Calculate the differential scattering cross-section, , using the Born approximation.
Suppose that the energy is slightly raised. Show that the angular distribution can then be written in the form
Obtain an approximate expression for .
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 .
where , and is the Bohr radius.