We show that a (generalized) function f has a nonnegative Wigner distribution $W(f,f)$ if and only if f is a Gauss function (possibly degenerate). We prove, more generally, that the convolution of $W(f,f)$ with certain Gauss functions is nonnegative if and only if f is of the special type mentioned. As a consequence we have that the only (generalized) functions whose Wigner distributions are concentrated on a curve of a particular type are delta functions or exponentials $\exp ( - \pi \alpha t^2 + 2\pi \beta t + \gamma )$ with $\alpha $, $\beta $, $\gamma $ complex, $\operatorname{Re} \alpha = 0$ . The main tool used is Moyal’s formula for the Wigner distribution together with Bargmann’s integral transform.We show that a (generalized) function f has a nonnegative Wigner distribution $W(f,f)$ if and only if f is a Gauss function (possibly degenerate). We prove, more generally, that the convolution of $W(f,f)$ with certain Gauss functions is nonnegative if and only if f is of the special type mentioned. As a consequence we have that the only (generalized) functions whose Wigner distributions are concentrated on a curve of a particular type are delta functions or exponentials $\exp ( - \pi \alpha t^2 + 2\pi \beta t + \gamma )$ with $\alpha $, $\beta $, $\gamma $ complex, $\operatorname{Re} \alpha = 0$. The main tool used is Moyal’s formula for the Wigner distribution together with Bargmann’s integral transform.

SIAM Journal on Mathematical AnalysisJan 1, 198429 citations

The Wigner distribution function

GERALD J. IAFRATE, H. L. Grubin, David K. Ferry

Physics Letters AJan 1, 198227 citations

Wigner distributions for n arbitrary operators

René Schwonnek, Reinhard F. Werner

We show that for any collection of hermitian operators A1...An , and any quantum state there is a unique joint distribution on R^n, with the property that the marginals of all linear combinations of the operators coincide with their quantum counterpart. We call it the Wigner distribution, because for position and momentum this property defines the standard Wigner function. In this note we discuss the application to finite dimensional systems, establish many basic properties and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behaviour under symmetry groups, and informational completeness.We show that for any collection of hermitian operators A1...An , and any quantum state there is a unique joint distribution on R^n, with the property that the marginals of all linear combinations of the operators coincide with their quantum counterpart. We call it the Wigner distribution, because for position and momentum this property defines the standard Wigner function. In this note we discuss the application to finite dimensional systems, establish many basic properties and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behaviour under symmetry groups, and informational completeness.

Feb 22, 20182 citations

On derivation of Wigner distribution function

Siamak Khademi

Wigner distribution function has much importance in quantum statistical mechanics. It finds applications in various disciplines of physics including condense matter, quantum optics, to name but a few. Wigner distribution function is introduced by E. Wigner in 1932. However, there is no analytical derivation of Wigner distribution function in the literatures, to date. In this paper, a simple analytical derivation of Wigner distribution function is presented. Our derivation is based on two assumptions, these are A) by taking the integral of Wigner distribution function, with respect to configuration space, the momentum space distribution function is obtained B) WDF is real. Similarly, and in addition to Wigner distribution function, the distribution function of Sobouti-Nasiri, which is imaginary, is also derived.Wigner distribution function has much importance in quantum statistical mechanics. It finds applications in various disciplines of physics including condense matter, quantum optics, to name but a few. Wigner distribution function is introduced by E. Wigner in 1932. However, there is no analytical derivation of Wigner distribution function in the literatures, to date. In this paper, a simple analytical derivation of Wigner distribution function is presented. Our derivation is based on two assumptions, these are A) by taking the integral of Wigner distribution function, with respect to configuration space, the momentum space distribution function is obtained B) WDF is real. Similarly, and in addition to Wigner distribution function, the distribution function of Sobouti-Nasiri, which is imaginary, is also derived.

arXiv:quant-ph/0608046Aug 4, 20060 citations

Some remarks on the wigner distribution

Felix Alexandrovich Berezin

Theoretical and Mathematical PhysicsDec 1, 197325 citations

We show that a (generalized) function f has a nonnegative Wigner distribution

W(f,f)

if and only if f is a Gauss function (possibly degenerate). We prove, more generally, that the convolution of

W(f,f)

with certain Gauss functions is nonnegative if and only if f is of the special type mentioned. As a consequence we have that the only (generalized) functions whose Wigner distributions are concentrated on a curve of a particular type are delta functions or exponentials

\exp ( - \pi \alpha t^2 + 2\pi \beta t + \gamma )

with $\alpha $, $\beta $, $\gamma $ complex,

\operatorname{Re} \alpha = 0

. The main tool used is Moyal’s formula for the Wigner distribution together with Bargmann’s integral transform.