Let us choose to eamine the momentum eignetate

momentum eignestate

expectation value

A measurement will yield .

position basis

Conclusion is that it is also an energy eigenvector for the FREE Hamiltonian.

Consider a state called a wavepacket. Now we will not address the question as to how one builds such a state. We will simply ask some questions as to what features it manifests.

Normalize

Just as with any probability density one can define a spread in the function in terms of an RMS .

is the width of a Gaussian function. Therefore the spread of the function in space can be characterized by this RMS.

You can also view this state wrt the momentum basis

Time evolution:

Time t is considered to be a parameter that can be measured and used to mark the status of the state. Thus the state evolves as a function of time. At this point the role of time in quantum mechanics is not properly treated. Indeed if you imagine a state that decays very quickly into another state then the energy of the initial state cannot be an exact value because of the “width” in time. So there is an uncertainty relationship between energy and time. However there is not operator . While we treat x as a measurable observable we treat time as a parameter.

Again there is some importance in looking at the time dependence on special states.

One way of achieving this set of solutions is separation of variables:

We get back a specific time dependence and the energy eigenequation.

So energy eigenvalues have the distinct property that there time dependence is straight forward.

Now we already discovered that the momentum eigenstates are also eigenstates for the free Hamiltonian. If the states are not confined by any potentials.

We can derive a time evolution operator.

If you want to be very careful you need to address what we mean by an operator such as . The Hamiltonian and the momentum operator take on the form of a differential operator that operates on the wavefunctions. Thus an operator that transforms vectors in the Hilbert space into new vectors becomes a differential operator that operates on the amplitude function. For the very careful student the meaning of these two operators is different and can lead to confusion. The point is that the impact of considering observables that are continuous leads to a derivative rather than a number. Discrete states lead to a matrix of complex numbers and the matrix represent the original operator. Continuous states lead to a derivative. {The derivative is an operator when considering its impact on functions}.

The student is referred to a discussion.