Help needed to understand whether multiple fermions can occupy same physical space or not

Question

As per my understanding:

1. Multiple fermions cannot have the same quantum state (as per Pauli exclusion principle)
2. Multiple fermions can occupy the same physical space as long as they have different quantum states (or numbers or properties such as spin)

If both these statements are true then, part of the second statement "as long as they have different quantum states (or numbers or properties such as spin)" doesn't become necessary. Because first statement implies that "multiple fermions always have different quantum states". Hence, the second statement simply becomes "Multiple fermions can always occupy the same physical space" (For a moment let's consider only fermions, their quantum state and physical space they occupy. And not other factors like electromagnetic repulsion etc)

However, at multiple places on the Internet it has been stated (and seems like widely accepted) that: Multiple fermions cannot occupy the same physical space as per Pauli exclusion principle, and that is why matter structures exists in the universe.

Can someone please help me trying to figure out where am I making mistake?

Answer 1

Both of these statements are technically true, though (2) is basically redundant.

The Pauli Exclusion Principle has been proven using advanced techniques. What it says is that fermions cannot have the same quantum numbers; it doesn't necessarily mean that they 'cannot occupy the same physical space'. There's more to a quantum system than simply position.

One example of this is the electron shell in an atom. Take the helium atom, for example, which has both electrons in the 1s subshell. In fact, these electrons have three of their four quantum numbers the same - the only difference is their spin.

But does this mean that matter has no structure? No - and that is because of the discrete property of quantum energy levels. There are only two possible spins for an electron, so a maximum of two electrons can occupy the same 'physical space'. As long as the fermions cannot bunch together without limit (like bosons), there must be fermions occupying different positions, and this gives rise to he structure of matter (what we used to learn as 'matter takes up volume').

Answer 2

Your first two statements are correct.

The third statement you quote,

Multiple fermions cannot occupy the same physical space as per Pauli exclusion principle, and that is why matter structures exists in the universe.

Isn't terribly wrong as a first approach to the exclusion principle at the level of science comunication to the general public. But it is not really correct in the details. If you see advanced enough to understand how the Pauli principle relates to quantum states, then you can completely discard the formulation in the third statement and move on.

(Though, that said, maybe it's relevant to mention that the first two formulations are still oversimplified and short of the full story, which is to do with the effect on the wavefunction of exchanging two indistinguishable particles.)

Answer 3

The true statement of Pauli says something quite different: Any pure $n$-electron state is represented by a wave function of $n$ 3-d space variables, that is completely antisymmetric on all $n$ entries.

For two electrons it reads: Any state function has symmetry

$$\psi(x_1,x_2) = - \psi(x_2,x_1)$$.

That means for 2 free electrons, that product states

$$\psi(x_1,x_2) = a(x_1) b(x_2)$$

with any $a,b$ are not allowed, only antisymmetric pairs are possible

$$\psi(x_1,x_2) = a(x_1) b(x_2)-b(x_1) a(x_2)$$.

States with any two equal factors do not exist.

This statement still seems to assume, that the electron coordinates can be numbered. But this is false categorically for identical particles. Enumeration is a mathematical vehicle only in the same sense as enumeration of the base in a vector space. Any observable entity is invariant with respect to permutation of the particle indices.

Now is coming in some fundmentals about Hilbert spaces: All Hilbert spaces with denumerable basis are isomorphic. Even states with denumerable particles have a state space, that is isomorphic to the simple Hilbert space of the one-dimensional harmonic oscillator.

By this simple statement, any n-particle state can be expanded in the product of 3n identical Hilbert spaces, eg of the oscillator, that has to be completely antisymmetric with respect to the permutation of the n groups of three coordinates belonging to one particle.

This amounts to saying: in expansions in any basis of one-particle Hilbert spaces no two factors can be equal.

What does it say with respect to the occupation of space? Two electrons far away from each other have orthogonal states by inspection. Symmetry does not play any role (fortunately, as the QM-fathers mentioned, we must not antsymmetrize with alle the electrons behind the moon).

If they are near to eachother in uits of the fundamental Compton wave length, the two state factors have to be orthogonal in Hilbert space. That means eg. in a box different wavelengths. And that means that adding another electron it has to be inserted into a higher energy state. The energy can be lowered by enlarging the box. In this sense the Pauli principle makes solid bodies with a definite volume for its light fermionic particles at low temperatures. The heavy ions in molecules and solids have much shorter Compton wave lenght and are always far apart from eachother, $H_+$-ions as an exception.