# Scattering Theory Tutorial 2

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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).