1. Theory of Scattering
Lecture 4

2. Scattering of identical spinless bosons particles:
S.E. For two particle system of equal masses in C.M. Frame
(1)

3. Classical result for cross-section:
(2)
Asymptotic solution for scattering:
(3)
Wave function corresponding to identical particles
in quantum mechanics must be symmetrised

4. For spinless bosons wave function must be
symmetric under the interchange of spatial
co-ordinates of two particles
implies replace
In polar coordinates
replaced by
Note that Eq. (3) is not sysmmetric under the above
conditions .

5. Symmetric combination has required symmetry
Above equation is also sol of S.E. (1)

6. Asymptotic form of sol is
Scattering amplitude:
Differential cross-section:

7. Last eq. Can be written as
Note the presence of interference term in above.
Total cross-section will be
For central potential:

8. For , quantum differential cross-section
will be
For the classical case it will be
Quantum cross-section is two times bigger than
the classical case.

9. Mott formula for coulomb scattering : dashed lines

10. Scattering of two identical spin ½ fermions
Two identical spin ½ fermions interacting through central force
Singlet state S = Triplet state S = 1
Full wave function (including spatial and spin part) for fermionic system must be
Anti symmetric.

11. For S = 0 state, Spin part will be antisymmetric
and therefore, spatial part must be symmetric
under the interchange of position vectors
Thus, symmetrised scattering amplitude will be
And differential cross-section will be

12. For S = 1 state, Spin part will be symmetric
and therefore, spatial part must be antisymmetric
under the interchange of position vectors
Thus, symmetrised scattering amplitude will be
And differential cross-section will be

13. If the particles of incident beams and target are
randomly oriented i.e. unpolarised particles
probability of particles in triplet state will be three
times in singlet state

14. Scattering for spin ½ fermions from coulomb potential