Quantum physics problem: SE and operators

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 [itex]\hat{H}[/itex] and another, time independent operator [itex]\hat{A}_{0}[/itex].
Construct a time dependent operator [itex]\hat{A}(t)[/itex] so that:
<ψ(t)|[itex]\hat{A}_{0}[/itex]|ψ(t)> = <ψ(0)|[itex]\hat{A}(t)[/itex]|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 [itex]\hat{U}(t)[/itex] to calculate the effect of time on the state ψ(0)...
So
ψ(t) = [itex]\hat{U}(t)[/itex] ψ(0) = exp(-i/[itex]\hbar \hat{H}[/itex] t) ψ(0).

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

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

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