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).

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