All Questions
Tagged with conventions fourier-transform
52
questions
1
vote
2
answers
152
views
Does the exponential representation of Dirac delta function depend on choice of Fourier convention?
Is it always true that
$$\delta(\omega) = \frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{i \omega t} dt , $$
regardless of your Fourier convention?
For example, if I choose to use the Fourier convention ...
- 101
0
votes
1
answer
80
views
Confusion on the signs in the complex scalar field [closed]
I saw there are different ways we can write down the complex scalar field. For example, in most textbooks I can find, this is defined as
$$\phi(x) =\int \dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2E_p}}\...
- 1,783
1
vote
2
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167
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Why can $ϕ(p)$ be Fourier expanded to $ψ(x)$ in quantum mechanics? [closed]
I know the Fourier transform is $$
F(\omega)=\int_{-\infty}^{\infty} f(x) e^{-i \omega x} \,d x
$$
$$
f(x)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) e^{i \omega x} \,d \omega,
$$
but in ...
- 45
1
vote
0
answers
49
views
Green function and Fourier transformation and inverse transformation
Let $G(x, y_1, \cdots, y_n)$ be my Green function
I want to expand Green function with Fouier transformation of $y_{i}$ and inverse Foruier on $x$ so that $$G(x, y_1,\cdots, y_n) = \int dk_{x} \int_{...
- 3,622
1
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0
answers
67
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Power spectrum and Fourier transform convention (cosmology, Gaussian random field)
I have been confused about the definition of the power spectrum in cosmology, mainly associated with the convention of Fourier transform. The most of literature I saw, they are using
$$ f(\vec{x}) = \...
- 11
1
vote
0
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48
views
Why we can Wick rotate momentum axis for correlation function?
In QFT writtern by Peskin and Schroeder, in page 293, PS wick rotate both time axis and momentum axis of correlation function of Klein-Gordon field, ie
$$D_F=<0|T\phi(x_1)\phi(x_2)|0>=\int\frac{...
- 326
2
votes
1
answer
139
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Do we wick rotate momentum axis on correlation function?
In QFT written by Peskin and Schroeder, it is discussed how correlation function is evaluated in Euclidean space, on page 292 to 293,
In (9.48)
$$<\phi (x_{E1})\phi(x_{E2})>=\int \frac{d^4k_E}{(...
- 326
5
votes
1
answer
390
views
Wave equation boundary conditions for an engineer versus physicist
The wave equation is:
$$\nabla^2 \mathbf{E} + k^2 \mathbf{E} = 0$$
Using separation of variables, we get a solution of $E = a(x)b(y)c(z)d(t)$. Say for the $x$-direction we get a solution of:
$$ a(x) = ...
1
vote
0
answers
36
views
Expression of Dirac delta function by integral of exponential function [duplicate]
In many QFT books including Peskin & Schroeder's and M.Schwartz's mention about Fourier transform and representation of Dirac delta function as
$\begin{align}
&f(x) = \int\frac{d^3p}{(2\pi)^3}...
- 169
3
votes
0
answers
75
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Fourier transform of continuous-mode quantum creation operators
As I was reading "The quantum theory of light" by Rodney Loudon, I reached the following part that defines the relationship between time-domain and frequency-domain continuous-mode ...
- 173
0
votes
2
answers
717
views
Fourier Transformation in 4D space
In mathematical physics course, I see Fourier transformation of function f(t) as
$$\bar{f}(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)\ e^{-i\omega t} dt.$$
I wanted to know the ...
- 69
1
vote
0
answers
162
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Factor of $(2\pi)^4$ in momentum space Feynman rule
I'm trying to figure out the momentum space Feynman rules using Peskin and Schroeder. For simplicity I'll ask about section 4.6 for case of the $\phi^4$ theory.
In section 4.5, we have $$\tag{4.72}S=1+...
- 789
0
votes
0
answers
186
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Question on the bounds for finding Fourier coefficients
In Griffit's E&M, when solving Laplace's equation for the potential, he uses the "Fourier trick" on Legendre polynomials, where
my question is, why are the bounds from -1 to 1?
because ...
- 460
7
votes
1
answer
1k
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Incoming and Outgoing Waves in Quantum Field Theory
I apologize if this seems like a simple question, but I have been agonizing over it recently. In nonrelativistic quantum mechanics, a plane wave of the form $e^{\pm i\vec p\cdot \vec x}$ is called ...
- 785
-4
votes
1
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156
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Derivation of Fourier Transform in Quantum Mechanics [closed]
I recently came across an expression for Fourier Transform in Quantum Mechanics given by:
$$
\psi(x)=\frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^{\infty} e^{\frac{ipx}{\hbar}} \phi(p) d p
$$
I tried ...
