For a homogeneous system the one body density matrix is
defined as the Fourier transform of the momentum distribution (1.42)
(1.45)
where the contribution of the condensate has been
extracted from the integral. After angular integration one gets
(1.46)
At the momentum distribution is given by (1.42).
By introducing the dimensionless variable , one
obtains the following result for the coordinate dependent part of
the one-body density matrix
(1.47)
where .
For ,
and the one-body density matrix coincides with the total density
. For (where
is the healing length)
(see derivation and comments in Appendix B).
Thus the asymptotic value of the one-body density matrix
coincides with the condensate density
.
For arbitrary values of the integral (1.47) can be
calculated numerically. Results for different values of
are shown in Fig. 1.1