Note on Born approximation: conditions of validity
The Born approximation for the scattering amplitude f (n; n0 ) is based on
general equation:
Z
0
m
e ikn x V (x) + (x) d3 x
((1))
f (n; n0 ) =
2
2 ~
and substitution of the exact solution + (x) by the incident wave eiki x . Let
us analyze when this approximation is valid. The exact wave function obeys an
integral equation:
+
iki x
(x) = e
m
2 ~2
Z
0
eikjx x j
V (x0 )
jx x0 j
+
(x0 ) d3 x0
((2))
The question is: at what condition the integral term is small in comparison
to the incident wave? Let consider …rst the case when ka . 1, where a is a
characteristic radius of the potential. The integration in (1) and (2) is limited
0
by the potential in a volume with the linear size a, the exponent eikjx x j in this
+
0
0
region is of the order of 1 as well as
(x ), the denominator jx x j is of the
order of a. Thus, the order of the value of the integral term in (2) in this case
2
2
1 or
is ma~2jV j and the criterion of the Born approximation validity is ma~2jV j
~2
~2
jV j
ma2 . The value ma2 has meaning of kinetic energy for a particle con…ned
in a volume with the linear size a.
Let consider now the short-wave limit ka
1. Then the exponent in equation (2) changes rapidly in the volume with the linear size a and the previous
0
estimate becomes invalid. To improve it let substitute eiki x instead of + (x0 )
and multiply and divide the integral by the factor eiki x . The result reads:
Z
0
eikjx x j
V (x0 )
jx x0 j
+
(x0 ) d3 x0
eiki x
Z
eikjx
x0 j iki (x x0 )
jx
x0 j
V (x0 ) d3 x0 ((3))
0
The exponent in the integral in eq. (3) can be rewritten as eikjx x j(1 cos ) ,
where is the angle between ki and x x0 . Since ka is very large and jx x0 j
has the order of magnitude a, the oscillating exponent cancel the integrand
0
everywhere beyond a region in which
p the argument of the exponent k jx x j (1 cos )
is less or equal to , i.e. at . 1= ka. The integration over gives the factor
1= (ka), the rest of the integral contributes a factor jV j a2 . Thus, the Born
ja
jV ja
approximation works if mjV
1, where v = ~k
~2 k = ~v
m is the velocity.
1