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Results tagged with fourier-transform
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user 9887
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.
4
votes
Meaning of a certain value at Fourier Transform
I'd like to ask what is the meaning of the value obtained from X(jω)
with certain frequency ω
Consider for a moment, the synthesis equation where we 'construct' $x(t)$ out of a weighted 'sum' ( …
- 60.3k
2
votes
String Theory and Fourier Analysis
So my question is, is Fourier Analysis essentially what String Theory
is?
Briefly, no.
String theory "is a theoretical framework in which the point-like particles of particle physics are repla …
- 60.3k
2
votes
Fourier series of single tone modulated wave
When a single-tone continuous modulating signal modulates a sinusoidal
carrier ... can't we apply fourier series and determine the harmonic
frequency components
$\cos(\omega_m t) \cdot \cos(\ …
- 60.3k
4
votes
Fourier Transform of 1
Why is the Fourier transform of 1 equal to δ(ω)
$$f(t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F(\omega)e^{i\omega t}\mathrm d\omega = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\delta(\o …
- 60.3k
9
votes
Continuous Fourier transform vs. Discrete Fourier transform
To be sure, it's the continuous (time) Fourier transform versus the discrete time Fourier transform (DTFT).
The former is a continuous transformation of a continuous signal while the later is a conti …
- 60.3k
3
votes
What is the physical interpretation of the Fourier transform $(\mathcal{F}Z)(t)$ an impedance?
For concreteness, assume a linear circuit element or network with driving point impedance $Z_a$:
$V_a(j \omega) = Z_a(j \omega) \cdot I_a(j \omega)$
Let:
$I_a(j \omega) = 1 \Leftrightarrow i_a(t) = …
- 60.3k
1
vote
Acausality in solving time-domain inhomogeneous differential equations with Fourier transforms?
This would imply that the solution depends on future values of the
input function.
This isn't true, your reasoning here is faulty.
First, an LTI system is causal if:
$h(t) = 0, t < 0$
where …
- 60.3k
2
votes
Accepted
What intermediate steps of the Dirac Delta Function and Fourier Series am I missing in findi...
From this step,
$$U_G=\frac1a\int_{-a/2}^{a/2}\mathrm{d}x
U(x)e^{-iGx}=\frac1a\int_{-a/2}^{a/2}\mathrm{d}x
\sum^\infty_{n=-\infty}A\cdot \delta(x-na)e^{-iGx}$$
note that the summation is an impu …
- 60.3k
4
votes
Accepted
Why does the mathematical constant $e$ enter into quantum mechanics so much?
Why is e useful in quantum mechanics?
Essentially, it is because $e^{ax}$ is an eigenfunction for the $\frac{d}{dx}$ operator:
$$\frac{d}{dx}e^{ax}= ae^{ax}$$
So, as an almost trivial example, …
- 60.3k
2
votes
Does the Fundamental Frequency in a Vibrating String NOT Necessarily Have the Strongest Ampl...
It all depends on the initial configuration. If the string's initial configuration (shape of string at $t=0$) includes a node at the center, the string will not vibrate at the fundamental frequency.
…
- 60.3k
8
votes
Accepted
How do we know that the Fourier transform of space is momentum?
What's the mathematical process and physical logic?
The Fourier transform of position space ($\vec x$ domain) is wave number space ($\vec k$ domain). This is an unambiguous, well understood math …
- 60.3k
0
votes
Accepted
How to prove that the position operator in momentum is $i\hbar \partial/\partial p$ - One Mi...
How to prove that the position operator in momentum is iℏ∂/∂p
Apply the following useful result.
If
$$F(k) = \int_{-\infty}^{\infty} f(x)e^{-ikx}dx$$
then
$$\int_{-\infty}^{\infty} xf(x)e^{-i …
- 60.3k
1
vote
Spacing of resonant modes in a (laser) cavity
I don't believe the section referenced is referring to the modes of a linear (1D) optical resonator but, rather, the modes of a 'volume' resonator analogous to the acoustic resonances in my home theat …
- 60.3k
2
votes
Accepted
How do I take take the partial derivatives of the general solution to the TDSE for a free pa...
$$\frac{\partial}{\partial t}\Psi(x,t) =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} \phi(k) e^{i\hbar kx}\left(\frac{\partial}{\partial t}e^{-\frac{1}{2}\frac{\hbar^2k^2}{m}t/\hbar}\right) dk$$
$$\ …
- 60.3k
2
votes
Same quantum states represented in different basis
Why is the vector |S⟩ represented as Ψ for both bases when working out
the components for the quantum mechanics case above?
The first of the final two equations is simply an expression for the …
- 60.3k