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

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... antum transition can be well characterized by the fidelity of the first excited state even for a small-size system.