All Questions
18
questions
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29
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Can we get quasiprobability distributions other than $P,Q,W$ from generalised characteristic functions?
It's a standard result that the three well-known quasiprobability distributions can all be expressed in terms of the "$s$-ordered characteristic functions" as
$$
W(\alpha) = \int\frac{d^2\...
- 14.8k
0
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0
answers
31
views
Unitary evolution of composite system in phase space
Given a quantum state $\rho$ in a Hilbert space $\mathcal H_S$, we can always write it in terms of the displacement operator $D_\alpha$ using the characteristic function $\chi_\rho(\alpha)=\text{Tr}[\...
- 1
0
votes
0
answers
62
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The Lorentz-non-covariance of the Wigner Function
What does the fact that the Wigner function is not Lorentz-covariant imply?
My analysis so far led me to the (probably naive) understanding that there really is nothing special about it, just that it ...
- 692
1
vote
1
answer
217
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What is the most general wave function of a minimum uncertainty (Gaussian) state in quantum mechanics?
For some state $|\psi\rangle$ it is possible to recover the uncertainty principle using the fact that $$\left|(\hat{\sigma_{Q}}-i\lambda\hat{\sigma_{P}})|\psi\rangle\right|^{2}\geq0,$$where$$\hat{\...
- 941
0
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0
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56
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Husimi $Q$-function of Infinite Square well
Eigen-Wavefunction of infinite square well is
$$\psi(x)=\sqrt{2/l}\sin(n\pi x/l).$$ I want to write Husimi $Q$ function for infinite square well. General expression of Q function is $$Q=(1/2)\pi \...
- 781
5
votes
1
answer
449
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How can the Wigner function of squeezed states be non-negative?
It is always said that when the Wigner function of quantum states takes a negative value, then it is a clear signature of non-classicality of this particular state. It is also well-known that the ...
- 135
3
votes
2
answers
562
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What does it mean for $P$ functions to be "more singular than a delta"?
Consider the Glauber-Sudarshan $P$ representation of a state $\rho$, which is the function $\mathbb C\ni\alpha\mapsto P_\rho(\alpha)\in\mathbb R$ such that
$$\rho = \int d^2\alpha \, P_\rho(\alpha) |\...
- 14.8k
2
votes
1
answer
444
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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 ...
- 261
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
$$f_\psi(x,p) = \frac{1}{2\pi} \int dy e^{-iyp} \psi(x+y/2)\psi^*(x-y/2)
\equiv \frac{1}{2\...
- 14.8k
0
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1
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273
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How does the Weyl transform take into account which quasiprobability distribution was used?
I'm trying to get a better understanding of the Weyl correspondence which, as described e.g. on Wikipedia, gives "an invertible mapping between functions in the quantum phase space formulation ...
- 14.8k
1
vote
1
answer
318
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Why does the star product satisfy the "Bopp Shift relations": $f(x,p)\star g(x,p)=f(x+\frac{i}{2}\partial_p,p-\frac{i}{2}\partial_x) g(x,p)$?
In (Curtright, Fairlie, Zachos 2014), the authors mention (Eq. (14) in this online version) the following relation, known as "Bopp shifts":
$$f(x,p)\star g(x,p)=f\left(x+\frac{i}{2}\...
- 14.8k
0
votes
1
answer
79
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Phase space formulation: "Representation" vs "function" vs "quasi-probability distribution"
In the phase space formulation, the terms "representation", "function, and "quasi-probability distribution" (as in Glauber–Sudarshan P representation, $P$-function) seem to be used interchangeably.
I ...
- 387
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\...
- 1,620
0
votes
1
answer
476
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Plotting quadrature uncertainties in phase space
In most books like in the picture given below, the uncertainties regarding quantum states like coherent and squeezed states are represented in phase space plot by some area enclosed within a circle or ...
- 119
4
votes
1
answer
916
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Why exactly is the Husimi-Q distribution not a real probability distribution?
From this question I understood that the uncertainty principle is causing a problem because two points $x,p$ and $x',p'$ in phase space can be confused. Why exactly is this a problem? I don't grasp ...
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