuum expectation value in presence of a source

If a vacuum is translationally invariant i.e., $P^\mu|0\rangle=0$ or $e^{(\pm ip\cdot x)}|0\rangle=0$, we can express the the vacuum expectation value of a field as $\langle 0|\phi(x)|0\rangle$ as ...
0
votes
1answer
26 views

Energy Transferred to a Spring by a Time Dependent Force (using Fourier Transformations)

I found an excersice in Byron-Fuller's: "Mathematics of Classical and Quantum Physics", about the energy which is transferred to a spring by a time depended force of the form: $F(t)=\left\{ ...
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0answers
17 views

Hankel transformation of Yukawa potential [on hold]

This is a Hankel transformation problem that is used to do the 2-d Fourier transformation of Yukawa potential. We already know that $H_0(\frac{1}{z^2 + r^2}) = K_0(kz)$. Then what should be the right ...
2
votes
1answer
63 views

Anomaly, Ward identity [on hold]

While studying notes on anomaly by Adel Bilal (http://arxiv.org/abs/0802.0634), I stuck in a calculation. Here it goes as follows: The three-current correlator in perturbation theory as a one-loop ...
0
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0answers
55 views

Asymptotic behaviours from Fourier transforms [migrated]

I have completely forgotten how one derives the asymptotic behavior in frequency space, given the asymptotic behavior of the function in real space (e.g. time). As an example example, it is often said ...
1
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0answers
17 views

Help with two dimensional polar axis Fourier transform

This is a problem that I met in real-life physics research. This question is related to Wick's theorem. The question is: 1. In two dimensional plane with polar axis, why do we have the following ...
2
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0answers
27 views

What is the meaning of the Fourier Transform of the electric field of a well guided mode in a dielectric waveguide?

I have been studying waveguiding in dielectrics for a while now; however, I cannot understand the meaning of the Fourier transformed electric field. I will first give some background information. ...
0
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0answers
43 views

Why is the Hermitian conjugate of the Fourier transform of an operator not the transform of the Hermitian conjugate? [migrated]

It is defined that: \begin{align} O(\omega)&=\frac{1}{\sqrt{2\pi}}\int O(t)e^{-i\omega t} \mathrm{d}t \tag{1} \\ O^{\dagger}(\omega)&=\frac{1}{\sqrt{2\pi}}\int O^{\dagger}(t)e^{-i\omega t} ...
0
votes
1answer
53 views

Interpreting group velocity of free particle wave packet

I am trying to understand the concept of group velocity of a free particle wave packet: $$\Psi(x,t) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty} \phi(k)e^{ikx}e^{-\frac{i \hbar k^2 t}{2m}}dk.$$ ...
0
votes
1answer
26 views

Which information do we get from the phase spectrum about the wave?

Let a wave is represented by an equation $$y=f(t)=10\sin(\frac{2\pi f_1t}{T} + \pi/6)+5\cos(\frac{2\pi f_2t}{T} +\pi/3)$$. Here, let us take $f_1=10 ,f_2=5 ,T=100$ Then, from the Fourier transform ...
0
votes
1answer
64 views

Quantum Mechanics: how exactly does “delta function normalization” work for eigenfunctions in 1-d free space case?

The definition of "delta function normalization" says a basis of eigenfunctions of a particle in free space are orthonormal when ...
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0answers
24 views

Weinberg Cosmology book Ch 5.2

I am working on Weinberg Cosmology book, and have one question about what contained in Ch 5.2 (page 229). Basically, this chapter is dealing with stochastic initial conditions. What he wrote is that ...
1
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0answers
26 views

Fourier Transform point force on a half-space [migrated]

I have to calculate the following Inverse Fourier-Transform, which describe the potential function for a point force on a half-space: ...
2
votes
4answers
141 views

Meaning of a certain value at Fourier Transform

Define the Fourier Transform of a certain signal in the time domain FT[$x(t)$]=$X(j\omega)$ $X(j\omega)$ = $\int$ $x(t)$ $e$^($j\omega$$t$)$ $dt I'd like to ask what is the meaning of the value ...
1
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0answers
24 views

Radial Distribution Function - Structure Factor relation, deriviation help

I'm attempting to prove the relation between the structure factor and the RDF, following the deriviation here (pg 92-94). The solution this source comes too disagrees with this paper which I trust ...
4
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3answers
106 views

Same quantum states represented in different basis

In literature on an introduction to quantum mechanics which I am working through, there is a section which explains that a vector has different representations based on the basis you choose and then ...
3
votes
2answers
205 views

Schrödinger equation in momentum space

In literature on an introduction to quantum mechanics which I am working through, there is a section which explains that a vector has different representations based on the basis you choose. It then ...
1
vote
0answers
36 views

Lippmann-Schwinger equation and time dependence

Consider the Lippmann-Schwinger equation (LSE) $$ |\psi\rangle = |\phi\rangle + \hat{G}_0(\epsilon) \hat{V} |\psi\rangle \tag{1}$$ where $\hat{G}_0(\epsilon) = \frac{1}{\epsilon - \hat{H}_0 + ...
4
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0answers
58 views

Interpreting the Fourier transform of a Gibbs measure

Recall that a Gibbs measure gives a probability distribution on states $x$ of the form $$ p_X(x) = \frac{1}{Z(\beta)}\exp(-\beta E(x)) $$ As I understand, the function $E$ is interpreted as the ...
2
votes
3answers
47 views

Position and momentum measurement effects on wave functions

I have a few short questions about an interpretation of what happens with position and momentum wave functions described in literature I am using. Given momentum space wave function and position space ...
2
votes
0answers
29 views

What is the shape of the MTF curve in coherent imaging?

For incoherent imaging, the shape of the diffraction-limited MTF curve would look roughly like a triangle, with normalized contrast starting at 1 for zero spatial frequency and decreasing to 0 at the ...
2
votes
0answers
59 views

Intervalley scattering in graphene in presence of impurities

A long range impurity like coulomb impurity does not induce an inter valley scattering between the two Dirac points. Is there any mathematical explanation for the same although this is explained by ...
1
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1answer
72 views

Deriving Schrodinger equation from QFT with the definition $\psi(\textbf{x},t)\equiv \langle 0|\phi_0(\textbf{x},t)|\psi\rangle$

In the book "Quantum Field theory and the Standard Model" by Matthew Schwartz, he uses the equation $$\partial_t^2\phi_0=(\nabla^2-m^2)\phi_0$$ (i.e., the Klein-Gordon equation for the free ...
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votes
0answers
23 views

How to get the vibratonal frequency of a bond using FFT of velocity autocorrelation function?

I guess there is some errors in the way I am calculating VAC since I ma ending up with a peak whose frequency is two times the actual frequency. I ran an MD simulation long enough with 60 molecules of ...
1
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2answers
56 views

Fourier transform of Hamiltonian for scalar field

In the Srednicki notes (http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf) page 36 he goes from $$H = \int d^{3}x a^{\dagger}(x)\left( \frac{- \nabla^{2}}{2m}\right) a(x) $$ to $$H = \int ...
1
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3answers
61 views

Normal mode analysis

I'm reading lots of texts about normal modes and I've seen that normal modes are solutions of the wave function produced by separation of variables. However, when most of authors I've read perform the ...
3
votes
3answers
234 views

How to measure an image's contrast?

I'm studying Fourier optics and Interferometry and I intend to determine the contrast of an image using computer software. My teacher of Experimental Physics didn't tell me how to do it, and so, I'm ...
0
votes
1answer
23 views

Help understanding the wave number in light

For my final optics project I want to implement the beam propagation method using Fourier transforms. I came across the following document ...
3
votes
3answers
59 views

Why is response of system same frequency as driving force frequency

Super basic question: why does a system (to be definite, perhaps assume a collection of coupled harmonic oscillators) respond (in the steady-state, after transient effects have dissipated) with all ...
-1
votes
1answer
43 views

Expectation value of position operator $X$ in momentum space [closed]

I'm solving the following question: If $\psi(p)$ is the wavefunction of a particle in momentum space, write down the expression for the expectation value of the position operator $\langle x\rangle$? ...
1
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0answers
41 views

Help normalising and taking the inverse Fourier transform of this wavefunction [closed]

Normalising Consider the wavefunction $$\psi(x,0)=Ne^{-\frac{|x|}{\lambda}}.$$ In order to normalise this I take the integral, which due to the modulus on the $x$ I evaluate just from zero to ...
1
vote
0answers
65 views

Expanding a wavefunction [closed]

I have a wave function that I have already normalised: $$ \psi(x) = \sqrt{\frac{30}{a^{5}}}x(a-x) $$ but now I have been asked to expand it to get: $$ \psi(x) = \sqrt{\frac{960}{\pi^{6}}}\sum_{k} ...
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0answers
46 views

Using Plancherel's theorem on delta function

Plancherel's Theorem states that for $f \in L^{2}(\mathbb{R})$ we have $$f(x) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty}F(k)e^{ikx}dk \Longleftrightarrow F(k) = \frac{1}{\sqrt{2 ...
-1
votes
1answer
79 views

Commutation relations in Quantum Field Theory [closed]

\begin{align} [a, a^\dagger] =& \left[\int d^3 x e^{-ikx} (\omega \phi(x) + i \Pi^\dagger(x)), \int d^3 x' e^{ikx'} (\omega \phi^\dagger(x') - i \Pi(x')) \right] \\ =& \int d^3x \, d^3x' \, ...
3
votes
1answer
46 views

Finding the noise spectral density of a quantity made from different noisy components

I'm looking for the expression of the noise spectral density of the magnetic flux $\Phi$ generated by a noisy voltage signal $V$ applied to a resistor with Johnson-Nyquist noise $R$ which produces a ...
0
votes
1answer
87 views

Momentum and position for free particle

In the section of 'The free particle' in 'Introduction to quantum mechanics, second edition' by Griffiths page 65. He has the wave equation as $$\Psi(x,t) = \frac{1}{\sqrt{2 \pi}} ...
1
vote
0answers
24 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 ...
0
votes
1answer
43 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\} = ...
0
votes
1answer
38 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 ...
3
votes
0answers
122 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} ...
1
vote
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
votes
1answer
50 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
votes
1answer
96 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
votes
0answers
59 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: $\langle p | x \rangle = \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \, .$ ...
0
votes
0answers
20 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}}\\ ...
1
vote
0answers
35 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
votes
3answers
251 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
votes
3answers
132 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 ...