Generalized Wigner Operator and Bivariate Normal Distribution in p-q Phase Space

A. K. Rajagopal, S. Teitler

Fan Hong-Yi, Wang Tong-Tong

We introduce a kind of generalized Wigner operator, whose normally ordered form can lead to the bivariate normal distribution in p-q phase space. While this bivariate normal distribution corresponds to the pure vacuum state in the generalized Wigner function phase space, it corresponds to a mixed state in the usual Wigner function phase space.

Ji‐Suo Wang, Xiang‐Guo Meng, Hong-Yi Fan

By extending the usual Wigner operator to the s-parameterized one, we find that in the process of the generalized Weyl quantization the s parameter plays the role of correlation between two quadratures Q and P . This can be exposed by comparing the normally ordered form of Ω s with the standard form of the Gaussian bivariate normal distribution of random variables in statistics. Three different expressions of Ω s and the quantization scheme with use of it are presented.

B. Lesche

J. Krüger

In order that a bivariate normal probability distribution in phase space with variances ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{p}}$ and covariance ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q},}$p may correspond to a Wigner distribution of a pure or a mixed state, it is necessary and sufficient that Heisenberg's uncertainty relation in Schr\"odinger form ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{p}}$-${\mathrm{\ensuremath{\sigma}}}_{\mathit{q},}$${\mathit{p}}^{2}$\ensuremath{\ge}${\mathrm{\ensuremath{\Elzxh}}}^{2}$/4 should be satisfied. The diagonalization of the corresponding density matrix entails a correspondence between the statistical and the physical properties of temperature-dependent oscillator states; the expansion of the density matrix into coherent states allows a physical interpretation in phase space.