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
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1answer
50 views

Quantization of a free field: Klein-Gordon case

I am a beginner and reading this course text on QFT. The author first introduces the KG equation: $$\partial_\mu\partial^{\mu}\phi+m^2\phi=0$$ [with Minkowski signature $(+,-,-,-)$]. Then the ...
0
votes
0answers
12 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) ...
1
vote
1answer
38 views

Why can't we define a unique wavelength for a short wave train? [duplicate]

Here we encounter a strange thing about waves; a very simple thing . . .namely, we cannot define a unique wavelength for a short wave train. Such a wave train does not have a definite wavelength; ...
-1
votes
0answers
62 views

What does basis function actually mean?

I want to know that how any signal can be represented as sum of basis vector and what basis function actually means? Can anybody explain with simple examples?
2
votes
1answer
87 views

Second Quantisation, Fourier Transform, minus sign [closed]

I want to expand a field \begin{equation} \Phi (x) = \int \frac{d^3 p}{(2 \pi)^3} e^{ipx} \end{equation} in terms of the second quantisation \begin{equation} \Phi = \frac{1}{\sqrt{2 E}} (a + ...
0
votes
1answer
20 views

a question regarding Fourier transform in electron microscopy

I have recorded a micrograph of a 2-D array at a magnification of 43,000x on my DE-20 digital camera, which has a 6.4 μm pixel size and a frame size of 5120 × 3840 pixels. This magnification is ...
0
votes
0answers
20 views

Prove the relation related to this Fourier Transformate [migrated]

I have this Fourier trasformate $$\hat{f}(p)=\int_{-1}^{+1}(1-x^2)e^{-ipx}dx$$ and I have to prove this relation $$\hat{f}(p)=\int_{-1}^{+1}(1-x^2)e^{-ipx}dx =2 \int_{0}^{1}(1-x^2)\cos(px) dx$$ ...
6
votes
2answers
186 views

Can the momentum eigenstates be non-orthogonal?

Consider the Hilbert space of a particle, whose position domain is confined to $q\in[0,1]$ (e.g. a particle in a box with unit width). Using $$ 1=\int_0 ^1 dq |q\rangle\langle q| $$ and the position ...
0
votes
1answer
27 views

Inverse Fourier Transfrom of a wavefunction

I was reading about how a Fourier transform yields the wave-function expressed in terms of the momenta which constitute it, i.e. the wave-function in momentum space. I'm not so good at calculus yet ...
0
votes
1answer
91 views

How to prove that the position operator in momentum is $i\hbar \partial/\partial p$ - One Missing Sign

I am trying to prove that the position operator in momentum space is $i\hbar \partial/\partial p$ but my derivation is missing one sign. Can someone spot the error? Start with $$<\hat x> ...
1
vote
1answer
110 views

What intermediate steps of the Dirac Delta Function and Fourier Series am I missing in finding a solution to the Kronig-Penney Model?

Intro We're looking at the Kronig-Penney model in class and one of the conundrums is related to the Kronig-Penney potential for a chain of $N$ atoms. I'm supposed to squeeze out some expression for ...
2
votes
1answer
70 views

What am I REALLY doing when I take the Fourier transform of the momentum operator

I was playing around with some equations and found a surprising relationship when I took the fourier transform of the momentum operator Define $\hat P = \frac{\hbar}{i} \partial_x$, then $F(\hat P) = ...
0
votes
0answers
26 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 ...
1
vote
0answers
28 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 ...
-2
votes
0answers
26 views

Expanding Operator - Renormalization Group for SIAM

im currently reading a paper about renormalization group theory, especially for the single impurity anderson model. There occurs the approach do expand some fermionic operators via a ...
0
votes
1answer
36 views

What does multi-periodicity mean in stellar pulsations?

How can there exist multi-periodicity in stellar pulsations? http://www.kitp.ucsb.edu/sites/default/files/kitp/preprints/moskalik2.pdf How can one visualize a multi-periodic pulsation or oscillation?
0
votes
1answer
34 views

Simulating of Fraunhofer Diffraction of Zigzags by FFT

I tried to study the diffraction pattern of the following zigzag grating by Matlab(FFT of this image).. And the result showed like this(please ignore the scale bar in this img) I think the ...
1
vote
4answers
243 views

Fourier Transform of 1 [closed]

Consider the following convention for defining the Fourier transform $\hat{f}(\omega) = \int f(x) e^{-2 \pi i x \omega } d\omega $. Why is the Fourier transform of 1 equal to $\delta(\omega)$. ...
1
vote
4answers
121 views

Why does the mathematical constant $e$ enter into quantum mechanics so much?

In A. Zee's book Quantum Field Theory in a Nutshell, he mentions on pages 11-12 the following formula which he assumes reader had encountered before: \begin{equation} \langle q | p \rangle ~=~ ...
1
vote
0answers
26 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: ...
2
votes
1answer
62 views

The contraction of fermion field in 1+1-dimensional massless QED

My question comes from the textbook by Peskin & Schroeder, the integral (19.26): $$\begin{align} \int \frac{d^2 k}{(2\pi)^2}\! e^{- i k\cdot (y-z)}\frac{i \not{k}}{k^2} = -\not\partial ...
1
vote
1answer
105 views

What is the advantage of using exponential function over trigonometric function in analyzing waves?

A.P.French in his book Vibrations and Waves writes: . . . Why should the exponential function be such an important contribution to the analysis of vibrations? The prime reason is the special ...
1
vote
1answer
94 views

What is the difference between real and imaginary parts of a sinusoid? [closed]

Can somebody explain, without using complicated mathematical formulas, what do real and imaginary parts of the sinus function represent? And what are relations between them? I cannot understand why ...
0
votes
1answer
56 views

Inverting the field creation operator $|\Psi\rangle$

In my lecture notes on second quantization it is written that the creation field operator is given by $|\Psi\rangle^{\dagger}_s (r) = \frac{1}{\sqrt{V}} \sum_{k} e^{-i k r} \hat{a}^{\dagger}_{ks}$ ...
1
vote
1answer
153 views

Shouldn't motion be represented as a Taylor series rather than a finite sum of functions or a polynomial? [closed]

Since the change in velocity of an object at rest prior to time $t_{0}$ implies a change in acceleration — that is, let's postulate, $ \mathbb{P} $, the object would have remained still, so there was ...
2
votes
2answers
119 views

Is it accurate to say “a wavefunction is a function of particle positions or momenta”?

Something has been bothering me for a while. I encounter this kind of statement everywhere: While a single particle is described by a wave function $\Psi({\vec r};t)$, a system of two particles, ...
5
votes
3answers
128 views

How do we know that the Fourier transform of space is momentum?

How do we know that the Fourier transform of real space $x$ is the momentum $p$ space or for energy and time, receptively? What's the mathematical process and physical logic?
1
vote
2answers
68 views

Proof that quantum Fourier transform is unitary

I'm trying to work through the proof that the quantum Fourier transform can be described by a unitary operator, i.e $F^{\dagger}F=\mathbb{1}$, where ...
1
vote
2answers
138 views

Numerically solving 2D poisson equation by FFT, proper units

The 2D Poisson equation is: (1)$$\frac{d^2\varphi(x,y)}{dx^2}+\frac{d^2\varphi(x,y)}{dy^2}=-\frac{\varrho(x,y)}{\epsilon_0\epsilon}$$ And in $k$-space it is in form of: (2)$$(k_x^2+k_y^2) ...
0
votes
2answers
69 views

Interpretation of the four-vector $k$ in scalar QFT

I'm studying the canonical quantization of the Klein-Gordon real scalar quantum field theory, given by the classical Lagrangian density $$\mathscr L = {1\over ...
0
votes
0answers
22 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 ...
5
votes
3answers
242 views

The ubiquitous Planewave Ansatz

In physics, the planewave ansatz (meaning: an educated solution guess) is very ubiquitously used, when solving differential equations, in different domains of physics. E.g. to solve the dispersion ...
0
votes
1answer
32 views

Scattering from a potential, matrix elements of momentum eigenstates, and the Fourier transform [closed]

I am working on my last quantum homework and don't know where to begin with part (i) in this question 4. Do I need to use a product rule in the FT and use convolution? Not sure how to go about the ...
2
votes
1answer
147 views

Fourier and inverse fourier transform in QFT

According to my lecture notes, the inverse Fourier transform of an operator $\phi(p)$ is given by $$\phi(x)=\int \frac {d^4p}{(2\pi)^4}\phi(p)e^{-ip\cdot x}.$$ As @WenChern pointed out below, Peskin ...
1
vote
1answer
35 views

Why are unilateral Laplace transforms suitable for causal systems and bilateral aren't?

https://en.wikipedia.org/wiki/Two-sided_Laplace_transform#Causality The above section says that bilateral transforms will not necessarily make sense for causal systems. In the course of advanced ...
1
vote
1answer
55 views

What does $σ$ equal to zero mean?

Consider the Laplace transform of an RC filter. For those who can't immediately summon it, refer equation (46) at this link: http://web.mit.edu/2.151/www/Handouts/FreqDomain.pdf for a refresher. In ...
0
votes
1answer
30 views

Calculating a resonance fluorescence spectrum (Mollow Triplet)

I was working through a lecture on quantum optics, in which we calculate the spectrum of electric field correlations of fields produced by two level emitters. Now, the part where I got stuck was ...
2
votes
1answer
87 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 ...
1
vote
1answer
32 views

Water wave packet variance

Consider the following quantity, $$I = \int x^2|\eta(x)|^2 \ dx,$$. For $\eta(x)$ a solution to some linear equation, we have $\eta(x) = \int a(k) e^{ikx} \ dk$ where, for $\eta$ to be real, we ...
2
votes
4answers
98 views

What's the difference between frequency domain and time domain spectra?

If I have a mechanical oscillator and want to observe the dynamical behavior of the oscillator, is there any additional information to observe it in time domain and frequency domain? Normally, we ...
0
votes
1answer
49 views

Integral over scalar product of eigenfunction of momentum operator and harmonic oscillator one

Recently I've met following expression: $$ \tag 2 \sum_{n}f(n)\int dp~ |\langle p | n\rangle|^{2} = 2\pi \sum_{n}f(n). $$ Here $|n>$ is eigenfunction of harmonic oscillator with energy $$E_{n} = ...
0
votes
3answers
581 views

Why frequency is inversely proportional to time-period?

Why frequency is inversely proportional to time-period? While studying about Fourier transform that shows frequency representation. A doubt that came to me was a set of signal with same wavelength ...
1
vote
0answers
26 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 ...
4
votes
2answers
141 views

Renormalization, integrating out high momenta Wilson way

In equation $(12.5)$ in Peskin and Schroeder, they write out the generating function but leave out all quadratic terms of the form $\phi\hat{\phi}$ arguing that they vanish since Fourier ...
3
votes
2answers
123 views

Is a wave packet physically realizable as a Fourier series?

In QM a wave packet is modeled as an infinite, or almost infinite, Fourier series, and the Fourier transform provides a transformation between momentum space and position space. To what extent is ...
4
votes
2answers
300 views

Position operator in QFT

My Professor in QFT did a move which I cannot follow: Given the state $$\hat\phi|0\rangle = \int \frac{d^3p}{(2\pi)^3 2 E_p} a^\dagger_p e^{- i p_\mu x^\mu}|0\rangle,$$ he wanted to show that this ...
1
vote
0answers
99 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 ...
8
votes
1answer
202 views

Why is there $1/2\pi$ in $\int\frac{dp}{2\pi}|p\rangle\langle p|$?

I'm reading Richard MacKenzie's lectures on path integrals and on page 7 he derives the propagator for the free particle as follows: $$ \begin{align} K &= \langle q'|e^{-iHT}|q\rangle \\ &= ...
2
votes
0answers
96 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 ...
0
votes
0answers
45 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 ...