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Equation (923) is not particularly useful, as it stands, because the
quantity
depends on the unknown ket .
Recall that
is the solution of the integral equation

(924) 
where is the wavefunction of the incident state.
According to the above equation the total wavefunction is a superposition
of the incident wavefunction and lots of sphericalwaves emitted from
the scattering region. The strength of the sphericalwave emitted at
a given point is proportional to the local value of the scattering
potential, , as well as the local value of the wavefunction, .
Suppose that the scattering is not particularly strong. In this case, it is
reasonable to suppose that the total wavefunction, , does
not differ substantially from the incident wavefunction, .
Thus, we can obtain an expression for
by making
the substitution

(925) 
This is called the Born approximation.
The Born approximation yields

(926) 
Thus,
is proportional to the Fourier transform
of the scattering potential with respect to the wavevector
.
For a spherically symmetric potential,

(927) 
giving

(928) 
Note that
is just a function of for a
spherically symmetric potential.
It is easily demonstrated that

(929) 
where is the angle subtended between the vectors
and . In other words, is the angle of
scattering. Recall that the
vectors and have the same length by energy conservation.
Consider scattering by a Yukawa potential

(930) 
where is a constant and measures the ``range'' of the
potential. It follows from Eq. (928) that

(931) 
since

(932) 
Thus, in the Born approximation, the differential crosssection
for scattering by a Yukawa potential is

(933) 
given that

(934) 
The Yukawa potential reduces to the familiar Coulomb potential as
, provided that
. In this limit the Born differential crosssection becomes

(935) 
Recall that is equivalent to , so the above
equation can be rewritten

(936) 
where is the kinetic energy of the incident particles.
Equation (936) is the classical Rutherford scattering crosssection formula.
The Born approximation is valid provided that is
not too different from in the scattering region.
It follows, from Eq. (907), that the condition for
in the vicinity of is

(937) 
Consider the special case of the Yukawa potential. At low energies,
(i.e., ) we can replace
by unity,
giving

(938) 
as the condition for the validity of the Born approximation.
The condition for the Yukawa potential to develop a bound state
is

(939) 
where is negative. Thus, if the potential is strong enough to
form a bound state then the Born approximation is likely to break
down. In the high limit, Eq. (937) yields

(940) 
This inequality becomes progressively easier to satisfy as increases,
implying that the Born approximation is more accurate at high
incident particle energies.
Next: Partial waves
Up: Scattering theory
Previous: The LipmannSchwinger equation
Richard Fitzpatrick
20060216