In order to describe a cold bosonic gas in a trap we use the model Hamiltonian of
type (2.122)
Our system consists of spinless bosons of mass
interacting through the
two-body interatomic potential
and is subject to the external field
which is taken to be harmonic and axially symmetric:
For the interatomic potential we use two different repulsive model potentials: the
hard-sphere (HS) potential (1.48) and the soft-sphere (SS) potential
(1.52). In the case of the HS potential the -wave scattering length
coincides with the radius of the sphere and in the case of the SS potential is
given by (1.59). For finite
one always has
, while for
the SS potential coincides with the HS one with
. The
height
of the potential is fixed by the value of the range
in units of
the scattering length, for which we choose
. It is worth noticing that
the HS and the SS model with
represent two extreme cases for a
repulsive interatomic potential. In the HS case, the energy is entirely kinetic,
while for the SS potential
, according
to Born approximation, and the energy is almost all potential. By comparing the
results of the two model potentials we can investigate to what extent the
ground-state properties of the system depend only on the
-wave scattering length
and not on the details of the potential.