As a non physicist, non scientist, I have a question, which may be really stupid. If quantum mechanics expounds that at an atomic level matter can be in 2 places, at one point in time, does this matter have mass in these 2 different places? If this matter can have a mass in more than one place, at one point in time, how can we attempt to calculate the mass of matter present in the universe as surely it would depend on what proportion of matter was in what number of places at any point in time? Does that mean its unit of measurement would need to include number of atoms, the proportions of this matter in what numbers of places, at a fixed point in time? Is there some basic reading that might help me understand this a bit more? Thank you.

It's not a stupid question. The way that popular accounts "explain" quantum mechanics leads naturally to your question. The moral is that those popular accounts are not to be trusted.

Quantum mechanics is unusual in that on the one hand, we understand very well how to apply it and what we should expect to find in experiments if it's correct, but on the other hand there is sharp disagreement over what quantum mechanics is telling us about the nature of the things we use it to predict and explain. The problem you're raising comes from the superposition principle. A quantum system can be in a superposition of being in two different, non-overlapping places, for example. When that happens, there's some probability that if we "look" (make an appropriate measurement) we'll find the system in one of the places, and some probability that we'll find it in the other. However, we can't understand this as a simple case of ignorance --- as a case where the system really is in one place or really is in the other and we simply happen not to know which until we look. Quantum mechanics allows this latter sort of case, but if we perform other experiments, we find measurable differences between the superposition case and the mere ignorance case. It's superposition that leads people to talk as though the system really is in two places at once, but the math doesn't force any such thing on us, and if we thought it did we'd be left with questions like the one you ask.

What quantum mechanics unarguably does is give us probabilities for experimental results. (I'm not saying that's all it does, but it at least does that much.) Those probabilities have some surprising features. They lead to experimental predictions that we can't squeeze out of classical theories or, perhaps more accurately, can mimic classically only if we make other assumptions that depart from our classical picture. The superposition principle is the source of the differences, and is what provokes people to talk in the ways that lead to your question. Unfortunately there's no way to set forth the superposition principle in a couple of sentences, but perhaps this will do for now: the mathematical basis for quantum probability is closer to adding the amplitudes of waves (which can interfere with one another) than it is to counting how often things happen. The lab data gives us counts (we find a particle "here" some percent of the time and "there" some other percent), but the counts come from talking amplitudes, which can be positive, negative or imaginary, and squaring. The probabilities don't come from garden-variety ignorance of exactly what's going on at the micro-level, and they also don't come from assuming that particles can be in many places at once.

At this point you're probably finding this maddeningly abstract and hard to picture. That's because it is. In fact, some of the earliest debates about the foundations of quantum mechanics were about this very issue of whether there's a way to "picture" the quantum world. Those debates haven't simply gone away, but we don't need to resolve them to take quantum theory seriously. Physicists learn how to think about quantum systems by learning how to connect the quantum math with what goes on in the lab. In order to do that, they don't need to think of superposition as a matter of contradictory things being true at once. Popularizers may think they do their readers a service when they talk about one thing being in two places at once, but my own sense is that in the long run this makes for more confusion than enlightenment.

As for basic readings, here are a couple of suggestions. One is a short little book by Valerio Scarani, called Quantum Physics: A First Encounter. The other is longer but still approachable. It's by David Z. Albert, and it's called Quantum Mechanics and Experience. Best of luck!

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