  Under the second-order degenerate perturbation theory, we show that the
physics of $N$ particles with arbitrary spin confined in a one dimensional trap
in the strongly interacting regime can be described by super-exchange
interaction. An effective spin-chain Hamiltonian (non-translational-invariant
Sutherland model) can be constructed from this procedure. For spin-1/2
particles, this model reduces to the non-translational-invariant Heisenberg
model, where a transition between Heisenberg anti-ferromagnetic (AFM) and
ferromagnetic (FM) states is expected to occur when the interaction strength is
tuned from the strongly repulsive to the strongly attractive limit. We show
that the FM and the AFM states can be distinguished in two different methods:
the first is based on their distinct response to a spin-dependent magnetic
gradient, and the second is based on their distinct momentum distribution. We
confirm the validity of the spin-chain model by comparison with results
obtained from several unbiased techniques
