  We investigate the Bose gas with repulsive or attractive interactions between
atoms in the scheme of Bethe Ansatz equation in a hard wall trap. Three typical
quantum phases in the current research of 1D interacting cold atoms are
clarified in terms of the energy spectrum, single particle density distribution
and two-particle correlation function. We identify two matching points in the
phase diagram, i.e. the TG and STG gas show the same profiles at the strongly
interacting point $-1/\gamma=0$, and in the weakly interacting limit $\gamma=0$
the ground states TG and BS join to each other smoothly.
