Brandon, the simple truth is that you have just asked one of the hardest and least understood questions in all of physics. So, don't feel bad if you don't understand it very well, because, er... no one else really does either?
It's not that we can't model this stuff mathematically. Shoot, Richard Feynman's version of something called Quantum Electrodynamics (QED), which is sort of quantum mechanics merged with Einstein's theory of special relativity, is arguably the most accurately predictive theory in all of physics. (Or was; I haven't kept track lately.) The problem is that whenever we use such precise theories, we can't help but toss in a bit of everyday life in the mix, sort of like a salad in which we mix things more by taste than by precise rules.
So, for example, Feynman's QED theory is incredibly precise in predicting how an electron in one place and state (e.g., velocity) gets to some other place and state. However, to set up the electron in a real experiment -- to create the location and state you are describing in the QED problem setup -- you must use real-world equipment. And that is the fly in the ointment (or is it the secret ingredient in that salad?): The real-world setup for any physics problem is unavoidably embedded at some points in everyday physics concepts like "ordinary," or irreversible time. Once you toss something like ordinary time into the mix, all the nicely reversible properties of time at the atomic scale no longer apply, at least not for the experiment as a whole. Or stated a bit differently: Everyday physics seems to beget more everyday physics. That is the flaw you will find at some level in every single experiment looking at the physics of very tiny scales. It has to be that way, since otherwise how would we as large-scale creatures every find know about the result in the first place?
So, as the amazing physicist John Bell once said while mulling over pretty much the same question you just asked (he could never really answer it; that's how hard it is!), folks who do experimental physicists just sort of develop a "feel" for when you stop applying quantum physics and start applying everyday (or "classical") physics. Time is a very big part of the transition: If time is reversible, it's almost certainly quantum, and if it's not, it's probably better treated as everyday (or classical). Size is less reliable, but for most phenomena at ordinary temperatures, classical physics starts to kick in at roughly the size of a medium-sized molecule, say a buckyball. That metric is very unreliable overall, though, since things as ordinary as a reflection off of a piece of silver are deeply quantum events that cannot be modeled using only classical physics. Shoot, size is a deeply quantum phenomenon, and so is chemistry. Without quantum mechanics stepping in, we'd just be part of some huge big black whole, and so would not be having this conversation.
I'll end by recommending a book: Richard Feynman's "QED: The Strange Theory of Light and Matter." It's paperback, cheap, uses almost no math, yet provides profound and accurate insights into that very precise quantum theory I mentioned above. I won't say it will answer your question, but at least it will present the remarkably non-intuitive features of quantum mechanics about as sharply and starkly as possible.