All Questions
17
questions
2
votes
2
answers
111
views
Dirac delta of operators multiplying matrix element
In playing around with the Wigner-Weyl correspondence, I found myself needing to perform an integral of exponential operators, which I am confused about. TLDR: help to evaluate $$\int d{x}dy\ \delta(x\...
- 173
2
votes
0
answers
89
views
Fourier Transform of $s$-ordered Characteristic Function
In the book, "Quantum Continuous Variables (A Primer of Theoretical Methods)" by Alessio Serafini, on page 70, he defines an $s$-ordered characteristic function to be:
$$
\chi_s(\alpha)=\...
3
votes
0
answers
156
views
Is there a canonical Taylor expansion for operators in terms of $X$ and $P$?
Consider the algebra of operators acting on wavefunctions ($L^2(\mathbb{R})$) generated by $X$ and $P = -i\hbar (\partial/\partial x)$. For some operator $A$ in this algebra, or possibly in a ...
- 3,684
1
vote
1
answer
122
views
Obtaining the star product from the Weyl quantisation of the product of two symbols
It can be shown (Groenewold 1946) that the Weyl quantisation of the product of two Weyl symbols is given by
$$ [A(\textbf{r})B(\textbf{r})]_{w}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{4}}e^{i\...
- 941
6
votes
1
answer
210
views
What is the Wigner representation of $\left(\hat{x}^2+\hat{p}^2\right)^n$?
I would like to calculate the Wigner representation of the operators $\left(\hat{x}^2+\hat{p}^2\right)^3$ and $\left(\hat{x}^2+\hat{p}^2\right)^4$. I know at least two ways to do it, but both rely on ...
- 73
1
vote
1
answer
110
views
Multiplicative inverse of Weyl symbol and invertibility of operator
If the Weyl symbol $A_W$ of an operator $\hat{A}$ has a multiplicative inverse at every point of the phase-space, can I conclude that $\hat{A}$ is invertible?
- 13
2
votes
2
answers
269
views
General quantum operator
Is it true that any operator can be expressed as
(e.g. in one dimension)
$$\hat{A}=\sum_{n=0, \, m=0}^{\infty}c_{n,m}\hat{x}^n\hat{p}^m \, ?$$
It seems true because any classical observable is a ...
- 33
1
vote
1
answer
192
views
Wigner-Weyl transform for a function of coordinates only
I am reading this paper by Tatarskii, which serves as an introduction to the Wigner representation of quantum mechanics.
There is a step in the paper involving the Weyl transform that does not seem ...
- 1,242
0
votes
1
answer
273
views
How does the Weyl transform take into account which quasiprobability distribution was used?
I'm trying to get a better understanding of the Weyl correspondence which, as described e.g. on Wikipedia, gives "an invertible mapping between functions in the quantum phase space formulation ...
- 14.8k
1
vote
1
answer
318
views
Why does the star product satisfy the "Bopp Shift relations": $f(x,p)\star g(x,p)=f(x+\frac{i}{2}\partial_p,p-\frac{i}{2}\partial_x) g(x,p)$?
In (Curtright, Fairlie, Zachos 2014), the authors mention (Eq. (14) in this online version) the following relation, known as "Bopp shifts":
$$f(x,p)\star g(x,p)=f\left(x+\frac{i}{2}\...
- 14.8k
0
votes
1
answer
440
views
Wigner-Weyl ordering in exponential
If the particle number is $\hat{a}^\dagger\hat{a}\leftrightarrow|\alpha_w|^2-1/2 $, it can be mapped on the Wigner fields by assuming symmetric ordering:$|\alpha_w|^2\leftrightarrow\hat{a}^\dagger\hat{...
- 1,620
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
$$
(AB)(x,p) = A\left (x-\frac{\hbar}{2i}\frac{\partial}{\partial p}, p+\frac{\...
- 793
4
votes
1
answer
441
views
Non-commutative Fourier transform of an operator
Wigner-Weyl transform relates an operator to its distribution function in phase space through an operator Fourier transform which is said to be non-commutative.
$$
\hat{\rho} \xrightarrow[non-comm]{...
- 199
1
vote
2
answers
172
views
Basic confusion with quantum mechanical operators
Given a classical observable, $a(x,p),$ Weyl quantization gives the correspondent QM observable as:
$$\langle x | \hat{A} | \phi \rangle=\hbar^{-3}\int \int a \left(\frac{x+y}{2},p\right)\phi(y) e^{2\...
- 1,262
1
vote
4
answers
1k
views
Definition of symmetrically ordered operator for multi-mode case?
As I know, Wigner function is useful for evaluating the expectation value of an operator. But first you have to write it in a symmetrically ordered form. For example:
$$a^\dagger a = \frac{a^\dagger ...
- 121