Computational Physics

2.1 Introduction 17

• V

r

V= 10

V= 0

V= 205

Figure 2.2. The radial wave functions forl=0 for various square well potential

depths.

We see thatulapproaches a sine-wave form for largerand the phase of this wave
is determined byδl, hence the name ‘phase shift’ forδl(forl=0,ulis a sine wave
for allr>rmax).
The phase shift as a function of energy andlcontains all the information about
the scattering properties of the potential. In particular, the phase shift enables us
to calculate the scattering cross sections and this will be done in Section 2.3; here
we simply quote the results. The differential cross section is given in terms of the
phase shift by

dσ

d

=

1

k^2

∣

∣∣

∣∣

∑∞

l= 0

( 2 l+ 1 )eiδlsin(δl)Pl(cosθ)

∣

∣∣

∣∣

2

(2.7)

and for the total cross section we find

σtot= 2 π

∫

dθsinθ

dσ

d

(θ )=

4 π

k^2

∑∞

l= 0

( 2 l+ 1 )sin^2 δl. (2.8)

Summarising the analysis up to this point, we see that the potential determines
the phase shift through the solution of the Schrödinger equation forr<rmax. The
phase shift acts as an intermediate object between the interaction potential and the
experimental scattering cross sections, as the latter can be determined from it.
Unfortunately, the expressions (2.7) and (2.8) contain sums over an infinite num-
ber of terms – hence they cannot be evaluated on the computer exactly. However,
there is a physical argument for cutting off these sums. Classically, only those waves