All Questions
18
questions
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The Lorentz-non-covariance of the Wigner Function
What does the fact that the Wigner function is not Lorentz-covariant imply?
My analysis so far led me to the (probably naive) understanding that there really is nothing special about it, just that it ...
2
votes
0
answers
89
views
Fourier Transform of $s$-ordered Characteristic Function
In the book, "Quantum Continuous Variables (A Primer of Theoretical Methods)" by Alessio Serafini, on page 70, he defines an $s$-ordered characteristic function to be:
$$
\chi_s(\alpha)=\...
1
vote
1
answer
81
views
What is the Weyl transform of narrow Gaussians and/or the Dirac delta?
Consider the family of Gaussians in $q$, $p$ with decreasing widths $σ$
$$Φ_σ(q,p) = \frac{2}{π σ^2} e^{-\frac{2}{σ^2}(q^2+p^2)}$$
or in complex plane coordinates
$$\tilde Φ_σ(α) = \frac{1}{π σ^2} e^{-...
1
vote
1
answer
217
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What is the most general wave function of a minimum uncertainty (Gaussian) state in quantum mechanics?
For some state $|\psi\rangle$ it is possible to recover the uncertainty principle using the fact that $$\left|(\hat{\sigma_{Q}}-i\lambda\hat{\sigma_{P}})|\psi\rangle\right|^{2}\geq0,$$where$$\hat{\...
5
votes
1
answer
449
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How can the Wigner function of squeezed states be non-negative?
It is always said that when the Wigner function of quantum states takes a negative value, then it is a clear signature of non-classicality of this particular state. It is also well-known that the ...
2
votes
1
answer
444
views
Understanding derivation of Wigner function for the Harmonic oscillator
In the document https://www.hep.anl.gov/czachos/aaa.pdf, they derive the Wigner functions, $f_n$ for the harmonic oscillator. However, I have some problems understanding some of the steps. On page 37 ...
0
votes
1
answer
351
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Prove that $f_\psi(x,p)$ is the Wigner Function of a pure state iff $H\star f_\psi= E f_\psi$
Given a pure state $|\psi\rangle$ with position wavefunction $x\mapsto\psi(x)$, define its Wigner function as
$$f_\psi(x,p) = \frac{1}{2\pi} \int dy e^{-iyp} \psi(x+y/2)\psi^*(x-y/2)
\equiv \frac{1}{2\...
0
votes
1
answer
273
views
How does the Weyl transform take into account which quasiprobability distribution was used?
I'm trying to get a better understanding of the Weyl correspondence which, as described e.g. on Wikipedia, gives "an invertible mapping between functions in the quantum phase space formulation ...
1
vote
1
answer
318
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Why does the star product satisfy the "Bopp Shift relations": $f(x,p)\star g(x,p)=f(x+\frac{i}{2}\partial_p,p-\frac{i}{2}\partial_x) g(x,p)$?
In (Curtright, Fairlie, Zachos 2014), the authors mention (Eq. (14) in this online version) the following relation, known as "Bopp shifts":
$$f(x,p)\star g(x,p)=f\left(x+\frac{i}{2}\...
1
vote
0
answers
509
views
What are the Fock-state probabilities of general Gaussian states?
A general (pure) Gaussian state has the form $\newcommand{\on}[1]{\operatorname{#1}}\newcommand{\ket}[1]{\lvert #1\rangle}\ket{\alpha,\xi}\equiv D(\alpha)S(\xi)\ket{\on{vac}}$, with $\ket{\on{vac}}$ ...
5
votes
2
answers
2k
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What is the Wigner function of a thermal state?
I am wondering how you would compute the Wigner Function of a Thermal State with
average phonon number $\bar{n}_{\mathrm{th}}$.
I know the result should be a Gaussian with variance in position $\...
1
vote
1
answer
124
views
Wigner phase space operator correspondence: how to order?
According to Gardiner-Zoller (Quantum Noise), operators acting on the density matrix can be mapped via e.g. (I'm taking Wigner space as an example, but the same holds for P and Q)
$$a\rho\...
1
vote
1
answer
347
views
Wigner map of the product of two operators
Does anyone know how to prove that for the product of two operators $\hat{A}\hat{B}$ the Weyl-Wigner correspondence reads
$$
(AB)(x,p) = A\left (x-\frac{\hbar}{2i}\frac{\partial}{\partial p}, p+\frac{\...
1
vote
1
answer
163
views
Fourier transform of cross-spectral density space matrix elements
In order to derive phase space like equation of motion (e.g. the equation of motion for the Wigner function of a single particle in one-dimension), it is an advantage to work with the Fourier ...
5
votes
1
answer
596
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Proof of "non-existence" of marginals of the Husimi $Q$-function
There are many ways to consider the Husimi ($Q$) quasi-probability distribution function, e.g. as the expectation of the density operator in a coherent state or as the Weirstrass transform of the ...