  Bosons in the form of ultra cold alkali atoms can be confined to a one
dimensional (1d) domain by the use of harmonic traps. This motivates the study
of the ground state occupations $\lambda_i$ of effective single particle states
$\phi_i$, in the theoretical 1d impenetrable Bose gas. Both the system on a
circle and the harmonically trapped system are considered. The $\lambda_i$ and
$\phi_i$ are the eigenvalues and eigenfunctions respectively of the one body
density matrix. We present a detailed numerical and analytic study of this
problem. Our main results are the explicit scaled forms of the density
matrices, from which it is deduced that for fixed $i$ the occupations
$\lambda_i$ are asymptotically proportional to $\sqrt{N}$ in both the circular
and harmonically trapped cases.
