  A simple Mathematica code based on the differential realization of hard-core
boson operators for finding exact solutions of the periodic-N spin-1/2 systems
with or beyond nearest neighbor interactions is proposed, which can easily be
used to study general spin-1/2 interaction systems. As an example, The code is
applied to study XXX spin-1/2 chain with nearest neighbor interaction in a
uniform transverse field. It shows that there are [N/2] level-crossing points
in the ground state, where N is the periodic number of the system and [x]
stands for the integer part of x, when the interaction strength and magnitude
of the magnetic field satisfy certain conditions. The quantum phase
transitional behavior in the ground state of the system in the thermodynamic
limit is also studied.
