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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