 0  0.2  0.4  0.6  0.8  1  1.2  0  1  2  3  4  5  6  7  8  9  10 Relative momentum distribution k A = 100 1000 A = 5000 = 10000 FIG. 6. (Color online) Plot of relative momentum distribution against k (in (o.u.) − $^{1}$)for vari- ous indicated values of particle numbers. The choice of a$_{s}$ = 0.00433 o.u. corresponds to $^{87}$Rb experiment in JILA trap. that different interatomic potential shapes corresponding to the same a$_{s}$ produce identical correlation properties, which indicates that the calculated property is independent of the shape of the two-body potential. The fact that the ground state energy is dependent on the shape of the two-body potential 29 whereas the one-body density and the pair distribution function are not, can be attributed to the fact that the many-body wave function remains in- variant irrespective of the variation of the shape of the two-body potential due to variation of ( C$_{6}$, r$_{c}$ ) for the same value of a$_{s}$ . Note that the condensate wave function in the hyperradial space ζ$_{0}$ ( r ) is obtained by solving the hyperradial equation (6) in the effective many-body potential ω$_{0}$ ( r ). The shape and stiffness of this effective potential remain unchanged for the same value of a$_{s}$ , irrespective of the variation of the shape of the two-body potential. How- ever, as discussed earlier, the position and value of the minimum of the effective many-body potential change with the variation of the parameter C$_{6}$ . To demonstrate this, we plot the effective many-body potential ω$_{0}$ ( r ) as a function of r for various values of C$_{6}$ in Fig. 8. The curves have been shifted both vertically and horizontally in order that the minima coincide (both in position and value). One observes that all the curves overlap completely (within numerical errors) showing that the shape and stiffness of the effective potential remain in- variant. Hence the kinetic energy and the wave function remain unchanged, while potential energy and the total energy change with change of C$_{6}$ . By direct numerical calculation, we have checked that the change of kinetic energy is very small compared with that of the total energy, as C$_{6}$ varies. Therefore the ground state energy is shape dependent, whereas the 13