The impact of an operator of the form

moves a function that depends on λ forward to λ+η . We refer to this as a translation. On could connect an operator with these transformations. Thus

can be defined as the generators for certain transformations: rotation, time evolution and spatial translation, respectively.

In addition we can write the Schrodinger equation that states that the kinetic and potential energy added together becomes the total energy and H is also equal to the total energy. Thus the time evolution is determined by the total energy of a system.

This relationship- identifying the total energy with the time derivative and therefore related to the time evolution- is the equation of motion or Newton’s law for the quantum systems.

In the text it is shown that the amplitudes for an energy eigenstates have a simple time dependence. One can see that this results in a magnitude that is unchanging in time. So the Gaussian wave packet which was generated with a width a and disperses therefore widening as time evolves has the same function .

Thus even though the wave function widens in x it doesn’t widen in k. Is this a problem? It doesn’t violate the uncertainty principle because remains valid since is increasing and isn’t changing.

One could build a state new state localized in space as a Gaussian with the same width as the time evolved state but with a narrower range of momenta. The phases of the amplitudes are what widens the state above not a change in the relative amounts of any momentum contribution because the magnitudes of each momentum are independent of time.