The idea of a wave packet is confusing a bit. First let me tell you what I have understood about it. A wave packet is obtained by the superposition of harmonic waves of nearby frequencies which represents a matter wave. It is well known fact that the width of a moving Gaussian wave packet in free space is time dependent due to uncertainty in momentum which leads to uncertainty in velocity. If the wave packet represented a microscopic particle its width will increase rapidly due Heisenberg’s uncertainty principle. If the same wave packet is moving in a dispersive medium, the spreading is further enhanced due to wavelength dependence of phase velocity. Is it right? Assume that time tends to infinity.