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Let us test out this scheme using a particularly simple example. Consider
scattering by a hard sphere, for which the potential is infinite
for
, and zero for
. It follows that
is
zero in the region
, which implies that
for all
.
Thus,

(995) 
for all
. It follows from Equation (988) that

(996) 
Consider the
partial wave, which is usually referred to as the
wave.
Equation (996) yields

(997) 
where use has been made of Equations (962)(963). It follows that

(998) 
The
wave radial wave function
is

(999) 
The corresponding radial wavefunction for the incident wave
takes the form

(1000) 
It is clear that the actual
radial wavefunction is similar to the
incident
wavefunction, except that it is phaseshifted by
.
Let us consider the low and high energy asymptotic limits of
.
Low energy corresponds to
. In this limit, the spherical Bessel functions
and Neumann functions reduce to:
where
. It follows that

(1003) 
It is clear that we can neglect
, with
, with respect to
. In other words, at low energy only
wave scattering
(i.e., spherically symmetric scattering) is important. It follows
from Equations (938), (979), and (998) that

(1004) 
for
. Note that the total crosssection

(1005) 
is four times the geometric crosssection
(i.e., the crosssection for classical particles bouncing off a
hard sphere of radius
).
However,
low energy scattering implies relatively long wavelengths, so we do not
expect to obtain the classical result in this limit.
Consider the high energy limit
. At high energies, all partial
waves up to
contribute significantly to
the scattering crosssection. It follows from Equation (981) that

(1006) 
With so many
values contributing, it is legitimate to replace
by its average value
. Thus,

(1007) 
This is twice the classical result, which is somewhat surprizing,
because we might expect to obtain the classical result in the short
wavelength limit. For hard sphere scattering, incident waves with
impact parameters less than
must be deflected. However, in order to
produce a ``shadow'' behind the sphere there must be scattering
in the forward direction (recall the optical theorem) to produce
destructive interference with the incident plane wave. In fact, the
interference is not completely destructive, and the shadow has a bright
spot in the forward direction. The effective crosssection associated with
this bright spot is
which, when combined with the
crosssection for classical reflection,
, gives the actual
crosssection of
.
Next: Low Energy Scattering
Up: Scattering Theory
Previous: Determination of PhaseShifts
Richard Fitzpatrick
20130408