Quantum effects are extremely important in many areas of molecular science, especially low temperature astrophysics. Yet most molecular calculations are performed using classi- cal trajectory simulations that include no quantum effects at all. This research focuses on a new quantum trajectory method approach without quantum wave functions that incorpo- rates many quantum effects while still retaining all of the practical benefits of the classical trajectory simulation (fast and low physical memory occupation). The research also exam- ines novel techniques for sampling the ensemble of trajectories to capture quantum structural effects such as zero-point energy. The above theory is then implemented numerically to com- pute various quantum quantities for various types of model chemical reactions of pedagogical interest to molecular scientists (Eckart barrier, Morse potential and London-Eyring-Polanyi- Sato potential) . These are then compared with the results of an exact quantum algorithm (discrete variable representation).