Confirmation concept questions eigenfunctions and operators

[black_hole]

Homework Statement

Are the momentum eigenfunctions also eigenfunctions of e free particle energy. Operator?
Are momentum eigenfunctions also eigenfunctions of the harmonic oscillator energy operator?
An misplayed system evolves with time according to the shrodinger equation with potential v. The wave function depends on time. Does the expectation value of energy depend on time?

Homework Equations

The Attempt at a Solution

I think not necessarily. Looking at the commutator it looks like this would only work if the potential was a constant.

I guess this is a no judging by the form of the eigenfunctions of the two.

Does this depend on the initial conditions? It snot clear wether or not this is a stationary state?

 

[DrClaude]

I think not necessarily. Looking at the commutator it looks like this would only work if the potential was a constant.

What potential?

I guess this is a no judging by the form of the eigenfunctions of the two.

Can you show this more rigorously?

Does this depend on the initial conditions? It snot clear wether or not this is a stationary state?

My understanding of the question is that you start from an eigenstate of the system and displace it (or conversely displace the potential) and then look at the time evolution of the system. The problem states that "The wave function depends on time", so is it a stationary state? If the answer is no, does this mean that the expectation value of the energy will vary in time?