What exactly is meant by a quantum state in QM?

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?

• Comments are not for extended discussion; this conversation has been moved to chat. – ACuriousMind Mar 14 '17 at 13:53

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).

I think that your problem is semantics, as the phrase

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?

suggests.

First, it is not true that a wave nature makes the position infinitely different. It is not a wave in spacetime first of all; and second, it is a probability amplitude. It gives you information about the likelihood of finding in particle in $dx$ region of space.

Second, a quantum state contains all information about the particle (or field) in qustion. The sentence is a combination of all properties of the quantum object makes no sense.

I emphasize: the fact that a state has all information of the system is not the same as to say that the system has all information that it can have. You are mixing these two notions.

Perhaps I can clarify this statement with an example.

If you consider a free particle, a definite mometum state is represented by $| k \rangle$. This means that

$$P | k \rangle = \, k \, | k \rangle$$

where $P$ is the momentum operator and $k$ is some real number. Notice, $| k \rangle$ is a vector, $P$ is an operator and $k$ is a number. There is no notion of all momenta available. It is one definite momentum $k$.

You may ask, however, what are its properties with respect to position. To do this, you must expand $| k \rangle$ in the position basis, that is, the basis where all vectors have definite positions. Schematically,

$$| k \rangle = \int c(x) | x \rangle.$$

And it may happen that some $c(x)$ are zero, so, again, there is no notion of all different positions. In this case, it is unfortunate that no $c(x)$ vanishes. But I think you get the logic.