QuantumPhysics.dvi

Using the time-dependent Schr ̈odinger equation,

i ̄h

∂ψ

∂t

=−

̄h^2

2 m

∆ψ+V ψ (12.34)

one shows conservation of this current,

∂ρ

∂t

+∇·j= 0 (12.35)

The global form of the conservation of probability equationis that the time variation of the total
probabilitypvin some volumevplus the fluxφ∂vthrough the boundary∂vof volumevvanishes,

dpv

dt

+φv= 0 (12.36)

where we have

pv=

∫

v

d^3 xρ(t,x) φ∂v=

∮

∂v

d^2 S·j(t,x) (12.37)

A scattering process is considered to be a steady state process in which some of the incoming flux
due to the plane waveeik·xis scattered off into a spherical wave. Thedifferential cross section
gives a quantitative evaluation of the strength of this scattering process, as a function of angle of
the outgoing beam.

Using the definition of the probability current, we readily find that the incoming probability
current density is given by the velocityvof the particles in the incoming planar wave beam,

jincident=

v

(2π)^3

(12.38)

wheremv= ̄hk. Thedifferential cross sectionis obtained by analyzing the probability flow of the
outgoing wave as a function of angle, far away from the interaction region. Thus, we consider the
flux through an infinitesimal surface elementd^2 S, at a pointxwhich is a long distancer=|x|
away from the target. Fixing the solid angle to bedΩ, we have

d^2 S=r^2 dΩ

x

r

(12.39)

The flux through the solid angledΩ is then given by

dφ

dΩ

= d^2 S·jscattered

= limr→∞r^2

x

r

·jscattered(x) (12.40)

The current density is obtained by substituting (12.31) into the general expression, and is given by

jscattered = −

i

2 m

f(k′,k)∗

(2π)^3

e−ikr

r

∇

(

f(k′,k)

eikr

r

)

+c.c. (12.41)