For the wave packet

The width of the peak in momentum space is two times the inverse of the width in position space.

First find the amplitudes for the momentum states.

Each momentum state has a know time dependence

So the wave function is a sum over momentum states with well prescribed time evolution.

The book proceeds to examine the implications of time evolution by algebraically factoring the time evolved wavefunction into a piece that reflects the amplitude, mean and width.

Each one of these has an explicit time dependence and the impact of the time evolution can be seen in terms of how the standard parameters that describe a Gaussian change in time.

What was the role of the propagator?