Thermodynamics, Statistical Physics, and Quantum Mechanics

QUANTUM MECHANICS 317

where is the range of the potential. The first condition derives

from the requirement that the perturbedwavefunction beveryclose to the

unperturbedwave function. Inequality (S.5.65.5) may also be considered

the requirement that the potential besmall compared to thekineticenergy

of the particlelocalized at thesource of theperturbation. Even if thefirst

condition is not satisfied, particleswithlargeenough willalsojustify the

Born approximation.

5.66 Scattering of Two Electrons (Princeton)

We evaluate thescattering in the Born approximation, which is validwhen

the kineticenergies aremuch largerthan the binding energy. TheFourier

transform of thepotential is

and the formulafor the totalcrosssection of electronswith initialwave

vector is

This crosssection is suitable forclassicalparticles,without regard tospin.

The specification to the spin statesS= 0, 1 is made below. Write

where is thesolidangle of thescattering. Thedifferentialcross

section is found by taking the functional derivative of the cross section with

respect tothissolid angle:

where we have used the fact that and is defined by
The magnitudes of the vectors and are the same,
so (seeProblem5.65 andFigureS.5.65). All
of the dimensional factors arecombined into the Bohrradius Now we
consider how thisformula isaltered by the spin of the electrons. Spin is
conserved in thescattering, so the pair of electrons has thesamespin state
before and after thecollision.