I’ve been reading a lot about massive stellar objects, degenerate matter, and how the Pauli exclusion principle works at that scale. One thing I don’t understand is what it means for two particles to occupy the same quantum state, or what a quantum state really is.
My background in computers probably isn’t helping either. When I think of what “state” means, I imagine a class or a structure. It has a spin field, an energy_level field, and whatever else is required by the model. Two such instances would be indistinguishable if all of their properties were equal. Is this in any way relevant to what a quantum state is, or should I completely abandon this idea?
How many properties does it take to describe, for example, an electron? What kind of precision does it take to tell whether the two states are identical?
Is it even possible to explain it in an intuitive manner?
In classical statistical theory, manipulating a probabilistic state is equivalent to picking a single initial state with whatever probability, and then manipulating it. In quantum statistics it’s provably not (at least if we’re measuring particles with as much free will as we think); you need the whole thing for it to make sense. Two likely trajectories can interfere and cancel out, for example.
So, sure, a position is a vector. But we can only meaningfully talk about functions from a (measurable) set of vectors to their probability amplitude (which is like a probability, but complex). Or, in practice, the infinitesimal density of probability amplitude at that given point
The uncertainty principal is just one manifestation of that. And, like in the uncertainty principal, entanglement might not stay confined to just position if there’s other parameters, so you really have to talk about functions on the whole state vector. I can’t speak too much to quantum field theory, but the actual dynamics of basic quantum physics is about (very “basic”) functions on those functions.