Radial size of the system

Finally, in Fig. 3.9, we show results for the mean square radius in the transverse direction. The cross-over from 3D to 1D is clearly visible in the case of $N=100$, for both the HS and SS potential, and for the HS potential in the case of $N=5$ and $a_{3D}/a_\perp =1$. For the system with $N=5$ and $a_{3D}/a_\perp =0.2$ we only see small deviations from $\sqrt{\langle r_\perp^2\rangle}=a_\perp$ for the largest values of $\lambda$. In the $a_{3D}/a_\perp =0.04$, as well as in the $a_{3D}/a_\perp =1$ case with the SS potential, the transverse density profile is well described by the harmonic oscillator wave function and we find $\sqrt{\langle r_\perp^2\rangle}\simeq a_\perp$ over the whole range of values of $\lambda$. It is worth noticing that for the $N=5$ system with SS potential, the largest deviations from $\sqrt{\langle r_\perp^2\rangle}=a_\perp$ are achieved for $a_{3D}/a_\perp =0.2$, corresponding to a transverse confinement $a_\perp=R$ where $R$ is the range of the SS potential.

Figure 3.9: Mean square radius in the radial direction as a function of $\lambda$. Solid symbols: HS potential; open symbols: SS potential. Down triangles: $N=100$ and $a_{3D}/a_\perp =0.2$; circles: $N=5$ and $a_{3D}/a_\perp =1$; squares: $N=5$ and $a_{3D}/a_\perp =0.2$; up triangles: $N=5$ and $a_{3D}/a_\perp =0.04$. Error bars are smaller than the size of the symbols.