In this chapter we present exact Quantum Monte Carlo results of the ground-state
energy and structure of a Bose gas confined in highly anisotropic harmonic traps.
Starting from a 3D Hamiltonian, where interparticle interactions are model by a
hard-sphere or a soft-sphere potential, we show that the system exhibits striking
features due to particle correlations. By reducing the anisotropy parameter
, while the number of particles
and the ratio
of scattering
to transverse oscillator length are kept fixed, the system crosses from a regime
where mean-field theory applies to a regime which is well described by the 1D
Lieb-Liniger equation of state in local density approximation. In the cross-over
region both theories fail and one must resort to exact methods to account properly
for both finite size effects and residual 3D effects. For very small values of
we find clear evidence, both in the energy per particle and in the
longitudinal size of the cloud, of the fermionization of the system in the
Tonks-Girardeau regime.