The states have special properties. Their probability amplitudes do not change in time. Do we need to see this ?

We can examine the general states evolution.

let us examine

The orthogonality and the exponential time behavior maintains the time independence. This isn’t not true for the amplitudes of other states. We have already said that we expect to change.

To see this let us examine a wavepacket in free space. The momentum states and the energy states can be chosen to be the same. Thus the amplitudes for momentum are expected to be independent of time. At all times the probability to find a certain momentum remains the same. However the location of the particle is not the same at different times. Thus the will change but will not change.

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 in x-space which was generated with a starting width a and then disperses therefore widening as time evolves has the same function independent of time.

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.

Examine a simple example the sum of two eigenstates:

The time evolved system looks almost the same as the original. You are adding two functions E1(x) and E2(x). The only difference is each time adds the functions together with a different phase and the phase of each term changes differently. You can consider two sine functions. If they have a different frequency then the addition of the two will result in regions where the waves are in phase and adding and regions of the where they are canceling. If you shift both waves by the same phase then these regions don’t change. For example if the two cosine waves with zero phase are added then there is constructive interference at x=0. If both waves are shifted by 2π then nothing has changed. If one wave is shifted by π the waves will be canceling and the overall sum will be diminished at x=0. There can be some confusion on this point due to the nature of the cosine function. Certainly shifting the cosine function by π/2 results in a function that has changed from 1 to 0 at x=0. However the phase factor included above is which always has an amplitude of 1 and merely changes the real and imaginary ratio. Mathematically a constant phase factor can be factored out of the sum and then this factor will not have an impact on the probability. This example is meant to illustrate that changing the relative phase of two waves changes their sum which can be seen in the case of two cosine functions that are changed or shifted relative to each other while changing each wave by the same amount does not result in a different probability for an event. The book uses beats as an example. Here waves added change from loud to soft. The times that are loud and those that are soft are determined by the relative phase not the absolute phases.

If one examine the probability density for both cases given above t=0, t=later

So the wave function changes wrt time even though the relative amounts of the energy eigenstates remain the same.

For the wave packet the most instructive view is to break the wave function into three pieces the amplitude, mean and sigma

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.

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.

The translation of a system in space, time and through some angle (rotation) can be performed using operators of the general form

where the operator is referred to as a generator.

Time translation has the Hamiltonian.

Space translation has the momentum.

Rotation has the angular momentum operator.

Specifying that the system should not change behavior when moves to a transformed frame means the operators must commute with the Hamiltonian. This then results in a conservation law.