  Quantum many-body systems in one dimension (1D) exhibit some peculiar
properties. In this article, we review some of our work on strongly interacting
1D spinor quantum gas. First, we discuss a generalized Bose-Fermi mapping that
maps the charge degrees of freedom to a spinless Fermi gas and the spin degrees
of freedom to a spin chain model. This also maps the strongly interacting
system into a weakly interacting one, which is amenable for perturbative
calculations. Next, based on this mapping, we construct an ansatz wavefunction
for the strongly interacting system, using which many physical quantities can
be conveniently calculated. We showcase the usage of this ansatz wavefunction
by considering the collective excitations and quench dynamics of a harmonically
trapped system.
