1000 Solved Problems in Modern Physics

154 3 Quantum Mechanics – II

show thatσ(θ)=a^2

[

1 −(ka)

2

3 +2(ka)

(^2) cosθ+···

]

andσ= 4 πa^2 [1−(ka)^2 /3]

3.110 Find the elastic and total cross-sections for a black sphere of radiusR.

3.111 Ramsauer (1921) observed that monatomic gases such as argon is almost
completely transparent to electrons of 0.4 eV energy, although it strongly
scatters electrons which are slower as well as those which are faster. How is
this quantum mechanical peculiarity explained?

3.112 What conditions are necessary before the Schrodinger equation for the inter-
action of two nucleons can be reduced to the form
d^2 U
dr^2

+

(

2 m

^2

)

[E−V(r)]U= 0

whereU(r)=rψ(r) and the other symbols have their usual meanings?

By solving this equation for a square-well potentialV(r) for a neutron –

proton collision show that the neutron – proton scattering cross-section, as

calculated for high energies is about 3 barns compared with the experimental

value of 20 barns. What is the explanation of this discrepeancy and how has

this explanation been verified experimentally?

[Adapted from the University of Durham 1963]

3.2.9 Scattering(BornApproximation).....................

3.113 In the case of scattering from a spherically symmetric charge distribution,
the form factor is given by

F(q^2 )=

∫∞

0

ρ(r)

sin

(qr



)

qr/

4 πr^2 dr

whereρ(r) is the normalized charge distribution.

(a) If the charge distribution of proton is approximated byρ(r)=Aexp(−r/a),

whereAis a constant andais some characteristic “radius” of the proton.

Show that the form factor is proportional to

(

1 +q

2

q^20

)− 2

whereq 0 is/a.

(b) Ifq 02 = 0. 71

(GeV

c

) 2

, determine the characteristic radius of the proton.

3.114 The first Born approximation for the elastic scattering amplitude is

f=−

(

2 μ

q^2

)∫

V(r)eiq.rd^3 r

Show that forV(r) spherically symmetric it reduces to

f=−

(

2 μ/q^2

)

∫

rsin(qr)V(r)dr