Density Operators and Quasiprobability Distributions

摘要: The problem of expanding a density operator $\ensuremath{\rho}$ in forms that simplify the evaluation important classes quantum-mechanical expectation values is studied. weight function $P(\ensuremath{\alpha})$ $P$ representation, Wigner distribution $W(\ensuremath{\alpha})$, and $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, where $|\ensuremath{\alpha}〉$ coherent state, are discussed from unified point view. Each these quasiprobability distributions examined as value Hermitian operator, an integral representation for associated with by one operator-function correspondences defined preceding paper. shown to be all whose eigenvalues infinite. existence infinitely differentiable found equivalent well-defined antinormally ordered series expansion powers annihilation creation operators $a$ ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$. $W(\ensuremath{\alpha})$ continuous, uniformly bounded, square-integrable symmetrically power-series operator. which differentiable, corresponds normally form Its use involve singularities closely related those occur representation. A parametrized introduced $W(\ensuremath{\alpha},s)$ may identified $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$ when order parameter $s$ assumes $s=+1, 0, \ensuremath{-}1$, respectively. analog $\ensuremath{\delta}$ This trace class $\mathrm{Res}l0$, has bounded $\mathrm{Res}=0$, infinite $s=1$. Marked changes properties exhibited varied continuously $s=\ensuremath{-}1$, corresponding $s=+1$, $P(\ensuremath{\alpha})$. Methods constructing functions using them compute presented illustrated several examples. One examples leads physical characterization appropriate.