PC 4201 QUANTUM MECHANICS (SCATTERING THEORY)
Tutorial 2
1.
Verify that

(r ) = e

4
 r ikr
is an exact solution of (
 2  
2
)

(
~ x ) = -

(
~ x )
2.
Computer, in the first Born approximation, the differential and total scattering cross section for an attractive potential.
U ( x
~
)
   e
 kr
~ x = r,
where
 and k are constants.

> 0, k > 0,
3.
Show, by using the optical theorem, that the first Born approximation cannot be applied to forward scattering.
4.
Prove that a repulsive potential leads to negative phase shifts


, an attractive potential to positive phase shifts.
5.
Obtain an expression for the phase shift

0
for S- wave scattering by a spherical square-well potential:
U (r ) = - U
0
if r > a
= 0 if r > a;
(U
0
> 0). Show that, though

0

0 as the bombarding energy tends to zero, the
cross-section tends, in general, to a finite limit .
6.
Use the born approximation to calculate neutron-proton scattering if their interaction potential is v
0
( e

(  / mc ) r
/ r ) where m is the mass of the

meson (137 Mev).