I have a question about the time evoltion operator.

I we define the time evolution operator as the operator the propagates states in time like:

|psi(t)> = U(t,t')|psi(t')>

and use it multiple times we get the following property:

U(t,t'')U(t'',t')=U(t,t')

We also have the inverse: U^(-1)(t,t') = U^(+)(t,t') = U(t',t)

I want to point out that I'm not dealing with time independent hamiltoinans so the evolution operator can not be written as: U(t,t') = e^(i(t-t')H/h).

Now comes my question. In the Heisenberg picture the operators are time dependent and the states are fixed in time. Assume that the Schrödinger and Heisenberg pictures coincide at time t=t_r.

Then we have that en operator in Heisenberg picture will be related to one in the Schrödinger picture by:

A_H (t) = U^(-1)(t,t_r) A_S U(t,t_r)

So that states in the Heisenberg picture are evolved from t_r to t and then acted on by the operator and evolved back to t_r.

But operators in H-picture of different times are related as:

A_H(t) = U^(-1)(t,t') A_H(t') U(t,t')

Rewriting A_H(t') in terms of the Schrödinger operator we:

A_H(t) = U^(-1)(t,t')U^(-1)(t',t_r) A_S U(t',t_r)U(t,t')

But acting on a state in the Heisenberg picture we will have the two time evolution operators on the left of A_S above acting on the Heisenberg state:

U(t',t_r) U(t,t') |psi_H> = U(t',t_r)U(t,t')|psi(t_r)>

So, how do I calculate this? Am I allowed to interchange the order of the two time evolution operators? The order of the two operators seems really strange to me. Is it really meaningful to act on a state in time t_r by the operator U(t,t')???

I hope someone can answer this. Please ask me if something I've written is not clear.

/Jezuz

Just a small point: the time evolution operator for a time independent Hamiltonian is U(t,t') = exp(-i(t-t')H), you have a plus sign in your definition. This is probably just a typo, but just in case remember that U satisfies i dU/dt = H U.

Now, everything you have written looks ok until you get to the point where you relate the Heisenberg operators at different times. The general result is actually A_H(t) = U^+(t',t_r) U^+(t,t') U(t',t_r) A_H(t') U^+(t',t_r) U(t,t') U(t',t_r) = U^+(t,t_r) U(t',t_r) A_H(t') U^+(t',t_r) U(t,t_r) = U^+(t,t_r) A_S U(t,t_r). Now, these time evolution operators evaluated at different times only commute if the Hamiltonian is time independent. With this information you can easily check that the more complicated expression I gave reduces to your simpler expression only for time independent Hamiltonians (or more properly only when [H(t),H(t')]=0). I think the reason why the operator order looks strange to you is that you apparently commuted the evolution operators without knowing it (presumably in coming up with your expression for relating Heisenberg operators at different times). You are correct to be suspicious since your expression is only valid in the case of time independent Hamiltonians. You can see in the general result above that the operator ordering comes out right.

Hope this helps.

Oh ok. Think I got it. So I actually should use the time-evolution operator written as:

U_H(t,t') = U^+(t',t_r) U(t,t') U(t',t_r)

and mulitply that on both sides of the operator in order to propagate it like:

A_H(t) = U_H^+(t,t')A_H(t')U_H(t,t')

have I got this correct then?