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Acta Physica Sinica

A family quasi-distribution function representation in phase space

Yuan Tong-quan, 北京理工大学理学院物理系,北京 100081

Published Jan 1, 2006• 1 citation

Abstract

A family quasi-distribution function representation is defined in this article. This family quasi-distribution function representation is constructed from the family wave function of the Schrdinger equation in phase space in which the definitions of the operators are α=αp-i?q and α=(1-α)q+i?p. Two interesting relationships are found. The first one is that the family wave function of the Schrdinger equation in phase space is a “Window” Fourier transform of the function φ(λ)exp［i(1-α)qp/?］. The second one is that different choices of the window functions result in different distribution functions. When the window function g(λ) is a Gaussian function the distribution function is the Husimi-like distribution function. When the window function g(λ) is a plural function representing an ellipse, the quasi-distribution function is the Ellipse distribution function; and finally when the plural function g(λ) is supplemented with the additional condition α=0, it will result in the standard ordering, anti-standard ordering distribution function and Wigner function. In this case g(λ) is a function depieting a rectangular window with width λ and height 1/12π?.

DOI: 10.7498/aps.55.5014

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A family quasi-distribution function representation is defined in this article. This family quasi-distribution function representation is constructed from the family wave function of the Schrdinger equation in phase space in which the definitions of the operators are α=αp-i?q and α=(1-α)q+i?p. Two interesting relationships are found. The first one is that the family wave function of the Schrdinger equation in phase space is a “Window” Fourier transform of the function φ(λ)exp［i(1-α)qp/?］. The second one is that different choices of the window functions result in different distribution functions. When the window function g(λ) is a Gaussian function the distribution function is the Husimi-like distribution function. When the window function g(λ) is a plural function representing an ellipse, the quasi-distribution function is the Ellipse distribution function; and finally when the plural function g(λ) is supplemented with the additional condition α=0, it will result in the standard ordering, anti-standard ordering distribution function and Wigner function. In this case g(λ) is a function depieting a rectangular window with width λ and height 1/12π?.

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.

Wigner phase space quasi-probability distribution function is a Fourier transform related to a given quantum mechanical wave function. It is shown that for the wave functions of type

ψ(q)=e^{-aq^2}ϕ(q)

, the Wigner function can be defined in terms of differential operators acting on a given function, independently from the integral formula which appears in the standard definition. Gaussian wave packet, harmonic and singular oscillators are given as the examples.

This paper discusses the relationship between the Wigner function, along with other related quasiprobability distribution functions, and the probability density distribution function constructed from the wave function of the Schr\"odinger equation in quantum phase space, as formulated by Torres-Vega and Frederick (TF). At the same time, a general approach in solving the wave function of the Schr\"odinger equation of TF quantum phase space theory is proposed. The relationship of the wave functions between the TF quantum phase space representation and the coordinate or momentum representation is thus revealed.

The Weyl-Wigner transformation enables one to construct a representation of quantum motion equations using functions in phase space. The states of the system are determined bv the quasi-probability distribution in phase space and the motion is described by an orthogonal integral operator. This formalism is employed in the study of the classical meaning of the discrete symmetry groups, in the problem of defining the current probability and in the proving of the expression of cross section from the S-matrix. (GE)

Quasiprobability distributions emerging in the stochastic wave-function method of Carusotto et al. [Phys. Rev. A 63, 023606 (2001)] are investigated. We show that there are actually two types of quasiprobabilities. The first one, the ``diagonal Hartree-Fock state projection'' representation, is useful in representing the initial conditions for stochastic simulation in the most compact form. It defines antinormally ordered expansion of the density operator and normally ordered mapping of the observables to be averaged. We completely characterize the equivalence classes of this phase-space representation. The second quasiprobability distribution, the ``nondiagonal Hartree-Fock state projection'' representation, extends the first one in order to achieve stochastic representation of the quantum dynamics. We demonstrate how the differential identities of the stochastic ansatz generate the automorphisms of this phase-space representation. These automorphisms turn the stochastic representation into a gauge theory. The gauge transformations of the quasiprobability master equation are described. In particular, it is the analyticity of the stochastic ansatz that allows one to transform the master equation into the genuine Fokker-Planck equation. We demonstrate how the different variants of the stochastic wave-function method can be constructed, first by choosing a certain optimality criteria or constraints, and then by satisfying these criteria with a suitable choice of gauge. The problem of boundary terms is considered. It is demonstrated that the simple scheme with Fock states of Carusotto et al. is not subjected to this problem.