## 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 ≡ |*), and 0 <

*x*|*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*is a generalized Bohr velocity. Here Λ

^{q}m^{1-q})^{1/(2-q)}_{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*, and if, near the s.c.l.,

^{q})*v*is regarded as a function not of

_{∞}/*u**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)}).

## Footnotes

This text was harvested from a scanned image of the original document using optical character recognition (OCR) software. As such, it may contain errors. Please contact the Royal Society if you find an error you would like to see corrected. Mathematical notations produced through Infty OCR.

- Received January 31, 1983.

- Scanned images copyright © 2017, Royal Society

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