Questions tagged [fourier-transform]

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 calculating the response of a time shift invariant linear system as the sum of its response to time harmonic excitation or for transforming a quantum state in position co-ordinates into one in momentum co-ordinates and contrawise. There is also a discrete, fast Fourier transform for discretised data.

Filter by
Sorted by
Tagged with
0
votes
2answers
102 views

What is The Heisenberg Uncertainty Principle Making Statements About?

Non-physicist here, trying to understand details about Heisenberg's uncertainty principle: Watching the The more general uncertainty principle, beyond quantum about the uncertainty principle I came ...
1
vote
0answers
27 views

Solving a PDE with Fourier transform [migrated]

[Homework problem:] In my assignment, I was given a problem where I had to solve a PDE using Fourier transform. It goes like this: Solve the PDE $$u_t=u_{xx}$$ subject to the initial conditions: $u(...
0
votes
0answers
16 views

Fourier Transform of finite time series

I have some signal $s(t)$ which is real data i.e. finite. The time runs from $-T$ to $+T$. The signal amplitude is large at $t=0$ and small ($\rightarrow 0$) at the $\pm T$ limits. I can do a finite ...
1
vote
0answers
22 views

Using FFT to find time-dependence of quantum mechanical free particle?

I am trying to visualize the time dependence of a free particle given an initial wave-function using Python and I just wanted to know if I could use the in built FFT implementation from NumPy to find ...
0
votes
0answers
32 views

In cosmology, what means the 'Fourier modes'?

I am reading papers of cosmology for my thesis and always I read 'Fourier modes' but I don't understand why is necessary that we work in the Fourier space and what means 'Fourier modes'? Maybe it is a ...
-1
votes
0answers
22 views

Is it (practically) possible to create non-uniform electrostatic in the form of a Fourier series?

Is it (practically) possible to create a non-uniform electrostatic field of the form $$E_x = \sum_{k=0}^N C_k e^{ikx}.$$ (keeping it in one dimension for simplicity) If yes, how?
0
votes
1answer
60 views

What is the Fourier Transform of the spatial portion of $Ψ(x,t)=A\exp(-b|x-2|)\exp(-i\omega t)$

What is the Fourier Transform of the spatial portion of $\Psi(x,t)=A\exp(-b|x-2|)\exp(-i\omega t)$? I'm not sure how to do it for the "spatial portion". I've only done Fourier transforms for ...
1
vote
1answer
40 views

Derivation of the Klein-Gordon solution via Fourier Transforms

I recently graduate with a bachelor's in physics, and I've been trying to take the next steps toward learning QFT. To this end, I have been working through Peskin and Schroeder's textbook step-by-step....
1
vote
0answers
44 views

Quantization of field with other complete orthogonal system

I've learned the quantization of Klein-Gordon field using Fourier expansion. I understand that this process is kind of exchanging complex fourier coefficients to operator and makes it satisfying the ...
3
votes
0answers
65 views

Interpretation of annihilation and creation operators

If we write some quantum field in a form using creation and annihilation operators we are, in a way, doing a Fourier series with annihilation and creation operators being coefficients. So, if they are ...
0
votes
0answers
27 views

Calculating a matrix element, $\langle x|p\rangle$? [duplicate]

I've often just assumed $$\langle x|p\rangle = \frac{1}{(2\pi\hbar)^{1/2}}e^{ipx/\hbar}$$ Is there a formal way to prove this? My guess is that, since $\langle x|x'\rangle = \delta(x'-x)$, and a ...
-1
votes
0answers
21 views

Thermal Conduction in Fluids

I am considering a project on radial thermal conduction in fluids. I am comparing radially inward and radially outward flow of heat in ideal fluids. Since it is fluids we are considering, it seems ...
0
votes
1answer
24 views

Why is the image complex in magnetic resonance imaging? (and how does partial Fourier imaging work?)

The MRI image is reconstructed using inverse Fourier transform from k-space data measured during a pulse sequence. According to some online sources the resulting image is complex-valued and usually (...
0
votes
0answers
13 views

Transformation to get rid of distribution multiplets

Are there any well-known distribution transformations that account for doublets, triplets, higher multiplets due to time series independent peaks occasionally being too close to resolve? I feel like ...
0
votes
1answer
38 views

Momentum in complex scalar field

Consider a complex scalar field $\psi(x)$ with Lagrangian density $$ \mathcal{L} = \partial_\mu\psi^* \partial^\mu\psi - M^2\psi^*\psi. $$ Expand the complex field operator as a sum $$ \psi = \int \...
1
vote
0answers
25 views

How do I calculate the resulting field of a 2f-arrangement?

I've been trying to proof that the 2f-arrangement for lenses causes a fourier-transform of the original field. After some research I found a hint how to pull it (see page 15), but all my calculations ...
0
votes
0answers
59 views

Solve time-independent Schrödinger equation in the momentum basis

So I was reading Henk Stoof's Ultracold Quantum Physics and there was this simple example of a particle in a finite space (not infinite well) with no potential whatsoever. It is solved in the position ...
0
votes
0answers
39 views

Feynman rules for scalar field with second order derivatives in the interaction term

Given the interaction term with $N$ scalars $\phi_i$, each massless, what would be the Feynman rules for an interaction term in the action as $$ \int d^dx (\partial^2 \phi^i)\phi_i(\partial_\mu \phi^...
2
votes
0answers
40 views

Integration domain of fourier series expansion on lattice

In solid state physics we define the Fourier expansion of a lattice periodic function $f$ as $$f\left(\vec{k}\right)=\sum_{\vec{R}_n} f_{\vec{R}_n} \mathrm{e}^{\mathrm{i}\vec{R}_n\cdot\vec{k}}$$ where ...
0
votes
1answer
25 views

What happens to a wave when the tube changes in diameter

everywhere they only show the single diameter tube, but they don't explain what happens to the wave when the tube changes its diameter Is someone kind enough to graph me how the wave changes when ...
0
votes
0answers
40 views

Harmonic oscillator eq. for complex amplitude ---field quantization

I am new to quantum optics and going through "Introductory quantum optics" by C. Gerry and P. Knight. In chapter 2 they are writing eq. (2.81), the harmonic oscillator eq. for complex amplitude A ...
1
vote
1answer
51 views

Should a wavefunction in momentum space be normalisable?

Is this a condition that the wavefunction in momentum space should be normalizable? Like we said that a particle has to be between ${-\infty}$ to ${\infty}$. Will the same argument also work for ...
2
votes
0answers
41 views

What properties of physical quantities make them Fourier transform pairs?

I understand the general notion of Fourier transform. I wanted to know what properties do a particular pair of physically measurable quantities have that such that the Fourier transform of a function ...
3
votes
1answer
215 views

Fourier transform in Hawking's paper

While calculating $\beta_{ij}$ for the case of a Schwarzschild black hole, Hawking uses the Fourier transform of the solution of the wave equation (Particle Creation by Black Holes, S.W. Hawking, ...
0
votes
2answers
49 views

Superimposed Waves

This question has been bothering me for a very long time. Imagine a wire carrying electric current. It carries two alternating current (AC) signals of different frequencies (say $50$ Hz and $60$ Hz). ...
1
vote
0answers
23 views

Derivation for number of photons assigned to to an electromagnetic field amplitude

I am trying to understand a derivation concerning the number of photons associated with an electromagnetic field which is irradiated on some probe. The point is to get a expression that connects the ...
4
votes
0answers
89 views

The frame of truncated momentum basis on a 1D lattice

$\def\ket#1{\left|#1\right\rangle } \def\bra#1{\left\langle #1\right|}$ (This is part of a research problem) The Setup: Consider a single particle on a finite 1D lattice with the Hilbert space ...
2
votes
1answer
38 views

How to take the Fourier transform of the propagator of a vector field?

In the paper Wilson Loops in N=4 Supersymmetric Yang--Mills Theory, the authors give the following generalized Fourier transform for a propagator in $d=2\omega$ dimensions: $$\int \frac{d^{2\omega}p}{...
0
votes
0answers
54 views

Crystal lattice Fourier coefficient

In solid state physics we have a periodic function over the lattice as $$f(\boldsymbol{r})=\sum_{\boldsymbol{G}}f_{\boldsymbol{G}}\mathrm{e}^{2\pi i\boldsymbol{G}\cdot\boldsymbol{r}}$$ where $$\...
1
vote
0answers
33 views

Vorticity of Fourier Expanded Velocity

I have been reading some papers which find all three components of the vorticity vector for a Fourier expanded (perturbation) velocity field i.e $\mathbf{u'}(x,y,z,t)=\int\mathbf{\hat{u}}(x,y,t)e^{...
0
votes
1answer
69 views

Finite-time Fourier transform of a wavefunction

Can someone explain this formula to me? Given a wave packet whose time evolution is $g(t)$, a partially resolved spectrum is found by Fourier transforming its overlap with the same wave packet at ...
0
votes
1answer
37 views

Calculation of fluctuation corrections to the saddle point approximation

From Statistical physics of fields by Mehran Kardar page 45 section 3.6 [...] the partition function including small fluctuations is $$\begin{align} Z \approx e^{-V\left(t/2 \cdot \overline m^2 +...
1
vote
1answer
25 views

Fourier Transform from lattice site into $k$-space in Hubbard-Holstein model

Say I have a one dimensional lattice with lattice constant $a$. With next nearest neighbor hopping (NNN) included, the hopping term that describe such system would be $$H_{hop} = -t\sum_j(\hat c_{j+1}...
0
votes
1answer
31 views

How to solve this problem involving the “longest interval”?

The problem is shown as follows: If one wants to make a digital record of sound such that no audible information is lost, what is the longest interval, $\Delta t$, between samples that could be ...
1
vote
0answers
60 views

The validity of some “applications” of the uncertainty principle

Given a $L^2$ function $f$ with $\int_\mathbb{R}xf(x)dx=0$, define its variance to be $\sigma_f^2=\int_{\mathbb R}x^2f(x)dx$. The uncertainty principle states that $\sigma_f\sigma_\hat f\geq 1/4\pi$,...
3
votes
1answer
132 views

Is there an explanation for this unexpected similarity between binomial coefficients and waves?

Background Binomial coefficients appeal mostly in probability, combinatorics number theory etc so were were surprised when we observed something that appeared to belong more to physics than pure ...
6
votes
1answer
83 views

Solving free particles with Fourier series

Here's a silly idea : take the action of a free particle, $$S = \int_{t_1}^{t_2} \dot{x}^2 dt$$ Our configuration space is the space of $C^1$ functions over $[t_1, t_2]$, which is spanned by the ...
0
votes
0answers
14 views

Triple infinite summation of a 3D Fourier series for Madelung Potential

I'm trying to evaluate the equation below excluding the case when $n_x=n_y=n_z=0$. I know this equation converges everywhere except where x,y, and z are all multiples of $2\pi$. I've attempted ...
0
votes
0answers
53 views

Projection into Lowest Landau Level and Fourier transform

I am studying Quantum Hall and therefore Laughlin wave functions and the Lowest Landau Level. States in the Lowest Landau Level have the form: $\phi_m(z,\bar{z}) \propto z^m exp( - z \bar{z} / 4 l^...
3
votes
2answers
205 views

Quantizing Klein Gordon Field: Sign Problem

I'm trying to re-derive the Quantization of the Klein Gordon Field but I'm running into sign problems. My starting point is: $$ \phi(x,t) = \frac{1}{(\sqrt{2 \pi})^3} \int \tilde{\phi}(k,t) e^{i kx}...
2
votes
0answers
43 views

Why is the optimum window length for a discrete fourier transform of a signal less than 100%?

I'm trying to determine the best settings for a discrete Fourier transform on a signal with noise. Now I've stumbled on something that I can't seem to explain, I'm hoping someone can give me some ...
2
votes
1answer
111 views

Momentum Wave Function gives strange expectation values

Suppose there's a particle with the wave function $\psi(x)=\frac{1}{\sqrt{L}}$ for $0<x <L$ and 0 everywhere else. One way to get the associated Momentum Wave function is direct integration on ...
0
votes
2answers
52 views

How many linear combinations of harmonics or normal modes can describe the same periodic function as a Fourier series?

Please note that I am not asking how many terms in a linear combination can describe a specific periodic function but if given that there exist a set or linear combination of normal modes that ...
1
vote
2answers
79 views

Impossibility of Monochromatic Light [duplicate]

Pages 24-25 of my textbook, Optics by Hecht, says the following: Using the above definitions we can write a number of equivalent expressions for the traveling harmonic wave: $$\psi = A\sin k(x \...
0
votes
1answer
26 views

Why does the resonant frequency disappear for a ball in a potential well being jiggled by multiple frequencies?

Here's what I'm doing: I'm using MATLAB to model a ball placed within a potential energy well. I'm then driving this ball with an external driving force. The function for the external driving force ...
0
votes
1answer
45 views

Can momenta eigenstate written in term of $x$ be an eigenfunction of position?

Being non-commuatable operators, momentum and position cannot have simultaneous eigenfunctions. But in "Theoretical Minimum: QM" by Lenny Susskind and Artsy Friedman, in explaining Heisenberg's ...
0
votes
0answers
9 views

Shifting identical but offset pulses in frequency domain

Supposes I have 2 identical pulses, but one is offset by some phase. I want to take fourier transform of the signal in time domain to frequency domain. In freq domain, their spectrums are identical, ...
1
vote
1answer
119 views

Quantum mechanics, Fourier transformation

Why do we use $p=-i\hbar\frac{\partial}{\partial x}$ in quantum physics? (I know the reason for $i\hbar$, quantization). Is this right to say we can't measure velocity and position of electrons at the ...
1
vote
1answer
32 views

Application in medical field

I have heard about many new developments in radiotherapy for treating cancer/tumors such as hadron therapy but why can't we use wave interference for it? Incoming waves could interfere destructively ...
1
vote
0answers
39 views

Wave equation calculation [closed]

when I have a 1 dimensional, linear wave equation: $\Box\ \psi(t,z)=(\partial_z^2-\frac{1}{v^2}\partial_t^2)\ \psi(t,z)=0$. $\Box$: d´Alembert operator So linear combinations of plane waves $e^{ik(...