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Communications in Theoretical Physics

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

Fan Hong-Yi, Wang Tong-Tong

Published Dec 1, 2008• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/50/6/10

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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.
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},}$p2{\mathit{p}}^{2}p2\ensuremath{\ge}${\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.
Combining quantum mechanics and the normal distribution in statistics we study the coherent state from the point of view of statistics and by using the integration method within ordered product of operators. We find that the pure coherent state |z >z| exhibits a bivariate normal distribution of randon variables in (q,p) phase space, z=(q+ip)/√2, with a real k-parameter which is related to the quantization scheme, and the correlation coefficient is ik. For k=±1, |z >z| respectively is arranged as P-ordering (all P stand on the left of all Q) and Q-ordering (all Q stand on the left of all P), while in the case of k=0, |z >z| is arranged as the Weyl-ordering. In the cases of P-ordering and Q-ordering, in the classical correspondence function of |z >z||z=(q+ip)/√2 the bivariates (q,p) are correlated, only in the case of Weyl correspondece, (q,p) are independent. In other words, the Weyl ordering of operators is liable to decouple the correlation in bivariates.
The modulus squared of a class of wavefunctions defined on phase space is used to define a generalized family of Q or Husimi functions. A parameter λ specifies orderings in a mapping from the operator |ψ⟩⟨σ| to the corresponding phase space wavefunction, where σ is a given fiducial vector. The choice λ = 0 specifies the Weyl mapping and the Q -function so obtained is the usual one when |σ⟩ is the vacuum state. More generally, any choice of λ in the range (−1, 1) corresponds to orderings varying between standard and anti-standard. For all such orderings the generalized Q -functions are non-negative by construction. They are shown to be proportional to the expectation of the system state with respect to a generalized displaced squeezed state which depends on λ and position ( p , q ) in phase space. Thus, when a system has been prepared in the state , a generalized Q -function is proportional to the probability of finding it in the generalized squeezed state. Any such Q -function can also be written as the smoothing of the Wigner function for the system state by convolution with the Wigner function for the generalized squeezed state.