Questions tagged [wigner-transform]

The Wigner transform is the bridge between Hilbert space operators to phase-space quantities (c-numbers). Use for issues relating to the Weyl correspondence (the inverse of the Wigner transform), the Wigner function (the Wigner-transform of the density matrix) and, in general, Quantum Mechanics in phase space issues, such as the *-product, the Wigner transform of the operator multiplication operation. May also use for distributions such as the Husimi.

Filter by
Sorted by
Tagged with
3 votes
0 answers
28 views

Are Wigner functions of any Unitary operator in $B(L^{2}(\mathbb{R}))$ in $L^{2}(\mathbb{R}^{2})$?

Are Wigner functions of any Unitary operator in $B(L^{2}(\mathbb{R}))$ in $L^{\infty}(\mathbb{R}^{2})$? i.e. Let $e^{iA} \in B(L^{2}(\mathbb{R}))$. Define the Wigner function (Wigner transform) as ...
user avatar
  • 133
1 vote
1 answer
26 views

Momentum probability density from Wigner distribution

I want to prove that $|\hat{\psi}(p)|^2= \frac{1}{2\pi} \int W_\psi \mathrm{d}x $ where $W_\psi $ is the Wigner function. Starting with the definition I get ($z=-y$ and $u=x+z/2$): $$\frac{1}{2\pi}\...
user avatar
  • 369
0 votes
1 answer
35 views

Derivation of Wigner Function of cosine with phase

The Wigner Function of $x(t)$ is $$ W(t,f) = \frac{1}{2\pi}\int x\Big(t+\frac{\tau}{2}\Big)x^*\Big(t-\frac{\tau}{2}\Big) e^{-j2\pi f\tau}\;d\tau $$ I know how to get the $W(t,f)$ of $$x(t)=\cos(2\pi ...
user avatar
0 votes
0 answers
22 views

Converting the complex Wigner function to its real form in terms of the quadrature operators

I noticed something that bugged me recently, the Wigner function which is defined for one mode in the complex plane as $$W(\alpha)=\frac{1}{\pi^2}\int e^{\lambda\alpha^*-\lambda^*\alpha} \operatorname{...
user avatar
1 vote
1 answer
114 views

Construct the Density Matrix of a Gaussian State from its First and Second Moments / Wigner Function

Borrowing some description for the setup from a question I posted earlier here; Suppose we have $N$ bosonic modes (or quantum harmonic ocsillators) with the usual commutation relations. Now define the ...
user avatar
5 votes
1 answer
132 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 ...
user avatar
1 vote
1 answer
129 views

Reconciling two descriptions of the Wigner function

A classical way to define the Wigner function ($\hbar=2$) of a density operator $\rho$ is as follows for $x=(x_{1}, x_{2})^{T}$: $$W(x) = \frac{1}{4\pi} \int^{\infty}_{-\infty} d\xi \exp(\frac{-i}{2}...
user avatar
0 votes
0 answers
39 views

Jordan-Wigner transformation particle exchnage

Suppose you have a spinless fermionic system: $\hat H = -t \displaystyle\sum_i (c^+_ic_{i+1} + h.c.)$ The Jordan Wigner transformation acts as: $c^+_i = Z_0\otimes..\otimes Z_{i-1} \otimes \sigma^+\...
user avatar
  • 63
0 votes
0 answers
44 views

Distributional derivative on shift operation

I am having trouble understanding the derivation of equations 47 in the paper by Zhuang, Heinz. Here, they start from the equations $$ D_t \varepsilon(t,\mathbf{p}) = \left( \frac{q}{12} (\partial_t \...
user avatar
1 vote
2 answers
270 views

Does the integral of a Wigner function over a finite region mean anything?

I've recently been dipping my toes deeper into the so-called "Wigner function" formalism for quantum theory, and what I am curious about is this: ostensibly, the Wigner function is the ...
user avatar
4 votes
1 answer
182 views

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 ...
user avatar
  • 115
1 vote
1 answer
64 views

Wigner Transform in momentum space coefficient

I am currently reading some about the Wigner transform, and I ran into a problem. The Wigner transform (in the literature I am reading) is defined as: $$\tilde{A} = \int \exp{\big[\frac{-ipy}{\hbar}\...
user avatar
1 vote
1 answer
106 views

Calculating the Wigner transform of operators

Recently I started to study the formulation of quantum mechanics in the phase space. So I was introduced to the concept of Wigner function and Weyl transform. I learned that if F is an operator, then ...
user avatar
2 votes
1 answer
165 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 ...
user avatar
  • 239
1 vote
1 answer
63 views

Effects of non-locality in the star-product of two fields

My question regards an argument appearing on page 19 of the review: Quantum Field Theory on Non-commutative Spaces - Szabo. The Fourier integral kernel representation of the star-product of two fields ...
user avatar
0 votes
0 answers
93 views

How is it possible to find the Wigner function for spin coherent states?

I studied Wigner function distribution for Glauber coherent state and I know that by using this function we can find the probability distribution for particle's position, but How can me find and ...
user avatar
  • 1
1 vote
1 answer
112 views

Wigner functional for fermionic fields (QFT in phase space)

I'm curently studying the Wigner functional formulation of Quantum Field Theory, which is derived from the Schrödinger picture: the operators which act on the states of the Fock space are functions of ...
user avatar
  • 180
6 votes
1 answer
322 views

Formulations of the Wigner function in Quantum Field Theory (QFT in Phase Space)

I'm studying the phase space formulation of quantum field theory for my final degree project, and I have found two very different ways to construct the Wigner funtion. In the first method, a phase ...
user avatar
  • 180
0 votes
0 answers
132 views

Intuition behind Wigner function of cat state

I have recently been studying quantum optics and have encountered Wigner functions of Schroedinger cat states and am a bit confused when it comes to interpret them. Why are the Wigner functions of cat ...
user avatar
  • 11
1 vote
1 answer
75 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?
user avatar
  • 13
2 votes
1 answer
137 views

Wigner transform of a thermal density matrix (Schroedinger Hamiltonian)

I am interested in computing the Wigner transform in $R^{2n}$ of $$e^{-\beta H}$$ where $$H = \sum_{k=1}^n P_k^2 + V(x_1, \ldots,x_n)$$ I am assuming that $V$ is a polynomial bounded below and ...
user avatar
2 votes
2 answers
268 views

Wigner transform & convolution

I'm trying to understand the gradient expansion within the Keldysh formalism. In particular, I am reading "Quantum Field Theory of Non-equilibrium States" by J. Rammer, section 7.2, ...
user avatar
2 votes
2 answers
194 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 ...
user avatar
0 votes
1 answer
82 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 ...
user avatar
  • 1,110
0 votes
1 answer
26 views

Change of variables for momenta [closed]

http://www.stat.physik.uni-potsdam.de/~pikovsky/teaching/stud_seminar/Wigner_function.pdf From the Appendix in the above PDF (page 945), below equation (A3) the following expressions are given: $$ u = ...
user avatar
3 votes
0 answers
83 views

Is the Wigner function a signed measure?

I have read in Wikipedia that quasiprobability distributions in phase space quantum mechanics may fail to be $\sigma$-additive, but I don't know in which sense this is true. If I have a Wigner ...
user avatar
  • 31
0 votes
0 answers
57 views

How does the Wigner function differ from quantum distribution functions?

I understand that Wigner functions are quantum-mechanical phase-space distribution functions (quasi-distribution to be more specific). For spin-1/2 particles the Wigner function is a $4\times 4$ ...
user avatar
0 votes
1 answer
99 views

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\...
user avatar
  • 12.4k
0 votes
1 answer
98 views

Wigner Function and Spin in the Classical Limit?

This is something I got curious about. Let's say I have the Wigner function for an $n$ particle system: $$W \equiv W(x_1,\dots,x_n,;p_1,\dots,p_n) $$ Now, let's say this system obeys has spin. As far ...
user avatar
1 vote
1 answer
169 views

Generalization of Wigner overlap formula

I want to generalize the Wigner overlap formula, $Tr( F G ) = 2 \pi \int_{-\infty}^{\infty} dq \int_{-\infty}^{\infty} dq W_F(q,p) W_G(q,p)$, where $W_F(q,p)$ and $W_G(q,p)$ are the Wigner functions ...
user avatar
  • 387
0 votes
1 answer
129 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 ...
user avatar
  • 12.4k
1 vote
1 answer
116 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}\...
user avatar
  • 12.4k
1 vote
0 answers
182 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 $\ket{\on{vac}}$ ...
user avatar
  • 12.4k
0 votes
1 answer
50 views

Is there a list of computed Wigner distributions of notable states?

I'm looking for a list of explicit expressions of Wigner distributions of notable functions (e.g., Fock states, Gaussian states, thermal states, etc). Is there a paper, book, or other online source ...
4 votes
1 answer
107 views

Bounding derivatives of the Wigner function using phase-space tails

Suppose I have a Wigner function that falls off faster than any polynomial for all directions in phase space. That is, for all $a,b>0$, $$\lim_{|x|\to\infty} |x^a p^b W(x,p)| =0=\lim_{|p|\to\infty} ...
user avatar
  • 3,458
4 votes
1 answer
411 views

Time Evolution of Wigner Function

The Wigner Function is defined as: $$W(x,p,t)=\frac{1}{2\pi\hbar}\int dy \rho(x+y/2, x-y/2, t)e^{-ipy/\hbar}\tag{1}$$ Where $\rho(x, y, t)=\langle x|\hat{\rho}|y\rangle$. I am supposed to find the ...
user avatar
  • 1,238
5 votes
2 answers
540 views

What is wrong with Weyl-Wigner representation?

The Weyl-Wigner representation is a useful tool to study QM from a semiclassical, phase-space point of view. My question is simple: if this method is so close to classical mechanics, why don't we use ...
user avatar
2 votes
1 answer
126 views

Free evolution of a density matrix in position space

I have a density matrix $\rho$ in momentum representation at time $t=0$: \begin{equation} \langle p' |\rho(0) |p\rangle = \sum_{n=1}^{1000} p_n \Psi_n^*(p',0) \Psi_n(p,0) \end{equation} ...
user avatar
  • 139
3 votes
2 answers
575 views

Intuitive derivation of the Husimi Q function from the Wigner function

The Wigner function is given by $$W(\alpha)=\frac{1}{\pi^2}\int \text{e}^{\alpha \beta^*-\alpha^*\beta}\text{Tr}\left(\hat \rho \hat D(\beta) \right) \text{d}^2\beta,$$ where $\hat D(\beta)=\text e^...
user avatar
  • 1,049
1 vote
1 answer
112 views

Post measurement state after heterodyne measurement

I want to understand the phase space formulation of quantum mechanics better. Specifically, I am considering the following situation: A quantum state $\rho$ on two modes can be described by its ...
user avatar
  • 337
1 vote
1 answer
81 views

Wigner-Weyl transformation for particle in magnetic field

I consider particle in external magnetic field, ${\bf A}=(-yB,0,0)$ and find wave functions (may be up to normalization factors), $$\psi(x,y,z)=\sum_n\sum_s\int\frac{dp_xdp_z}{(2\pi)^2}f_s\left(eBy+...
user avatar
0 votes
0 answers
131 views

Wigner distribution harmonic oscillator

I am interested in the Wigner distribution of a quantum harmonic oscillator \begin{equation*} W_n(q,p) = \int_{\mathbb R} \mathrm d x e^{ipx}\;\psi_n(q-x/2)\psi_n(q+x/2) \end{equation*} with ...
user avatar
  • 365
0 votes
1 answer
45 views

Trace in correlations to compute Wigner transform

In the derivation of Wigner-transformed quantum time correlation functions, the following identity is used (in the case of a one-dimensional particle, for simplicity): \begin{align} C(t) &\equiv \...
user avatar
  • 365
0 votes
1 answer
46 views

Continuous variables: Probability of outcome in a quadrature measurement

In the phase-space formulation of QM over continuous variables, how can I determine the probability of obtaining a particular measurement outcome $m$ in the following setting. Given a quantum state $\...
user avatar
  • 337
0 votes
0 answers
101 views

Density matrix and wigner function from first and second moments

Let's say I know the first and second moments of position and momentum for all times. $\langle x\rangle $, $\langle p\rangle $,$\langle x^2\rangle $, $\langle p^2\rangle $, $\langle xp\rangle $, $\...
user avatar
  • 139
3 votes
1 answer
1k views

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 $\...
user avatar
  • 139
0 votes
1 answer
260 views

Obstruction in quantization. Weyl Ordering

What is an obstruction in quantization? I've found that obstructions object of the study of a mathematical theory, previously concerned with homotopy. The problem is that to explain what an ...
user avatar
0 votes
1 answer
297 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{...
user avatar
  • 1,350
1 vote
0 answers
81 views

Computation of Wigner Functions

The Wigner function can be computed as the Fourier transform of the Weyl-ordered characteristic function: $$ W(\alpha) = \frac{1}{\pi^2} \int e^{\lambda^* \alpha - \lambda \alpha^*} C_W(\lambda) d^2\...
user avatar
  • 11
1 vote
1 answer
74 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\...
user avatar
  • 1,350