Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 5.5
Cross sections and decay rates — Schwartz M.D. z-lib.org
Physics — QFT · rutherford-scattering
Problem
One of the most important scattering experiments ever was Rutherford's gold foil experiment. Rutherford scattering is αN→αN, where N is some atomic nucleus and α is an α-particle (helium nucleus). It is an almost identical process to Coulomb scattering (e−p+→e−p+).
(a) Look up or calculate the classical Rutherford scattering cross section. What assumptions go into its derivation?
(b) We showed that the quantum mechanical cross section for Coulomb scattering in Eq. (5.41) follows either from the Born approximation or from quantum field theory. Start from the formula for Coulomb scattering and make the appropriate replacements for αN scattering.
(c) Draw the Feynman diagram for Rutherford scattering. What is the momentum of the virtual photon, kμ, in terms of the scattering angle and the energy of the incoming α-particle?
(d) Substitute in for k4 and rewrite the cross section in terms of the kinetic energy of the α-particle. Show that Rutherford's classical formula is reproduced.
(e) Why are the classical and quantum answers the same? Could you have known this ahead of time?
(f) Would the cross section for e−e−→e−e− also be given by the Coulomb scattering cross section?