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- Consider a scattering potential of the form

Calculate the differential scattering cross-section,
, using the Born approximation.

- Consider a scattering potential that takes the constant value
for
, and is zero
for
, where
may be either positive or negative. Using the method of partial waves, show
that for
, and
, the differential cross-section is isotropic, and that
the total cross-section is

Suppose that the energy is slightly raised. Show that the angular distribution can then
be written in the form

Obtain an approximate expression for
.

- Consider scattering by a repulsive
-shell potential:

where
.
Find the equation that determines the
-wave phase-shift,
, as a function of
(where
).
Assume that
,
. Show that if
is not close to zero then the
-wave phase-shift
resembles the hard sphere result discussed in the text. Furthermore, show that if
is close to zero then resonance
behavior is possible: i.e.,
goes through zero from the positive side as
increases. Determine the
approximate positions of the resonances (retaining terms up to order
). Compare the resonant
energies with the bound state energies for a particle confined within an infinite spherical well of radius
.
Obtain an approximate expression for the resonance width

Show that the resonances become extremely sharp as
.

- Show that the differential cross-section for the elastic scattering of a fast electron by the ground-state of a hydrogen
atom is

where
, and
is the Bohr radius.

** Next:** Identical Particles
** Up:** Scattering Theory
** Previous:** Resonance Scattering
Richard Fitzpatrick
2013-04-08