The trial wavefunction for the pure system was constructed in
chapter 3.2.1. In this section the same approach
will be extended to systems in the presence of quenched impurities.
The wave function of the system is chosen as the product of
one-body and two-body wavefunctions.
(4.1)
where stands for the particle-particle wavefunction,
which has already been obtained in section 3.2.1
and is defined by (3.37), (3.41) and (3.42).
In eq. (4.1) describes the effect of the
impurities on each particle.
To construct we use a similar procedure as for ,
i.e. we solve the particle-impurity Schrödinger equation
(4.2)
where the reduced particle-impurity mass is equal to the mass of a
particle, because the quenched impurity is infinitely massive Let
us look for the symmetric solution in spherical coordinates
(4.3)
The particle is modeled by a hard sphere of diameter and the
impurity by a hard sphere of diameter . The particle-impurity
interaction potential is
(4.4)
where is the particle-particle -wave scattering length
The dimensionless Schrödinger equation has form (lengths in units
of and energies in units of
)
(4.5)
and the differential equation which has to be solved is
(4.6)
The solution is
, with
being an arbitrary constant.
Let us construct the particle-impurity wave function in
the same way as it was done for the particle-particle wavefunction.
We introduce a matching point and choose
(4.7)
The function must be smooth at the matching point. The
request of continuity for , its derivative and
the local energy
is fulfilled