15
votes
Accepted
What is wrong with Weyl-Wigner representation?
Wrong? There is nothing "wrong".
The phase-space formulation of QM, for which the Wigner-Weyl ordering prescription is a particularly popular flavor, is, indeed, completely equivalent to the ...
- 66.3k
12
votes
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Is the Moyal-Liouville equation $\frac{\partial \rho}{\partial t}= \frac{1}{i\hbar} [H\stackrel{\star}{,}\rho]$ used in applications?
"Used in anger" or "killer ap"? To my knowledge, no problem has been solved in the phase-space quantization language that was not solvable in the other two formulations/pictures (Hilbert space or path ...
- 66.3k
10
votes
What is the Wigner function of $|n\rangle\langle m|$?
It is, explicitly, in terms of associated Laguerre polynomials, (actually the Wigner transform of $|n\rangle \langle m|$, given its (idiosyncratic/Moyal) flipped notation, $H\star f_{mn}=E_{n}\,f_{...
- 66.3k
6
votes
Show that the Husimi Q function equals $Q(\alpha)=\frac 1 \pi\langle \alpha |\hat \rho|\alpha\rangle$ from its relation with the Wigner function
We take the Wigner function
$$W(\alpha)=\frac{1}{\pi^2}\int \text{e}^{\alpha \beta^*-\alpha^*\beta}\text{Tr}\left(\hat \rho \hat D(\beta) \right) \text{d}^2\beta,$$
and write the displacement ...
- 1,219
6
votes
What is wrong with Weyl-Wigner representation?
The $\star$-product is highly non-trivial except for HW. Even for $SU(2)$ this is quite complicated. And without a $\star$-product you are reduced to pictures or to the TWA, which only does so much. ...
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6
votes
General quantum operator
It is literally (mathematically) false. E.g., the observable defined by the operator $|X|$ cannot be written that way. This observable has also a direct operational definition: I measure the position ...
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6
votes
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What is the Wigner representation of $\left(\hat{x}^2+\hat{p}^2\right)^n$?
Both of your methods are correct but impossibly inefficient. The fundamental theorem of phase-space quantization is that the Wigner image of operator products is the star-product of the Wigner image ...
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5
votes
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Solving the *-genvalue equation of a free particle
Basically there is, for the real part of the Wigner function, Lemma 3 (pp 27-29) of my book, CTQMPS. This is the basic exercise any student of that structure should do, whether instructed to, or not.
...
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5
votes
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Wigner function of position eigenket?
Why should you be stuck, if you know how to take factors in the argument of δ-functions downstairs out of this distribution, and collapse, e.g., the first one inside the integral into the second?
$$W(...
- 66.3k
5
votes
Classical limit in deformation quantization
The following comments seem relevant to OP's post:
In deformation quantization an associative star product
$$\star:~~C^{\infty}(M)[[\hbar]]\times C^{\infty}(M)[[\hbar]]\to C^{\infty}(M)[[\hbar]] \...
- 213k
5
votes
Accepted
What is the Wigner function of a thermal state?
I believe the correct Wigner function for the eigenstates is half yours, so take it to be
\begin{equation}
W_n(x,p) = \frac{(-1)^n}{\hbar \pi} e^{- z/2} L_n(z ),
\end{equation}
where $z=4 H/\hbar\...
- 66.3k
5
votes
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Understanding derivation of Wigner function for the Harmonic oscillator
Product support.
Recall, crucially, f was shown to be a function of z only, f(z), so, acting on it,
$$\partial_x = \frac{\partial z}{\partial x } \partial_z \leadsto \\
\partial_x^2 = \left (\...
- 66.3k
5
votes
Accepted
How can the Wigner function of squeezed states be non-negative?
There is no contradiction because positivity of the Wigner is not enough to guarantee classicality: squeezed states are precisely examples of this. What is known for pure states (Hudson’s theorem) is ...
- 47.9k
4
votes
Fourier transform in the complex plane
You have $i\Im (\alpha^*u) = \frac{1}{2}(\alpha^*u - u^*\alpha)$ from complex calculus; remember $$z-z^* = (a+ib)-(a-ib) = 2ib.$$ Therefore, the exponent is purely imaginary. The integration element $\...
- 3,518
4
votes
Is the Moyal-Liouville equation $\frac{\partial \rho}{\partial t}= \frac{1}{i\hbar} [H\stackrel{\star}{,}\rho]$ used in applications?
The Moyal-Liouville equation is widely used in condensed matter problems, where it is at the heart of the so-called transport formalism, or kinetic theory. It is also widely used in quantum optics. In ...
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4
votes
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Classical limit in deformation quantization
It might be worth reviewing formulas such as (122,34,5; 131) of our book, but, frankly, I am not sure I understand what underlies and follows the question, and especially the classical limit ...
- 66.3k
4
votes
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Proof of "non-existence" of marginals of the Husimi $Q$-function
The stuff is all in Husimi's original 1940 paper, but you have to work at it... So I'll just illustrate the point based on our concise summary of the material in Wolfgang's outstanding treatise, eqns (...
- 66.3k
4
votes
Negative probabilities with Wigner quasi-probability distributions
I'm not sure what you are getting wrong... Your expression, indeed, should go to 1 as β goes to 0. There must be some error in your implementation.
The Wigner transform of the canonical ensemble ...
- 66.3k
4
votes
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Efficient numerical evaluation of Wigner function
This is a bit more general since it's for density matrices, and perhaps it is not the fastest way for pure states, but here goes:
Suppose you have a finite difference grid for your spatial coordinate (...
- 3,430
4
votes
Accepted
Non-commutative Fourier transform of an operator
There are very compelling, and in my opinion enlightening reasons to call the Weyl transform a noncommutative Fourier transform.
Background
Classical theories can be seen as (classical) probability ...
- 12.2k
4
votes
Bounding derivatives of the Wigner function using phase-space tails
Felipe Hernández and I ended up showing that the answer is yes for both pure and mixed states: "Rapidly Decaying Wigner Functions are Schwartz Functions" [arXiv:2103.14183].
We were able to ...
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4
votes
Formulations of the Wigner function in Quantum Field Theory (QFT in Phase Space)
There are some problems in trying to define a quasi-probability distribution in phase space for quantum fields, à la Wigner.
The foremost complication is given by the lack of a universal reference ...
- 12.2k
4
votes
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Reconciling the expression for the Wigner function involving $\langle x+\xi/2|\rho|x-\xi/2\rangle$ with the one using the characteristic function
You have jammed up your shift variables, making them do double duty; and, importantly, you have used different conventions for $\hbar$: (1) requires that it be equal to 1, while (2) to 2.
In any case,...
- 66.3k
4
votes
Classical limit of Moyal bracket in integral representation
There are no compact integral forms of the PBs, of course, unless you consider conversions of derivatives into powers of the Fourier conjugate variable, as you might be insinuating in the comments. ...
- 66.3k
3
votes
Accepted
Measurements in the phase space picture of quantum mechanics
Conceptually, little has changed for 20 years to moot this outstanding Physics Today article on the subject.
Typically, in the iconic double-slit interference experiment, wave patterns on a screen ...
- 66.3k
3
votes
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Groenewold's derivation of the star product (On the principles of elementary Quantum Mechanics)
One can see it directly, but Cosmas Zachos' suggestion "to working in multi-Fourier space so there are no derivatives present" is the most systematic & fool-proof approach. Here is a sketched ...
- 213k
3
votes
How to plot Wigner functions for a quantum state
The so-called Wigner transform turns your density matrix into the corresponding Wigner function. An explicit formula is given at "What is the Wigner function of $|n\rangle\langle m|$?"
- 31
3
votes
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Wigner image of the product of two operators
You just use the fundamental theorem of phase space quantization, formulated by Groenewold in 1946: for Wigner maps $\hat{A}\mapsto a(x,p)$ and $\hat{B}\mapsto b(x,p)$,
$$
\hat{A}\hat{B}~~~~\...
- 66.3k
3
votes
Accepted
Time Evolution of Wigner Function
Starting from the von Neumann equation:
$$i\hbar\partial \hat{\rho} / \partial t=[\hat{H}, \hat{\rho}]$$
We now take the Weyl Transform on both sides and noting that the partial derivative commutes ...
- 1,518
3
votes
Accepted
Generalization of Wigner overlap formula
You seem to be profoundly misunderstanding the fundamental isomorphism of phase-space quantum mechanics. What you call "Wigner functions" are but Weyl symbols,
$$f(x,p) = \hbar\int\!\!dy ~ e^...
- 66.3k
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