where is a sphere of vanishing radius centered on . Hence, deduce that is the required Green's function.
where .
where is a Dirac delta function.
where , is
Hence, deduce that and correspond to outgoing and ingoing spherical waves, respectively.
Demonstrate, using the Born approximation, that
and
where , and is the Bohr radius.
in each of the following ways:
and
where . Show that the S-wave phase-shift is given by
Assuming that , demonstrate that if then the solution of the previous equation takes the form
Of course, in the limit , the preceding equation yields , which is the same result obtained when particles are scattered by a hard sphere of radius . (See Section 10.8.) This is not surprising, because a strong repulsive -shell potential is indistinguishable from hard sphere as far as external particles are concerned.
The previous solution breaks down when , where is a positive integer. Suppose that
where . Demonstrate that the S-wave contribution to the total scattering cross-section takes the form
where
Hence, deduce that the net S-wave contribution to the total scattering cross-section is
Obviously, there are resonant contributions to the cross-section whenever . Note that the are the possible energies of particles trapped within the -shell potential. Hence, the resonances are clearly associated with incident particles tunneling though the -shell and forming transient trapped states. However, the width of the resonances (in energy) decreases strongly as the strength, , of the shell increases.
where is the scattering angle, the proton mass, and the magnitude of the incident momenta of protons in each beam.