A quantum-mechanical Boltzmann equation for one-particle Γs-distribution functions

A unified derivation of the classical and quantum Boltzmann equation for the one-particle Γ and Γs distribution functions, respectively, is presented. The starting point is a Boltzmann-like equation for density operators in B2(Γ). In the context of classical mechanics this equation turns into the Boltzmann equation once it is supplanted by the assumption of small spatial inhomogeneity of the Γ-distribution function. Under the same assumption, and after replacing certain δ-like functions with δ-functions, the same procedure yields (for a certain range of values for the instrument characteristic s) a Boltzmann equation for the Γ distribution function. In this last equation a point-target differential cross-section σs appears. The relation of σs to the conventional quantum-mechanical differential cross-section σ is discussed.