  The many-body wave-function of an interacting one-dimensional electron system
is probed, focusing on the low-density, strong interaction regime. The
properties of the wave-function are determined using tunneling between two
long, clean, parallel quantum wires in a GaAs/AlGaAs heterostructure, allowing
for gate-controlled electron density. As electron density is lowered to a
critical value the many-body state abruptly changes from an extended state with
a well-defined momentum to a localized state with a wide range of momentum
components. The signature of the localized states appears as discrete tunneling
features at resonant gate-voltages, corresponding to the depletion of single
electrons and showing Coulomb-blockade behavior. Typically 5-10 such features
appear, where the one-electron state has a single-lobed momentum distribution,
and the few-electron states have double-lobed distributions with peaks at $\pm
k_F$. A theoretical model suggests that for a small number of particles (N<6),
the observed state is a mixture of ground and thermally excited spin states.
