  In this paper, we investigate the fidelity for the Heisenberg chain with the
next-nearest-neighbor interaction (or the $J_1-J_2$ model) and analyze its
connections with quantum phase transition. We compute the fidelity between the
ground states and find that the phase transition point of the $J_1-J_2$ model
can not be well characterized by the ground state fidelity for finite-size
systems. Instead, we introduce and calculate the fidelity between the first
excited states. Our results show that the quantum transition can be well
characterized by the fidelity of the first excited state even for a small-size
system.
