Unifying distribution functions: some lesser known distributions

Abstract

We show that there is a way to unify distribution functions that describe simultaneously a classical signal in space and (spatial) frequency and position and momentum for a quantum system. Probably the most well known of them is the Wigner distribution function. We show how to unify functions of the Cohen class, Rihaczek's complex energy function, and Husimi and Glauber-Sudarshan distribution functions. We do this by showing how they may be obtained from ordered forms of creation and annihilation operators and by obtaining them in terms of expectation values in different eigenbases.

Wigner function and the successive measurement of position and momentum

“…As for the relation between Wigner's function and Kirkwood's quasi-distribution, we also refer the reader to Refs. [11][12][13].…”

Time evolution of a Gaussian class of quasi-distribution functions under quadratic Hamiltonian

“…α α terms, we define new integration variables (this notation is similar to the notation of references [3][4][5]; Lee [1]…”

“…The idea behind quasi-distribution functions (QDF) (e.g. [1][2][3]) is to use a tool that resembles a classical distribution function in phase space and can be used to calculate expectation values of observables. In classical mechanics in phase space, expectation values are calculated as an integral ( ) ( ) , , , A dqdpA q p F q p t = ∫∫ , (1.1) where A is the expectation value of the observable A ,…”