We consider a Resonating valence Bond (RVB) state as a variational estimate for the ground state of S=1/2, antiferromagnetic Heisenberg model in two dimensional square and honeycomb lattices. We use Quantum Monte Carlo (QMC) simulation technique based on Metropolis algorithm and calculate energy and spin-spin correlation functio also we derive staggerd magnetization and we show there is still some magnetic order left in these lattices. We show that the quantum fluctuations on honeycomb lattice are much stronger than on square lattice. Our results are in good agreement with other methods such as spin-wave calculation, series expansion and exact diagonalization calculation. In addition we calculation the low-lying energy for triplet excitations in square and honeycomb lattices and we find that the excitation spectrum is linear and gapless near k=0 on both lattices.