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
121 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 + ...
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1answer
33 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 ...
6
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2answers
239 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 ...
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1answer
60 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 ...
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1answer
176 views

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

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> ...
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1answer
174 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 ...
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1answer
111 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) = ...
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0answers
64 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 ...
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0answers
63 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 ...
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1answer
74 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?
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1answer
202 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 ...
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4answers
292 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 $\delta(\omega)$. ...
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4answers
135 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 ~=~ ...
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1answer
146 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 ...
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1answer
291 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 ...
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1answer
363 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 ...
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1answer
70 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 $|\Psi\rangle^{\dagger}_s (r) = \frac{1}{\sqrt{V}} \sum_{k} e^{-i k r} \hat{a}^{\dagger}_{ks}$ ...
2
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1answer
167 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 ...
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2answers
141 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, ...
6
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3answers
470 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?
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2answers
263 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 ...
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2answers
385 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) ...
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2answers
94 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 ...
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3answers
356 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 ...
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1answer
89 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 ...
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1answer
340 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 ...
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1answer
146 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 ...
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1answer
67 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 ...
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1answer
286 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 ...
4
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1answer
199 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
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1answer
44 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 ...
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5answers
285 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
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1answer
67 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} = ...
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3answers
4k 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 ...
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0answers
34 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
170 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
304 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 ...
8
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2answers
703 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 ...
2
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0answers
204 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 ...
9
votes
1answer
235 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
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0answers
217 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
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0answers
61 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 ...
4
votes
2answers
310 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
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1answer
118 views

Momentum Representation vs Position Representation

We are given an operator $g$ from $\mathcal{l}^2(\mathbb{Z})$ to $\mathcal{l}^2(\mathbb{Z})$, i.e., the space of functions that are square summable over $\mathbb{Z}$ such that ...
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0answers
390 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 ...
2
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1answer
217 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 ...
5
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1answer
253 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 ...
3
votes
1answer
92 views

How does one plot the frequency and time domain of a distribution on a 3D plot?

I have seen this plot on Wikipedia's Fractional Fourier Transform, where it discusses the rotations between frequency and time domains of a distribution, however I do not understand how to plot a ...
1
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2answers
381 views

Volume of Brillouin zone is the same as Fourier primitive cell?

In Kittel's solid state text, problem 2.3, he says that the volume of the Brillouin zone is the same as a primitive parallelepiped in Fourier space. Somehow I can't see why this is true. Can someone ...
1
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1answer
203 views

Autocorrelation function for deterministic nonlinear dynamical systems

I am quite puzzled with the problem that spectral analysis has been either applied to noisy dynamical systems or to chaotic ones. I was wondering why nobody makes analysis of non-linear dynamical ...