Relation of partial charge with wavefunction

Suppose we have an one electron system. The probability $P$ that electron's position is in some interval $x + dx$ is given by:


$$P = \int\_x^{x+dx}|Ψ(x)|^2 dx$$


So if we measure $N$ times the electron's position the number of times that is in the interval $x+dx$ is $n$ and it holds that:


$$P = \frac{n}{N}$$


Lets move now into a polar molecule e.g. $\ce{HCl}$. In order to find the partial charges we must know the wavefunction of the bonding molecular orbital. Based on electronegativity differences we should expect $\ce{Cl}$ to have a negative partial charge and $\ce{H}$ to have a positive charge. (I will work with one dimension for sake of simplicity).


The value of $|Ψ(x)|^2$ should be higher in the region of $\ce{Cl}$ and lower in $\ce{H}$. I was thinking that this partial charge could be explained by "cutting" the distance between the two nuclei and then integrating $-e|Ψ(x)|^2$ at the new seperated spaces. So the partial charge at $\ce{Cl}$ would be:


$$δ^- = \int\_{x\_0}^{\infty}-e|Ψ(x)|^2dx$$


where $x\_0$ is defined as the point of seperation of two nuclei. In the same way we can find the partial charge of $\ce{H}$. The problem is that I can't interpret it. I mean $-e|Ψ(x)|^2$ gives the probability of finding a charge of size $-e$ at a given region (in analogy of finding electron at a given position).


**Edit**


First I would like to apologize for being unclear. Introducing time was a bad idea. What comes to my mind is "Shut up and calculate". In QM all physical quantities are calculated with expectation values so the same must holds for partial charges (dipole moment more correctly). Now why that must be the case (expectation values) may be related to Ergodic hypothesis as Ian pointed out in the comments.


I don't know if that is correct so I can write it as an answer. The question is about how assymetric wafevunction translates into partial charges.


