  By measuring the field and temperature dependence of magnetization on
systematically doped $La_{2-x}Sr_xCuO_4$ thin films, the critical current
density $j_c(0)$ and the collective pinning energy $U_p(0)$ are determined in
single vortex creep regime. Together with the published data of superfluid
density, condensation energy and anisotropy, for the first time we derive the
doping dependence of the coherence length or vortex core size in wide doping
regime directly from the low temperature data. It is found that the coherence
length drops in the underdoped region and increases in the overdoped side with
the increase of hole concentration. The result in underdoped region clearly
deviates from what expected by the pre-formed pairing model if one simply
associates the pseudogap with the upper-critical field.
