Computational Physics

16 Quantum scattering with a spherically symmetric potential

This is a partial differential equation in three dimensions, which could be solved
using the ‘brute force’ discretisation methods presented in Appendix A, but exploit-
ing the spherical symmetry of the potential, we can solve the problem in another,
more elegant, way which, moreover, works much faster on a computer. More spe-
cifically, in Section 2.3 we shall establish a relation between thephase shiftand the
scattering cross sections. In this section, we shall restrict ourselves to a description
of the concept of phase shift and describe how it can be obtained from the solutions
of the radial Schrödinger equation. The expressions for the scattering cross sections
will then be used to build the computer program which is described inSection 2.2.
For the potentialV(r)we make the assumption that it vanishes forrlarger than
a certain valuermax. If we are dealing with an asymptotically decaying potential,
we neglect contributions from the potential beyond the rangermax, which must be
chosen suitably, or treat the tail in a perturbative manner as described in Problem 2.2.
For a spherically symmetric potential, the solution of the Schrödinger equation
can always be written as

ψ(r)=

∑∞

l= 0

∑l

m=−l

Alm

ul(r)

r

Ylm(θ,φ) (2.2)

whereulsatisfies the radial Schrödinger equation:
{

^2
2 m

d^2

dr^2

+

[

E−V(r)−

^2 l(l+ 1 )

2 mr^2

]}

ul(r)=0. (2.3)

Figure 2.2shows the solution of the radial Schrödinger equation withl=0 for
a square well potential for various well depths – our discussion applies also to
nonzero values ofl. Outside the well, the solutionulcan be written as a linear
combination of the two independent solutionsjlandnl, the regular and irregular
spherical Bessel functions. We write this linear combination in the particular form

ul(r>rmax)∝kr[cosδljl(kr)−sinδlnl(kr)]; (2.4)

k=

√

2 mE/
.

Herermaxis the radius of the well, andδlis determined via a matching procedure
at the well boundary. The motivation for writingulin this form follows from the
asymptotic expansion for the spherical Bessel functions:

krjl(kr)≈sin(kr−lπ/ 2 ) (2.5a)

krnl(kr)≈−cos(kr−lπ/ 2 ) (2.5b)

which can be used to rewrite(2.4)as

ul(r)∝sin(kr−lπ/ 2 +δl), larger. (2.6)