Quantum physics problem: SE and operators

[max_jammer]

Hello.
I have this problem at hand:

Homework Statement

A quantum mechanical system has a hamilton operator $\hat{H}$ and another, time independent operator $\hat{A}_{0}$.
Construct a time dependent operator $\hat{A}(t)$ so that:
<ψ(t)|$\hat{A}_{0}$|ψ(t)> = <ψ(0)|$\hat{A}(t)$|ψ(0)>
for all states ψ(t) that develop in time according to the SE.

The Attempt at a Solution

In the derivation of the Schrödinger equation, we use the unitary operator $\hat{U}(t)$ to calculate the effect of time on the state ψ(0)...
So
ψ(t) = $\hat{U}(t)$ ψ(0) = exp(-i/$\hbar \hat{H}$ t) ψ(0).

In other words:
<ψ(t)|$\hat{A}_{0}$|ψ(t)> = <$\hat{U}(t)$ ψ(0)|$\hat{A}_{0}$|$\hat{U}(t)$ ψ(0)>
=<ψ(0) |$\hat{U}(t)^{+} \hat{A}_{0} \hat{U}(t)$ | ψ(0)>.

so my "solution" is that
$\hat{A}(t)$ = $\hat{U}(t)^{+} \hat{A}_{0} \hat{U}(t)$...

But this is way too simple to be correct...

So what am I missing?

Thanks

/D

 

[vela]

Looks fine to me. I suppose you could write explicitly what $U(t)^\dagger$ is equal to.

 

[kloptok]

It's correct, and what you have done is actually to find the correspondence between the Schrödinger and the Heisenberg picture. See e.g.
http://en.wikipedia.org/wiki/Heisenberg_picture
http://en.wikipedia.org/wiki/Schrödinger_picture