a r t Keywords: Fast Gaussian wavepacket transforms

address the first question, we introduce fast Gaussian wavepacket transforms and develop evolu mean momentum follow the Newton laws, up to quantum correction. This results in the field of semi-classical quantum mechanics, which provides a short-wavelength link between classical mechanics and quantum mechanics and maintains contact with the structure imposed by classical mechanics without sacrificing quantum mechanical accuracy. On the com-putational side, numerical methods based on the semi-classical approaches are also sought as efficient alternatives to capture the highly oscillatory phenomenon arising from quantum mechanics