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# 9.P: Exercises

1. Consider a scattering potential of the form

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

2. 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 .

3. 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 .

4. 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.