Do attractive scattering potentials concentrate particles at the origin in one, two and three dimensions? II. Potentials diverging as — C/rq
G. Barton
Abstract In classical mechanics (c.m.), and near the semi-classical limit h→0 of quantum mechanics (s.c.l.), the enhancement factors α ≡ ρ0/ρ∞ are found for scattering by attractive central potentials U(r); here ρ0,∞ (and v0,∞) are the particle densities (and speeds) at the origin and far upstream in the incident beam. For finite potentials (U(0) > — ∞), and when there are no turning points, the preceding paper found both in c.m., and near the s.c.l. (which then covers high v∞ ), α1 = v∞/v0, α2 = 1, α3 = v0/v∞ respectively in one dimension (1D), 2D and 3D. The argument is now extended to potentials (still without turning points), where U(r→0) ~ ─ C/rq, with 0 < q < 1 in ID (where r ≡ |x|), and 0 < q < 2 in 2D and 3D, since only for such q can classical trajectories and quantum wavefunctions be defined unambiguously. In c.m., α1(c.m.) = 0, α3(c.m.) = ∞, and α2(c.m.) = (1 —½q) N, where N = [integer part of (1 ─½q)-1]is the number of trajectories through any point (r, θ) in the limit r → 0. All features of U(r) other than q are irrelevant. Near the s.c.l. (which now covers low v∞) a somewhat delicate analysis is needed, matching exact zero-energy solutions at small r to the ordinary W.K.B. approximation at large r; for small v∞/u it yields the leading terms α1(s.c.l.) = Λ1(q)v∞/u, α2(s.c.I) = (1 ─½q)-1, α3(s.c.l.)= Λ3(q)u/v∞, where u ≡ (C/hqm1-q)1/(2-q) is a generalized Bohr velocity. Here Λ1,3 are functions of q alone, given in the text; as q →0 the α (s.c.l.) agree with the α quoted above for finite potentials. Even in the limit h = 0, α2 (s.c.l.) and α2 (c.m.) differ. This paradox in 2D is interpreted loosely in terms of quantal interference between the amplitudes corresponding to the N classical trajectories. The Coulomb potential ─ C/r is used as an analytically soluble example in 2D as well as in 3D. Finally, if U(r) away from the origin depends on some intrinsic range parameter α(e.g. U = ─ Cexp (─r/a)/rq), and if, near the s.c.l., v∞/u is regarded as a function not of h but more realistically of v∞, then the expressions α (s.c.l.) above apply only in an intermediate range 1/a ≪ mv∞/h ≪ (mC/h2)1/(2-q) which exists only if a ≫ (h2/mC)1/(2-q)).
Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Aug 8, 1983•2 citations