How does the Pauli exlusion principle work?

[Cibek]

Hello!

From what I have understood, there are two different states that an electron can have (Spin up and spin down), and if two electrons are in the same state their wavefunction collapse. So far so good. In a video I saw, they claimed that because of this, only two electrons can exist in the same shell, because after that they need to jump to the next shell, or energy level. I get the idea of it, but I've been taught in school that the shell closest to the nucleus (K-shell) only has two electrons, but that the rest of the shells have eight. I don't doubt this because that's what the periodic table of the elements is built up from, but the thing I don't understand is:
How can more than two electrons be in the same shell and therefore in the same state?

 

[PeterDonis]

For a more detailed answer, you might want to post this question in the Atomic Physics forum. But here's a quick and dirty response:

The Pauli exclusion principle, applied to electrons bound in an atom, says that no two electrons can have all of their quantum numbers the same. Spin is one quantum number, which can take on only two values (up or down), so a pair of electrons with *all* other quantum numbers the same must have opposite spins. That's where the rule that "only two electrons can exist in the same shell" comes from.

However, the word "shell" is not really correct in the rule as I just stated it above, because a "shell" does not necessarily refer to a single set of values for all of the quantum numbers except spin. That's only true in the lowest shell, the K shell (also called a 1s orbital). In higher shells, there can be more than one set of values for the other quantum numbers in the same "shell", so there can be more than two states for electrons to occupy. There are indeed eight such states in the second "shell", but in higher shells there are more. The following Wikipedia page gives more info:

http://en.wikipedia.org/wiki/Quantum_number

 

[Cibek]

For a more detailed answer, you might want to post this question in the Atomic Physics forum. But here's a quick and dirty response:

The Pauli exclusion principle, applied to electrons bound in an atom, says that no two electrons can have all of their quantum numbers the same. Spin is one quantum number, which can take on only two values (up or down), so a pair of electrons with *all* other quantum numbers the same must have opposite spins. That's where the rule that "only two electrons can exist in the same shell" comes from.

However, the word "shell" is not really correct in the rule as I just stated it above, because a "shell" does not necessarily refer to a single set of values for all of the quantum numbers except spin. That's only true in the lowest shell, the K shell (also called a 1s orbital). In higher shells, there can be more than one set of values for the other quantum numbers in the same "shell", so there can be more than two states for electrons to occupy. There are indeed eight such states in the second "shell", but in higher shells there are more. The following Wikipedia page gives more info:

http://en.wikipedia.org/wiki/Quantum_number

Thanks for the reply! What are all the quantum numbers? Is position and momentum part of those? And if so, is that why the K-shell only has two electrons, because it is so tight that they have the same position value (or quantum number) and therefore only can have two "ways to be different", and that being spin? Maybe I'm totally wrong, I'm just speculating. :P

 

[Bill_K]

Cibek, I believe if you take a look at the Wikipedia page that PeterDonis referred to, it will tell you exactly what quantum numbers.

 

[JamesOrland]

From what I have understood, there are two different states that an electron can have (Spin up and spin down), and if two electrons are in the same state their wavefunction collapse.

No, that's not what wavefunction collapse means at all. See here for my explanation of it (and if anyone has a better one, please do post it there).

Two electrons simply cannot be in the same state at the same time. There is no 'if they are.' It's physically impossible. The exclusion principle says exactly this, that it is simply impossible for two electrons to occupy the same state at the same time.

And as was explained above, the only shell that shares all quantum numbers except for spin is the first one. All the other ones have 'subshells,' and the electrons can occupy that. Each 'subshell' represents a different set of quantum numbers, and can house exactly two electrons, one with spin up and another with spin down.

Furthermore, to answer your question just a little bit (do refer to the Wikipedia post), no, position and momentum are not quantum numbers.