Do Attractive Scattering Potentials Concentrate Particles at the Origin in One, Two and Three Dimensions? I. Potentials Finite at the Origin
Abstract
Paradoxically, in beta decay, for instance, the finalstate Coulomb forces pulling the electron inwards accelerate the emission. Quantum mechanics (q.m.) makes the rate proportional to α equiv ρ_0/ρ_∞ ρ_{0,∞} (and v_{0, ∞}) are the particle densities (and speeds) at r = 0 and far upstream in the scattering state which describes the electron. Hence, as regards the effects of finalstate interactions, one must base one's physical intuition on this ratio α. It is shown that according to (nonrelativistic) classical mechanics, if the origin is accessible, then any central potential U(r) where ν_0 < ∞ (i.e. where U(0) > ∞) gives in 1, 2 and 3 dimensions, α_1 = v_∞/v_0, α_2 = 1, α_3 = v_0/v_∞ the remaining course of U(r) is irrelevant to α. The same results hold also in q.m. in the semiclassical regime, i.e. in the W.K.B. approximation which for such potentials becomes valid at high wavenumbers; in 2D it needs rather careful formulation, and in 3D one must avoid the Langer modification. (The W.K.B. results apply even if dU/dr diverges at r = 0, provided U(0) remains finite; these cases are covered by a simple extension of the argument.) The squarewell and exponential potentials are discussed as examples. Potentials which diverge at the origin are treated in the following paper.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 August 1983
 DOI:
 10.1098/rspa.1983.0089
 Bibcode:
 1983RSPSA.388..401B