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
1
vote
1answer
56 views

Fourier transform integral in QFT

I was wondering if there exists a general formula for Fourier transform integrals of this type, which appear frequently in qft $$ I(m,n)=\int \frac{d^d k}{(2\pi)^d} \, e^{i\vec{k}\cdot \vec{x}} [\log ...
0
votes
0answers
11 views

What distribution shape does this broadband signal have if the bandwidth changes?

In the Lumerical tool, I'm simulating a setup using a broadband dipole signal. It is (or should be) centered at $\lambda=$ 922.5 nm and has various bandwidths $\Delta \lambda$ for different ...
0
votes
1answer
38 views

Energy eigenvalue in 2D hydrogen atom

I encountered the following issue. Consider a 2-dimensional Coulomb potential $$\Phi(x,y) = \frac{1}{\sqrt{x^2 + y^2}}.$$ It is important to know that all the analytical results in this question were ...
0
votes
1answer
24 views

Solving a classical damped free particle via the Fourier transform (and residue calculus)

Deriving the transfer function Suppose we have a free particle in one dimension with position $x$ and momentum $p$, and some damping $\Gamma$: \begin{equation} \begin{aligned} \dot{x} &= p/m, \\ \...
0
votes
0answers
8 views

Modeling of Spatial Filtering of aberrated beam (~10mm) with Pinhole (~10μm) in 4f-Configuration -> size-constraints

I am trying to simulate a spatial filter in 4f-Configuration with an aberrated input beam to get a feeling for the remaining intensity and wavefront error depending on the pinhole size & ...
0
votes
1answer
30 views

A Oscillatory integral in light-cone coordinates

I am trying to evaluate an integral in light-cone coordinates Where light-cone coordinates in 1+1D are defined by $x^+=\frac{x^0+x^1}{\sqrt 2}$ and $x^-=\frac{x^0-x^1}{\sqrt 2}$. The integral that I ...
3
votes
1answer
104 views

Vacuum energy of a free scalar field from path integral

My question has been asked two other times: Spinor vacuum energy (misleading title) and Vacuum Energy Calculation using Path Integral. I am not completely satisfied with the answers and it looks like ...
0
votes
1answer
38 views

Why multiplying complex current $\hat{I}(\omega)$ with $e^{-i\omega t}$ and taking the real part gives actual current?

In modern electrodynamics by Andrew Zangwill chapter 14, section 14.13.2 an analysis of RLC circuit is shown where Fourier transform of current, EMF, and impedance is used. And equation is $\hat{E}(\...
1
vote
1answer
68 views

Conventions: $e^{ikx}$ vs. $e^{-ikx}$ [closed]

What are pros and cons to use a negative or positive sign.
1
vote
2answers
104 views

Use of Dirac delta in expectation value

In looking at page 2 of these lecture notes: \begin{align} \langle p | X | \bar p \rangle &= \iint \frac{1}{\sqrt{2 \pi \hbar}} \, e^{-i p \bar x / \hbar} \, X \, \delta(\bar x - \hat x) \frac{1}{...
0
votes
0answers
30 views

How to decompose the Coulomb potential into Fourier series in a 3d periodic space?

I want to decompose the Coulomb potential $W\left(\mathbf{r}\right)=1/\left(4\pi r\right)$, in a periodic 3d space of size $L^3$ into a Fourier series $\hat{W}(\mathbf{k})=\int_{\left[-L/2,L/2\right]^...
0
votes
2answers
55 views

1-dimensional Heat Equation

I have to solve the following differential equation: $$ \partial _t u(x,t) = D \partial ^2_x u(x,t) $$ with the initial condition $ u(x,0)=\exp \left( -100^2 \left( x-\frac{1}{2} \right) ^2 \right) $. ...
0
votes
0answers
26 views

Contradiction? Getting a complex Fourier variable when solving the Helmholtz equation in lossy media

I want to find the general solution of the Helmholtz equation in the context of electromagentism $$(\nabla^2+n(\omega)^2\omega^2)\hat{E}(\omega, \vec{r})=0$$ For this I tried Fourier transforming the ...
0
votes
0answers
25 views

Vertex Feynman rules for $ZZH$ coupling

I have a Lagrangian term $$\mathcal{L}=gHZ_{\mu \nu} Z^{\mu \nu}.$$ Where $$Z_{\mu \nu}={\partial}_\mu Z_\nu - {\partial}_\nu Z_\mu.$$ For this interaction, vertex Feynman rule is $$g (g^{\mu \nu}p.q -...
0
votes
0answers
25 views

Fourier optics, classical optics and complex field?

I am studying fourier optics on Goodman bible. For example one of the most useful formula is this one (pag. 96): where Uf is the final field distribution,d is the distance of the input from lens, f ...
4
votes
1answer
182 views

The Ward-Takahashi identity in Peskin and Schroeder (page 311)

I'm working on the Ward-Takahashi identity in Peskin (page 311), but I canʻt obtain Eq.(9.105) from Eq.(9.103) According to Eq.(9.103) \begin{align} &i \partial_{\mu}\left\langle 0\left|T j^{\mu}(...
1
vote
1answer
33 views

Duffing equation

I have a differential equation: $$\ddot{\omega}+2k\dot{\omega}+2k^2{\omega}-2{\omega}^3=0$$ As I understand it's a Duffing equation, but I can't find the first integral. How can I do it? I didn't find ...
0
votes
0answers
60 views

Quantum Field theory question just conceptually grasping from Sean Carroll's “Biggest Ideas in the Universe”!

Quantum Field Theory from Sean Carroll's Biggest Ideas in the Universe. I’m just checking to see if I’m on the right track of what he's explaining. He talks about a free field (non-interacting field), ...
1
vote
1answer
98 views

Fourier Transform of $1/k^4$

I am dealing with a higher derivative theory problem and I have to perform the following integral, \begin{equation} \int \dfrac{d^3k}{(2\pi)^3}\dfrac{e^{i{\bf k}\cdot {\bf r}}}{k^4} \end{equation} ...
0
votes
1answer
23 views

Transverse displacement profile of the string using Fourier series

A string of length $L$ fixed at $x=0$ and $x=L$ and released at time time $t=0$, the transverse displacement at a position $x$ along the string is given by: $y(x,0)=Ax(L-x)$ Assuming that the string ...
-1
votes
1answer
40 views

What are the real world applications involving Laplace transforms? [closed]

Laplace transform is really interesting. Speaking about Fourier transform, there are many real world applications like we use in removal of noise and Laplace transform is again the extension of ...
1
vote
1answer
63 views

Fourier Optics: Far Field Image

I have a question about computing the far field diffraction pattern of a laser beam: If $L_{1}$ is large enough, then at $z=L_{1}$ we see the Fourier transform of the input $f(x, y)$. If $L_{2}$ is ...
0
votes
0answers
18 views

Normalization for the Fourier series of a Gaussian Electric field

I have a function $f(x) = \exp(-\frac{x^2}{\sigma^2})$ on a box of size $L$. I have the given relation between the size of the box and the width of the Gaussian $\sigma = \sqrt{\frac{2}{\pi}}L$. I use ...
1
vote
1answer
70 views

Trouble solving partial differential equation with Laplacian squared

I am working in extensions of General Relativity Theory and at the moment of taking the Newtonian limit of this extension theory (essentialy, mathematically speaking, this is just linearizing the ...
1
vote
1answer
72 views

4-dimensional Fourier transform of $(k\cdot v)^{-1}$

I have been trying to compute, without much success, the following Fourier transform in 4-dimensional Minkowski space $$ I=\frac{1}{(2\pi)^4}\int d^4 k \,\frac{e^{ik\cdot x}}{k\cdot v}, $$ where $v^\...
0
votes
0answers
25 views

Math for physics - Fourier inverse transform [migrated]

I have basic problem. I am not able to find (in Google) and derive on myself the inverse Fourier transform integral formula. $$Fourier\space defined:$$ $$F(\omega)=\int\limits_{-\infty}^{+\infty} f(t) ...
1
vote
1answer
55 views

Time evolution of the Gaussian packet

I am trying to get the time evolution for the following initial condition: $$ \Psi(x,0) = \left(\frac{1}{2\pi \sigma^2} \right)^{\frac{1}{4}} e^{- \left(\frac{ x-x_{0}}{2 \sigma}\right)^{2}} e^{i\frac{...
2
votes
1answer
45 views

Difficulty with Kallen-Lehmann spectral functions

The following is for $D=4$. The correlators at a fixed point are power laws of the form $x^{-2\Delta}$, where $\Delta$ is the scaling dimension. Suppose I wish to find the nature of the spectrum at ...
3
votes
1answer
58 views

Solving wave equations with Fourier transform: where are the time-independent solutions?

One typically solves waves (fields) equations in Fourier space. For example, the 1D wave equation $\frac{\partial^2\phi(x,t)}{\partial t^2}-\frac{\partial^2\phi(x,t)}{\partial x^2} = 0$ in Fourier ...
3
votes
1answer
53 views

What is 'degrees of freedom' when using Fourier series to express a periodic waveform?

We can express any desired periodic waveform using Fourier series. In the book I am studying from it's said: 'We see that with Fourier series, we can produce any desired periodic waveform and extract ...
1
vote
1answer
38 views

Arfken, Weber, and Harris Fourier transform properties inconsistent?

Does anybody know where to find the erratum page for Arfken, Weber, and Harris' Mathematical Methods in the Physical Sciences seventh edition? In Arfken, Weber, and Harris' Mathematical Methods in the ...
1
vote
1answer
58 views

How to move from the aperture to the far screen (from identity to Fourier transform in a smooth way)?

In diffraction experiments, the pattern on a screen is the fourier transform of the aperture when it is a Fraunhofer diffraction (when the screen is far from the aperture). When the distance is null (...
3
votes
1answer
123 views

Yukawa potential in higher dimensions

I am trying to calculate the integral \begin{align} E_n(\mathbf{r}) = \int \frac{d^n \mathbf{k}}{(2\pi)^n} \frac{ e^{i\mathbf{k}\cdot\mathbf{r}} }{ \mathbf{k}^2 + m^2 } \end{align} for $n > 2$ (the ...
0
votes
0answers
40 views

Dot product between momentum and position $ \langle p|x \rangle$ [duplicate]

I've been trying to derive the result of the following dot product in position space: $\langle p|x\rangle$. The information I'm supposed to be using: i) Any state can be written as a linear ...
0
votes
0answers
9 views

Measuring momentum in a 2 height well

In a 2 height finite square well we know that qualitatively the position wavefunction will have a lower amplitude in the side of the well with a smaller potential energy (since it will be travelling ...
1
vote
3answers
55 views

Expressing a wave on a string using Fourier series

Where does time function of wave on string go when expressed in the Fourier series? A Standing wave on string of length $L,$ fixed at its ends $x=0$ and $x=L$ is: $\quad y(x, t)=A \sin (k x) \cos \...
4
votes
3answers
680 views

Why doesn't Gaussian wavepacket broadening in position mean there will be a shortening in momentum?

Many sources that say in free broadening of a Gaussian wavepacket, the momentum uncertainty (I think defined in terms of the range of 'significant' momentum amplitudes) is time invariant even as the ...
0
votes
0answers
19 views

Can I distinguish hexagonal close packing (HCP) from face centred cubic (FCC) arrangement based on Fourier transform

First of all I would say that I'm not a physicist, but I have recently been given the task of distinguishing a hexagonal close packing (HCP) from a face centred cubic (FCC) arrangement in a set of 3D ...
0
votes
0answers
52 views

Mechanical impedance and dynamic stiffness of a mass, spring, damper system including Coulomb friction

I'm trying to understand the concepts of mechanical impedance and dynamic stiffness, what do they mean and if/how they differ. Consider the very simple system below: Image curtesy of Joshua ...
1
vote
1answer
46 views

Plane Wave Decomposition of Electric Field

I've tried to understand the decomposition of an HF electrical field in a series of plane waves. $$\vec{E}(\vec{r}, t) = \int\int\int \hat{\vec{E}}(\vec{k}) \cdot\mathrm{e}^{\mathrm{i}(\vec{k}.\vec{r}-...
0
votes
3answers
37 views

What Plane Waves Make a Gaussian Beam?

When I think of a light beam, what first comes to mind is this: Black lines are axes (both spatial axes, at a snapshot in time), and blue lines represent the surfaces of constant phase of a plane ...
0
votes
0answers
38 views

The quantisation of the harmonic oscillator applied to the free Klein-Gordon field

In David Tong's lecture notes on quantum field theory, at the bottom of page 23, we are applying the quantisation of the harmonic oscillator to the field to obtain expressions for the field operators ...
-2
votes
1answer
68 views

Dispersion relation in $\frac{d^2y}{dt^2}=c^2\frac{d^2y}{dx^2} + ac^2\frac{d^4y}{dx^4}$

So I was reading through some lectures notes and I found this: The equation of motion of a non-ideal string is $\frac{d^2y}{dt^2}=c^2\frac{d^2y}{dx^2} + ac^2\frac{d^4y}{dx^4}$ and we are asked to ...
2
votes
1answer
58 views

Bounding derivatives of the Wigner function using phase-space tails

Suppose I have a Wigner function that falls off faster than any polynomial for all directions in phase space. That is, for all $a,b>0$, $$\lim_{|x|\to\infty} |x^a p^b W(x,p)| =0=\lim_{|p|\to\infty} ...
0
votes
1answer
34 views

Group velocity in solid state physics force equation

When reading about electron holes here https://en.wikipedia.org/wiki/Effective_mass_(solid-state_physics), the group velocity is introduced as the reciprocal space gradient of the dispersion relation. ...
0
votes
1answer
41 views

Continuous spectra and measurement

Suppose I have a particle whose momentum I measure to be $p$ with uncertainty $\delta p$. Right after the measurement we know that its wave function is given by $\psi(x)=\int g(p)e^{ipx/\hbar}dp$ (...
0
votes
1answer
52 views

Evaluating Fourier transform of $ f(t) = \sinh ^{-q}(a t) \left(1-\frac{1-a t \coth (a t)}{B}\right) $

I am interested in evaluating Fourier transform of the following function analytically, $$f(t) = \sinh ^{-q}(a t) \left(1-\frac{1-a t \coth (a t)}{B}\right) $$ where $a, B, q$ are some real parameters ...
0
votes
0answers
41 views

Fourier expansions of Klein Gordon field not Lorentz invariant?

I’m working in Peskin and Schroeders book on QFT and noticed that they expanded a solution to the Klein Gordon equation in a manner that seems to me not to be be Lorentz invariant even though the ...
11
votes
1answer
683 views

How can I compute the derivative of delta function using its Fourier definition?

I am wondering if it's possible to compute the derivative of the Dirac Delta function using the definition obtained from Fourier transformation: $\delta(x-x')=\frac{1}{\sqrt{2\pi}}\int e^{-ik(x-x')}dk$...
0
votes
1answer
29 views

Fourier transform of field variables rearrangement

I’m working in Peskin and Schroeders book on QFT These are the Fourier transforms of the field solutions to the Klein-Gordon equation: I don’t understand how to get from (2.25) to (2.27) The logical ...

1
2 3 4 5
23