1. Theory of Scattering
Lecture 2

2. Green’s Function: The Green function will be used in Born
approximation of scattering amplitude and therefore
first we try to understand a little about Green function.
The Green’s function of an operator L is defined by following
relation,
(1)
Once one is able to find the Green function, then it can be
shown that the solution of inhomogeneous equation of the
form
(2)
is given by
(3)

3. Integral form of Schrodinger Eq. and determination
of scattering amplitude:
As discussed earlier in the scattering problem we want to
find the solution of Schrodinger Eq.
-----(1)
Above Eq. can be written in the form
-----(2)
where, k2 =

4. The complete solution of Eq. (2) will be some of two part.
First will be the general solution of homogenous Eq.
(3)
in Eq. (3) is incident plane wave.
And 2nd part will be the particular solution of Eq. (2). The
Particular solution of inhomogeneous Eq. (2) can be
written using the Green function (Read Green’s function slide).
This we write the following solution of Eq. (2),
------(4)

5. In Eq. (4) (5)
Also the Green’s function in Eq. (4) corresponds to
the operator The Green’s function corresponding
to this operator is obtained by solving the Eq.
(6)
and is given by
-----(7)
Exercise: Find Eq. (7) by Solving Eq. (6).

6. Using Eq. (7) in Eq (4), we get
(8)
Above Eq. is integral form of Schrodinger equation.
Note that still our desire of finding the solution of Eq. (2)
is not fulfilled. In Eq. (8), the unknown is also present
on the right side in integrand.

7. Green function for operator
Consider point source equation satisfied by Green function
(1)
Green function and Dirac-delta function are defined using
Fourier transforms
(2)
(3)

8. Using (2) and (3) in (1), we get
(4)
Using (4) in (2), we get
(5)
Solving integral
(6)

9. Using cos(θ) = x, for solving integral,
____(7)
Using (7) in (6), we get
(8)

10. Integral in (8) is solved using contour integration.
Contour is closed in upper half plane
Integration Value = 2πi * residue of integration at
poles
Two Poles, q = ±k

11. At q = +k, Outgoing spherical wave
----(9)
(10)
At q = -k, incoming spherical wave
For our scattering case, only outgoing wave will be of
use

12. Asymptotic limit of wave function: In scattering
experiments the detectors are located at distances
which are much larger as compared to the size of
target i.e.
Note that r is distance of detector from the target and
r’ is the size of target.

13. In light of above discussion, we can write
----(9)
------(10)
In above Eq is the wave vector in the direction of
scattered particle.
Now we use Eq. (9) and (10) in Eq. (8) and obtain the
Asymtotic form.

14. So we get,
(11)
-----(12)
where, is the plane wave.
The differential cross-section can be written as,
---(13)

15. Born Approximation
First Born Approximation: If the scattering potential is weak
it will effect very little the incident plane wave.
The first Born approximation then corresponds to the iteration
in which in Eq. (4) on right side, inside the integral,
is replaced by
So we have,
(14)

16. Also under Born’s 1st approximation the scattering amplitude
and differential Eq. of equation (12) and (13) can be
written as,
(15)
and 
------(16)
respectively is the momentum transfer.

17. and are the linear momentum of incident and
scattered wave respectively.
In elastic scattering the magnitude of and are
equal and therefore, we write
(17)

18. For spherical symmetric potential, and
Choosing z – axis along , then
(18)

19. Using (18) in (15) , we get following expression for scattering amplitude
(19)
And also for differential cross-section, we have
(20)

20. Born Series:
First born approximation or 1st order solution
2nd Born approximation

21. Nth order approximation will be a series known as
Born series

22. Diagram represent Born series
V is vertex factor
g is green function act as propagator, propagating
the disturbance from one vertex to another.
Born series was inspiration for Feynman formulation
of relativistic QM using Feynman diagrams.

23. Validity of Born Approximation: Note that the first Born’s
Approximation is valid when scattered wave is very little
Different from incident wave.
We know,
The first Born approximation is valid if
(21)

24. (22)
------(23)

25. Now note that the energy of incident particle is
which is directly proportional to momentum k.
Keeping in mind this line and looking back into Eq. (23), we
observe that the first Born approximation is valid for
large incident energies and weak scattering potentials.
When the interaction energy between incident particle and
scattering potential is much smaller than the particle’s
incident kinetic energy the scattered wave can be considered
as plane wave

26. Exercise 1:
Exercise 2: