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Results tagged with fourier-transform
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user 83835
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.
1
vote
Accepted
Superposition of waves with different initial phase in Quantum Mechanics
There are some correct things and some incorrect things in your question. But let me just give the main points.
First of all, if you add two states together with different relative phases, you get di …
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1
vote
Accepted
In trouble with QFT field operators
You are quantizing inside a box of volume $V$. Therefore in this context, $$\int d^3x \neq \int_{\textrm{all space}}d^3x = \infty\,.$$
Instead,
$$\int d^3x = \int_{\textrm{box of volume } V}d^3x = V\, …
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7
votes
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Why use Fourier series instead of Taylor?
Many of the other answers are addressing the practicalities of expanding in Fourier series versus Taylor series. But there is at least one physical reason for choosing one over the other, and that is …
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42
votes
Accepted
Why do we hear frequencies in the basis of sine waves?
The "reason" that the Fourier decomposition is the "correct" one has to do with the fact that both signal detectors (microphones, ears, etc.) operate as driven harmonic oscillators. Or rather, it is t …
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7
votes
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Fourier transform of Hamiltonian for scalar field
Starting from
$$
H = \int d^{3}x a^{\dagger}(x)\left( \frac{- \nabla^{2}}{2m}\right) a(x)
$$
and
$${a}(x) = \int \frac{d^{3}p}{(2\pi)^{\frac{3}{2}}}e^{ipx}\tilde{a}(p)$$
(the second of which follows b …
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1
vote
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How is the initial condition term $\frac{1}{\sqrt{2\pi}}\phi(k)$ derived for free particles?
Interpreting the Fourier transform as an expansion in a basis
Ignore the time-dependence for the moment. Some pure maths stuffs from Fourier analysis allows us to write any square-integrable function …
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6
votes
Accepted
Applying measurement postulate to a continuous sum of eigenvectors (by analogy)
I've changed your bullet points: In the continuous sum
$$ \Psi(x,0)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\Phi(k)e^{ikx}dk\,,$$
$\frac{e^{ik_0x}}{\sqrt{2\pi}}$ plays the role of the eigenstate …
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2
votes
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"Stationary" vs. moving wave packet
If you look at your distribution in momentum space, you can see that it is an even function about $p=0$. For this reason, the average value of the momentum is zero, and so the center of the wave packe …
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6
votes
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Is there a momentum representation of the atomic stationary states?
This was apparently done by Podolsky and Pauling in 1929. Some of the details below are taken from this paper. I also found this comment on another paper that has these details in a more modern nota …
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How does the wave function in the momentum basis evolve over time?
The uncertainty principle does not guarantee that the product of uncertainties stays the same. It merely guarantees that the product of uncertainties must be greater than $\hbar/2$. So it is perfectl …
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