Concerning the scaling behavior in the classical mechanics of non-reactive collisions: an analytic investigation

Abstract

The classical mechanical treatment of non-reactive scattering processes is cast into a convenient formalism for describing the changes in the internal linear momentum affected by a collision. The dependence of these changes on the initial internal linear momentum and coordinate is shown to be weak. Combining this result with the classical virial theorem leads to a classical scaling theory for internal energy changes. In its simplest form, this theory explicitly exhibits the dependence of the internal energy transfer on four factors: (1) the initial internal energy; (2) the initial internal coordinate; (3) the bound state potential; and (4) the average momentum transfer. At a given initial kinetic energy, these four quantities suffice to characterize the systematic variation of the internal energy transfer with initial internal energy and coordinate. The combination of the classical scaling theory with the quasiclassical histogram quantization method is determination of all other transition probabilities P_n→Δ, Δ = 1, 2,...., n. This includes classically forbidden processes for which the scaling theory yields an exact value of zero