My question may seem quite esoteric given the title, but I think it's relatively straightforward when explained properly. Imagine a relatively simple situation of 2 hydrogen atoms (numbered 1 and 2), which we treat semi classically (first quantization QM). If we look at each atom individually and ignore the other one, we have a clear cut analytic solution that describes the electron orbital. On the other hand, if we consider the impact that proton 2 has on electron 1 (and vice versa), we need perturbation theory. Correct me if I am mistaken, but I believe we will end up with an oscillating solution whereas the electrons will trade places over time. If this is indeed the case, then these electrons' wave function will be greatly correlated at some point (in particular when it is equally likely for either of them to be orbiting proton 1 or 2). Here then are my questions :
1) Despite the approximate nature of the perturbation theory solutions, is it fair to say that such significant entanglement, or at least some reasonable measure of it, is likely ubiquitous to the exact description of such systems ? Is the back and forth "switch" from proton 1 to proton 2 for electron 1 (and vice versa) also a ubiquitous behavior ?
2) If this is the case, and if the timescale of such oscillations for typical separation size between proton 1 and 2 (say the distance between protons in the air) is much smaller than say the age of the earth's atmosphere, wouldn't it lead to all the electrons of the earth's atmosphere to be entangled in a giant mess resembling some nightmarish correlation web ? If so, shouldn't such entanglements be readily observable ? I limited the case to the earth's atmosphere almost as a gimmick to help visualize my question, though obviously the conclusion (if correct) would apply on global scales.
3) Does a QFT theoretical description substantially change the conclusions arrived at above ? My understanding of QFT is shaky at best, and breaks down completely when I try to apply it to this question.