QM: Writing time evolution as sum over energy eigenstates

[Muizz]

Suppose I have a 1-D harmonic oscilator with angular velocity $\omega$ and eigenstates $|j>$ and let the state at $t=0$ be given by $|\Psi(0)>$. We write $\Psi(t) = \hat{U}(t)\Psi(0)$. Write $\hat{U}(t)$ as sum over energy eigenstates.

I've previously shown that $\hat{H} = \sum_j |j>E_j<j|$ with $E_j = \hbar\omega(\frac12+j)$ (part one of the exercise I got this from) but with this second part I'm drawing a complete blank. Any help would be hugely appreciated.

 

[BvU]

Hello,

Make life simple and imagine $|\Psi(0)\rangle = |j\rangle$.
What would $\hat U(t)$ look like in that case ?