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
)
).
Plato’s cube and the natural geometry of fragmentation
