# 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.

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### 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 ...
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### 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 ...
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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+... 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 ... 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 \... 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 \... 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 , \... 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 \... 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 ... 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{... 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\...
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\...