An efficient Monte Carlo method is proposed to calculate the elementary excitation spectrum of quantum systems. The lowest energy with arbitrary momentum is obtained by the projector Monte Carlo method. This is applied to the spin-(1/2 and -1 Heisenberg antiferromagnetic chain with length 32. For the S=(1/2 case, the spectrum coincides completely with the spectrum of des Cloiseaux and Pearson. For the S=1 case, the spectrum has a gap at momentum \ensuremath{\pi} as was predicted by Haldane. The value of the gap coincides with the calculation of Nightingale and Blote. The spectrum satisfies a variational relation with the structure factor.