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Consider timeindependent, energy conserving scattering in which the Hamiltonian
of the system is written

(1250) 
where

(1251) 
is the Hamiltonian of a free particle of mass , and
the scattering potential. This potential is assumed to only be
nonzero in a fairly localized region close to the origin. Let

(1252) 
represent an incident beam of particles, of number density , and
velocity
. Of course,

(1253) 
where
is the particle energy.
Schrödinger's equation for the scattering problem is

(1254) 
subject to the boundary condition
as
.
The above equation can be rearranged to give

(1255) 
Now,

(1256) 
is known as the Helmholtz equation. The solution to this
equation is wellknown:^{}

(1257) 
Here, is any solution of
.
Hence, Eq. (1255) can be inverted, subject to the boundary condition
as
, to give

(1258) 
Let us calculate the value of the wavefunction well outside the
scattering region. Now, if then

(1259) 
to firstorder in , where is a unit vector
which points from the scattering region to the observation point.
It is helpful to define
. This is the wavevector
for particles with the same energy as the incoming particles (i.e.,
) which propagate from the scattering region to the observation
point. Equation (1258) reduces to

(1260) 
where

(1261) 
The first term on the righthand side of Eq. (1260) represents the incident particle
beam, whereas the second term represents an outgoing spherical wave
of scattered particles.
The differential scattering crosssection
is
defined as the number of particles per unit time scattered into
an element of solid angle , divided by the incident
particle flux. From Sect. 7.2, the probability flux (i.e., the
particle flux) associated with a wavefunction is

(1262) 
Thus, the particle flux associated with the incident wavefunction is

(1263) 
where
is the velocity of the incident
particles. Likewise, the particle flux associated with the scattered
wavefunction is

(1264) 
where
is the velocity of the scattered particles.
Now,

(1265) 
which yields

(1266) 
Thus,
gives the differential crosssection
for particles with incident velocity
to be scattered such that their final velocities are directed into a range of
solid angles about
. Note that the scattering
conserves energy, so that
and
.
Next: Born Approximation
Up: Scattering Theory
Previous: Introduction
Richard Fitzpatrick
20100720