Trying to understand why electrons fill orbitals the way they do

[physicurious]

TL;DR Summary

Trying to physics

Ok, I'm a little hesitant to post in the quantum physics forum here because I don't want anyone to make the mistake that I know much about quantum physics

But anyways I've been reading some bits lately about electricity and magnetism trying to understand why permanent magnets work the way they do, why conductors are conductors, what reactions are happening in batteries, etc

This lead me to looking at different elements, their electron configurations, and what kinds of spin electrons of different elements have in their outermost shells.

And that got me thinking - why do electrons first fill orbitals 1 at a time with all electrons having the same spin, then pair with opposite spins?

I remember back in my college chemistry course it was really briefly explained by the pauli exclusion principle - simply that no two things can have the same quantum state. So two electrons together must then have opposite spin.

Ok sure! But wait! This really doesn't explain anything. It's just a rule, or perhaps an observation. Thinking about it more I'm feeling like there must be a better conceptual reason for why this happens.

I'm thinking that the elctron's spin will be accompanied by a magnetic field and this is probably why two electrons are always paired with opposite spins. I wonder if they had the same spin then would they be repelled like trying to press together two north poles of a magnet? Although if I think about this a bit more then it seems like if they did have the same spin that would create a magnetic field that would attract the both of them together.

I dont know am I on the right track here? Right direction, wrong track? Feeling like there's something here but just not quite grasping it.

 

[PeroK]

And that got me thinking - why do electrons first fill orbitals 1 at a time with all electrons having the same spin, then pair with opposite spins?

It's more accurate to say that the electrons form a system of indistinguishable fermions. To which the PEP (Pauli Exclusion Principle) applies.

Heuristically, this means no two electrons can have the same overall quantum state, which is defined by four quantum numbers : principal/energy level ($n$), orbital angular momentum ($l$), magnetic ($m$) and spin ($s$).

Moreover, there is a restriction on the values of $l$ and $m$ corresponding to the energy level $n$, which limits the number of electrons at each energy level.

For example, for the lowest energy level ($n = 1$), both $l$ and $m$ must be zero. That allows only two electrons: one with each spin.

There is lots of university level material about this online.

Ok sure! But wait! This really doesn't explain anything. It's just a rule, or perhaps an observation. Thinking about it more I'm feeling like there must be a better conceptual reason for why this happens.

The PEP can be derived from Quantum Field Theory. But, in basic QM it is an axiom.

I dont know am I on the right track here? Right direction, wrong track? Feeling like there's something here but just not quite grasping it.

The wrong track. You need QFT to explain the PEP any further.

 

[physicurious]

Well at least I know now where the conceptual understanding for the Pauli exclusion principle would be found. Not nearly as satisfying as the answer I was hoping for. Can't help but think of that Oscar Wilde quote "The pure and simple truth is rarely pure and never simple"

 

[WernerQH]

why do electrons first fill orbitals 1 at a time with all electrons having the same spin, then pair with opposite spins?

It is called Hund's rule. (Just looked it up and discovered that there are actually three Hund's rules, this being the first one, see for example Gordon Baym: "Lectures on Quantum Mechanics", pp 456ff.) When the spin-part of the multi-electron wave function is symmetric, the spatial part must be anti-symmetric under exchange of two electrons. Because the Coulomb interaction is much stronger than the magnetic interaction between the spins, it is energetically favourable that the anti-symmetry of the spatial wave function tends to keep the electrons farther apart.

 

[maximdogger]

on a similar vein of thought..does the electrons wave form need to meet and "mesh" neatly so that a whole number of sine waves are around its orbits circumference?

 

[PeterDonis]

I've been reading some bits lately about electricity and magnetism

Where? Please give specific references.

 

[PeterDonis]

on a similar vein of thought..does the electrons wave form need to meet and "mesh" neatly so that a whole number of sine waves are around its orbits circumference?

The electron waves aren't sine waves and there aren't any orbits. (These were features of Bohr's 1913 model of the hydrogen atom, but that model has long since been discarded.) Electron orbitals are 3-dimensional probability distributions.

 

[maximdogger]

i am trying to understand the harmonics. much easier if electron waves "join" on the ends, and are a whole number in a orbit.

 

[PeterDonis]

i am trying to understand the harmonics.

The relevant harmonics are spherical harmonics, not sine waves.

much easier if electron waves "join" on the ends, and are a whole number in a orbit.

Again, there are no orbits and the mental picture you have of sine waves "joining" at the ends of an orbit is the wrong one.

 

[vanhees71]

The electron waves aren't sine waves and there aren't any orbits. (These were features of Bohr's 1913 model of the hydrogen atom, but that model has long since been discarded.) Electron orbitals are 3-dimensional probability distributions.

In atoms with more than 1 electron orbitals are $3N \times 2$-dimensional probability distributions.

 

[Ken G]

Ok sure! But wait! This really doesn't explain anything. It's just a rule, or perhaps an observation. Thinking about it more I'm feeling like there must be a better conceptual reason for why this happens.

Good for you, there always has to be a conceptual reason, usually multiple versions!

I'm thinking that the elctron's spin will be accompanied by a magnetic field and this is probably why two electrons are always paired with opposite spins.

We know magnetic field cannot be the issue, since it connects with charge but we have the same principle for uncharged fermions, and we do not have the principle for integer spins. It has to be the half integer spin that does it.

I wonder if they had the same spin then would they be repelled like trying to press together two north poles of a magnet? Although if I think about this a bit more then it seems like if they did have the same spin that would create a magnetic field that would attract the both of them together.

Don't look for any force at all, the PEP is not about some kind of "repulsion" between particles. One way to look at it is a kind of interference phenomenon. You have probably heard a lot about how particles must follow the rules of waves, and the main rule is that things that destructively interfere are prevented from happening. The PEP is like that, it is a prevention from happening that the two indistinguishable half integer spin particles could be in the same state. Integer spins don't interfere like that, half integer does. The great interpreter Richard Feynman once described it as being like what happens if you grab a coffee cup by its handle and twist your arm around a full twist, you can return the coffee cup to its original configuration but something is different, your arm is twisted. The cup has some kind of twisted connection to its environment, that is like what happens for half integer spin particles. So it's not their charge, it's that weird kind of twisted connection when two such particles are interchanged, this somehow causes the destructive interference that disallows them from being in the same state in the first place.

Feeling like there's something here but just not quite grasping it.

It's nothing we're used to, as it doesn't happen on the scales of our experience. It's closer to things that mathematicians have envisioned, symmetry principles and the like. But then, why should the "quantum world" seem normal to us?