How far does the Pauli Exclusion Principle apply?

[Sojourner01]

We know that two particles can't exist in identical quantum states in the same place, fair enough.

However, no particle can be sitting directly on top of another to infinite precision. Therefore, you can always say they're some minimum distance away from one another.

Now suppose you have two atoms sitting in a crystal lattice next to one another. One has its electrons sitting around doing their thing, and so does another, so they're to all intents and purposes identical.

Identical. Isn't this forbidden? The two atoms have electrons sitting in the same quantum states in their electron clouds. They're some small distance away from one another as before, so what's changed? What's fundamentally different about the distance between two electrons in one atom's electron cloud - which forbids identical quantum states - and the distance between electrons in neighbouring clouds?

 

[Mr Virtual]

The principal quantum number (n) defines where an electron is, with respect to the nucleus. Suppose we have two hydrogen atoms, A and B. Though both the electrons lie in K shell ( and 1s orbital), the electron of A lies in K shell with respect to A, not B. The electron of B does not lie in K shell of *A*. Similarly, the electron of A does not lie in K shell of B. Pauli exclusion principle comes into play only when the principal, azimuthal and magnetic quantum numbers of the same atom are same. Here, the first and most important quantum number- the principal quantum number, is not same for both the electrons. Pauli principle is used keeping in mind the nucleus we are taking into consideration. That is, with respect to the nucleus taken into account, no two electrons can have same four numbers. Every atom has its own set of quantum numbers.
I hope my post helped. If I am wrong, please correct me.

regards
Mr V

 

[Mr Virtual]

You can also understand it as follows:
There are two electrons, both having identical set of Q-numbers, but in different atoms. Each electron has its own probability density cloud. Pauli exclusion principle prohibits the two probability clouds to coincide with each other. And the principle Q-number is the one that makes sure that the two clouds have different location in space.

 

[olgranpappy]

If I am wrong, please correct me.

you are wrong. two electrons and two protons are not equivalent to two hydrogren atoms and the wavefunctions of the two-electron system can not simply be labelled at products of independent hydrogen-like wavefunctions. There is a huge body of work on the specific topic of the *hydrogen molecule* (i.e. H_2) which is exactly what you are discussing and is very much different that just two non-interacting hydrogen atoms

 

[Mr Virtual]

Can you please elaborate (in non-mathematical terms) what you are saying? Where am I wrong?

Mr V

 

[olgranpappy]

Sure. In the case of hydrogen there is no "electron-electron" interaction. The electron interacts with the proton which is assumed to be so much more massive than the electron that is can be taken as a fixed point charge. Thus in the hydrogen problem we simply solve the single-particle Schrödinger equation for a particle in a central potential.

When you treat H_2 you have the potential due to one proton and the potential due to the other proton that both electrons "see" and if this was the case then you really would be able to build up the H_2 solution from hydrogen-like solutions... but there is another interaction, the electron-electron interaction, that you must now also take into account which is what makes this problem much much harder than the hydrogen atom.

 

[Sojourner01]

What I am getting at is that the potential of the two atoms (say) is merely a periodically varying scalar field, correct? You don't 'have' to have point charges, for the sake of the mathematics, to create an electric potential, you can just say it's 'there' as the Hamiltonian doesn't care about the nature of the object creating the field.

Furthermore, the quantum numbers are merely a label for the states of the electrons - they're not physical things, merely a convenience for physicists.

I digress though - I actually had a more sensible example of what I mean which doesn't have such a blindingly obvious answer, but I'm afraid I don't recall it anymore. Obviously, for the hydrogen molecule, what the principle is saying is that probability distributions cannot precisely overlap at every point in space. Fair enough.