Computational Physics - Department of Physics

298 9 Two point boundary value problems

Compare these results with those obtained by solving the above differential equation as a
set of linear equations, as done in chapter 6. Which method would you prefer?

9.2.We are going to study the solution of the Schrödinger equation for a system with a
neutron and a proton (the deuteron) for a simple box potential. This potential will later be
replaced with a realistic one fitted to experimental phase shifts.
We begin our discussion of the Schrödinger equation with theneutron-proton (deuteron)
system with a box potentialV(r). We define the radial part of the wave functionR(r)and
introduce the definitionu(r) =rR(R)The radial part of the SE for two particles in their center-
of-mass system and with orbital momentuml= 0 is then

−

h ̄^2

2 m

d^2 u(r)

dr^2

+V(r)u(r) =E u(r), (9.26)

with
m= 2
mpmn
mp+mn

, (9.27)

wherempandmnare the masses of the proton and neutron, respectively. We use herem= 938
MeV. Our potential is defined as

V(r) =







0 r>a

−V 00 <r≤a

∞ r≤ 0

, (9.28)

displayed in Fig 9.1.

✲ x

0 a

−V 0

V(x)

Fig. 9.1Example of a finite box potential with value−V 0 in 0 <x≤a, infinitely large forx≤ 0 and zero else.

Bound states correspond to negative energyEand scattering states are given by positive
energies. The SE takes the form (without specifying the signofE)