From what I have read, a quantum state of a quantum object contains all the properties of the quantum object. But I have read that the Pauli Exclusion principle states that two identical particle in a system cannot have the same quantum state simultaneously.
But since the quantum state is the combination of all the properties of the quantum object and that due to a quantum object's wave nature its position can be infinitely different, why does the PEP work? Is a quantum state really represented by a collection of variables or is it represented by one fundamental property? If is a collection of variables, then the simple change of position can make the quantum object unique, why can't that work? If its a single variable, then which one and why?
Answer
A quantum state is an element of a projectivized complex Hilbert space. A quantum object has (at any given moment) one state, and that state is a complete description of the object.
In practice, we often simplify matters by choosing to ignore certain observables in order to be able to work with a more manageable Hilbert space. So if you're interested in position or momentum and willing to ignore spin, you might take your Hilbert space to consist of square integrable functions from ${\mathbb R}^3$ to ${\mathbb C}^1$. If you're interested only in spin, you might get away with something finite dimensional. But those are clearly understood to be approximations. The basic setup of the theory is that if you choose the right Hilbert space, your object has (at any given moment) a state that is a single element of the projectivization of that Hilbert space, and that state fully describes everything about the object.
Of course every element of every Hilbert space can be written as a sum in infinitely many ways. So can, for example, the number 8. This does not make quantum states "infinitely different" from anything, any more than it makes the number 8 "infinitely different" (whatever that might mean).
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