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9.P: Exercises
 Last updated
 08:41, 16 Nov 2014

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 Consider a scattering potential of the form
Calculate the differential scattering crosssection, , 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 crosssection is isotropic, and that the total crosssection 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 phaseshift, , as a function of (where ). Assume that , . Show that if is not close to zero then the wave phaseshift 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 crosssection for the elastic scattering of a fast electron by the groundstate of a hydrogen atom is
where , and is the Bohr radius.