Quantum Mechanics 1D harmonic Oscillator

[dcuk86]

Hi, I'm currently working through some exam papers from previous years before an upcoming module in Quantum Mechanics.

Homework Statement

See the attached image

Homework Equations

The Attempt at a Solution

I'm a little stumped with this one, I'm assuming that I'm looking at a 1D harmonic Oscillator and the wording of the question suggests that the wave function just needs to be stated and not actually proven (?).
In your opinion is this question looking for the wavefunction for n=1 or a general wavefunction in terms of n? its mostly the time dependence which has thrown me.

 

[betel]

I would say you can take the eigenfunction $$u_0,u_1$$ as given.
Eigenstates to different energies evolve differently in time.
The exact behaviour can be extracted from the full Schrödinger equation. Although I think you have probably written it at the start of solving the H.O., some Ansatz involving separation of variables.

The question is then looking for the time evolution of this composite state.