What is "The probabilities of each state of $O_A$ form a Gaussian distribution" supposed to mean? $O_A$ is an observable, it doesn't have states. What exactly are you saying when you say the momentum $P_B$ "becomes entangled with $O_A$"? States can be entangled, not observables or values. When you are "measuring the probe system", what observable are you measuring, and why do you think this has anything to do with measuring $O_A$?

Maybe $O_A$ can be the charge of a two parallel plate capacitor (in an LC circuit) that has no side walls and the probe is an electron that passes between those plates. The states of the charge are just the possible charge values (1 coulomb, 2 coulomb, etc.) . By $O_A$ I was not referring to an operator. Also, I meant measuring the momentum of the probe, whose possible momentum values (states) are entangled with the charge states.

You call $O_A(t)$ an "observable". A quantum-mechanical observable is a self-adjoint operator by definition of an observable. If you just want to say that you want to assume you have an entangled state of two systems, what is the Gaussian distribution supposed to add here?

The Gaussian distribution is part the specifics of the problem I am trying to solve. Maybe it is not critical to state. It is explaining how the expectation value is changing over time. In any case, if it is not important, then please ignore it.

@ ACuriousMind Instead of finding every possible flaw, and dinging from your position of superiority, could you please try to understand and constructively comment on the spirit of the question?

Or better yet, just refrain from commenting on this question.

I don't know what the "spirit" of this question is supposed to be because you seem to misuse a lot of technical terminology. In particular, I don't know what you mean by "getting an indirect measurement" of $O_A(t_0)$ or $O_A(t_1)$. Let's say you have some entangled state $\chi = \sum_{i,j} c_{ij}\phi_i\otimes \psi_j$ for $\phi_i$ an eigenbasis for $O_A$ and $\psi_j$ some basis of $B$, and it evolves to $\chi' = \sum_{i,j} c'_{ij}\phi_i\otimes\psi_j$. You measure $O_A$, i.e. project this state onto one particular $\phi_k$. What's the question about the resultant state?

Is there a chat room to go to?

@David There is now ;)

If you need to activate TeX rendering for chat, go here:

@ acuriousmind I asked the question because I was confused about something and that confusion may show in the wording. Howver, I don't think the question, once it is clear enough, is bad and does not deserve a (-) vote, in my view.

I think the answer is to also include the self hamiltinian of system a, not just the interaction hamiltonian that entangles systems a and b. Theae (-) votes for questions that are not obviously rediculous are what drives people from this site.

@David We've had the discussions about downvoting a million times on Physics Meta, my own position is most clearly here, I'm not going to rehash those discussions here.

I don't think tbis question fits yiur downvote criteria. I did research tgis and put effort into it. Anyway, as I said, I think the composite WF will evolve correctly wothout the wjrstion I Ask if I inckude the self-hamiltonians.

@David I still don't actually understand what your question is, or why you are now talking about Hamiltonians when you did not mention them in your question at all. You have an entangled state, you measure part of it. What exactly do you want to know about that?

Did you downvote me twice?

@David That is not possible (and you do not even know that I cast any of those two votes). Apparently at least one other person agrees with me that it is quite unclear what you are trying to ask, and I don't understand why you seem more interested in the votes on your question than in improving it so that people understand what you are asking and can answer it.