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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).