## Abstract

Paradoxically, in beta decay, for instance, the final-state Coulomb forces pulling the electron inwards accelerate the emission. Quantum mechanics (q.m.) makes the rate proportional to $\alpha \equiv \rho_0/\rho_\infty; \rho_{0,\infty}$ (and $v_{0, \infty}$) 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 final-state interactions, one must base one's physical intuition on this ratio $\alpha$. It is shown that according to (non-relativistic) classical mechanics, if the origin is accessible, then any central potential $U(r)$ where $\nu_0 < \infty$ (i.e. where $U(0) > -\infty$) gives in 1, 2 and 3 dimensions, $\alpha_1 = v_\infty/v_0, \alpha_2 = 1, \alpha_3 = v_0/v_\infty$; the remaining course of $U(r)$ is irrelevant to $\alpha$. 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 wave-numbers; in 2D it needs rather careful formulation, and in 3D one must avoid the Langer modification. (The W.K.B. results apply even if d$U$/d$r$ diverges at $r = 0$, provided U(0) remains finite; these cases are covered by a simple extension of the argument.) The square-well and exponential potentials are discussed as examples. Potentials which diverge at the origin are treated in the following paper.