Do attractive scattering potentials concentrate particles at the origin in one, two and three dimensions? II. Potentials diverging as — C/r q

Abstract

In classical mechanics (c.m.), and near the semi-classical limit h →0 of quantum mechanics (s.c.l.), the enhancement factors α ≡ ρ 0 /ρ ∞ are found for scattering by attractive central potentials U(r) ; here ρ 0,∞ (and v 0,∞ ) 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 ∞ ), α 1 = v ∞ / v 0 , α 2 = 1, α 3 = v 0 / 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/r q , 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., α 1 (c.m.) = 0, α 3 (c.m.) = ∞, and α 2 (c.m.) = (1 —½ q ) N , where N = [integer part of (1 ─½ q ) -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 α 1 (s.c.l.) = Λ 1 (q) v ∞ / u , α 2 (s.c.I) = (1 ─½ q ) -1 , α 3 (s.c.l.)= Λ 3 ( q ) u/v ∞ , where u ≡ (C/h q m 1-q ) 1/(2-q) is a generalized Bohr velocity. Here Λ 1,3 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, α 2 (s.c.l.) and α 2 (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 = ─ C exp (─r/a)/r q ) , 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/h 2 ) 1/(2- q ) which exists only if a ≫ ( h 2 / mC ) 1/(2- q ) ).