Thermodynamics, Statistical Physics, and Quantum Mechanics

QUANTUM MECHANICS 321

5.69 Scattering from Neutral Charge Distribution

(Princeton)

a) The particle scatters from thepotential energy which is related to

the chargedistribution

where is the Fourier transform of and is theFourier trans-

form of The differentialcrosssection in theBorn approximation is

b) In forwardscattering we take In order that thecross sectionhave

a nondivergent result inthis limit, we need to find

To obtain this result, we examinethe behavior of at small values of

Consider the three terms inbrackets: (i) the 1 vanishes since the dis-
tribution is neutral; (ii) thesecondtermvanishes since thedistribution
is sphericallysymmetric; (iii) the last term gives an angular average
and the integral of isA.The crosssection in forward
scattering is

c) The charges in a hydrogen atom are thenucleus, which is taken as a
deltafunction at the origin, and the electron, which is given by thesquare