How to mathematically write down a probability of measuring a particle

[fluidistic]

Consider a system of 2 identical electrons that are confined in a region so that there is a single wavefunction describing the whole system. In several textbooks one can read that the probability to measure the position of an electron in region near $r_1$ and the other in a region near $r_2$ as $|\Psi(r_1, r_2, t)|^2dV_1dV_2$. However I have never, ever seen written the probability of measuring the position of only one of those electrons. Is it because it is impossible to make a measurement on a single of these electrons? If not, then how would I write down the probability to measure of one these electrons near $r_1$ but without measuring the position of the other electron?

 

[PeterDonis]

I have never, ever seen written the probability of measuring the position of only one of those electrons. Is it because it is impossible to make a measurement on a single of these electrons?

The electrons are indistinguishable, so there is no such thing as "measure the position of electron #1" vs. "measure the position of electron #2". But there would be a valid operation of "measure the position of one electron".

how would I write down the probability to measure of one these electrons near $r_1$ but without measuring the position of the other electron?

You would have to average over all of the possible positions of the other electron to obtain an effective "wave function" for the one-electron position measurement, and then take the squared modulus of that. (I put "wave function" in quotes because the thing you get from this process is not quite the same as an ordinary wave function for a single particle.) Schematically, you would have something like $| \psi(r_1, t) |^2$, where

$$
\psi(r_1, t) =\int \Psi(r_1, r_2, t) dr_2
$$

where the range of integration would be over all possible values for $r_2$.