TY - JOUR
T1 - Do attractive scattering potentials concentrate particles at the origin in one, two and three dimensions? I. Potentials finite at the origin
JF - Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
JO - Proc R Soc Lond A Math Phys Sci
SP - 401
LP - 418
M3 - 10.1098/rspa.1983.0089
VL - 388
IS - 1795
AU -
Y1 - 1983/08/08
UR - http://rspa.royalsocietypublishing.org/content/388/1795/401.abstract
N2 - 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 α ≡ ρ0/ρ∞; ρ 0, ∞ (and v0, ∞) 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 (non-relativistic) classical mechanics, if the origin is accessible, then any central potential U(r) where v0 < ∞ (i. e. where U(0) > -∞) gives in 1, 2 and 3 dimensions, α1 = v∞/v0, α2 = 1, α3 = v0/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 square-well and exponential potentials are discussed as examples. Potentials which diverge at the origin are treated in the following paper.
ER -