# Quantum physics problem: SE and operators

1. Jun 29, 2012

### max_jammer

Hello.
I have this problem at hand:

1. The problem statement, all variables and given/known data
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.

3. 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

2. Jun 29, 2012

### vela

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

3. Jun 29, 2012