A family quasi-distribution function representation in phase space

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?.