Exploring Solids: Classical Physics & Quantum Mechanics

[arupel]

TL;DR Summary

The distance between atoms is huge. The question was that with so much space between them, why don,t solids simply penetrate each other. The answer is electricall repulsion of the outer orbital electrons. The inner orbital electrons act as a foundation. I would like to go further in this vein with a guess how this can work at a basic quantum mechanical level.

The question will be formulated as a hypothetical question in classical physics. I will offer what I think is an answer using basic quantum mechanics.

The classical question: Since it is the electrical repulsive force on the outer orbital elections, even with the inner orbital electrons serving as a foundation, then it would still seem that the orbits of the outer orbital electrons as well as the inner electrons would be distorted by the repulsive force.

They are not. Using the Bohr atom as the model, Any repulsive electrical force would push against the outer orbital electron. Being pushed back it cannot go between its outer orbit and the next quantum level. It must 'jump" to the next lower quantum level.

This would results in 3 electrons at this next lower level with two with the same spin. By the Pauli Exclusion principle, 2 fermions cannot occupy the same quantum state (please excuse the spelling). The outer orbital electron must remain at the outer orbit.

As a consequence of these two factors the orbits of the electrons are not distorted though the orbital electrons suffer the repulsive electrical force.

This leads to a second question. A quantum level with two electrons (spin up and spin down) does not violate the Pauli Exclusion principle. The second question is a classical one. Granted that the Pauli Exclusion principle is not violated, these two electrons are basically in the same orbit. The repulsive force between them must be severe. The only answer I can think of that in quantum mechanics there is no repulsive force between them. Then why not?

A third question. Can the the two electrons in the same orbit be considered to be in a single quantum entangled state?

I would appreciate any corrections to the distortions in my thinking.

Thanks,
Arthur Rupel

 

[PeterDonis]

The question will be formulated as a hypothetical question in classical physics.

You can't formulate the question in classical physics, because in classical physics atoms are not stable: the electrons would spiral into the protons and release huge amounts of EM radiation.

The classic treatment of this subject is Dyson & Lenard, 1967:

https://aip.scitation.org/doi/pdf/10.1063/1.1705209
As the paper shows, while the EM interaction plays a role, the primary factor that ensures the stability of solid objects is the Pauli exclusion principle. Note that their analysis does not just take into account the electrons in atoms, but the nuclei as well.

A quantum level with two electrons (spin up and spin down) does not violate the Pauli Exclusion principle. The second question is a classical one. Granted that the Pauli Exclusion principle is not violated, these two electrons are basically in the same orbit. The repulsive force between them must be severe. The only answer I can think of that in quantum mechanics there is no repulsive force between them. Then why not?

This is not a classical question; the Pauli exclusion principle is not classical, nor are electrons in "orbits". They are in quantum states that are called "orbitals" but are nothing like classical orbits. There is still repulsive force between the electrons, so your suggested quantum answer is wrong. But the repulsive force between them is not the only force acting; there is also the attractive force of the nucleus, and there is an attractive magnetic force between the electrons because of their opposite spins.

Can the the two electrons in the same orbit be considered to be in a single quantum entangled state?

Yes. In fact, all of the electrons in an atom, even ones in different orbitals, are in a single quantum entangled state, because the electrons are indistinguishable. For example, in a lithium atom in its ground state, there are two electrons in the 1s orbital and one in the 2s orbital. But there is no way to pick out two of the three electrons and say that those two are in the 1s orbital and entangled, while the third one is in the 2s orbital and is separable from them. The correct quantum state of the electrons is a three-electron entangled state. (Note that this leaves out the nucleus; strictly speaking the electrons are also entangled with the nucleus, but for many purposes this can be ignored.)

 

[PeterDonis]

It's also worth noting that for individual atoms, the key factor ensuring stability is not the Pauli exclusion principle (which, if you think about it, can't be all there is to it because the hydrogen atom has only one electron and it is stable), but the uncertainty principle: heuristically, as you make the atom smaller, you make the electron's momentum more uncertain, so its average kinetic energy increases, which sets a lower bound on the total energy of the atom, because the potential energy decreases as the electron gets closer to the nucleus, so there will be some particular state in which the total energy is minimized, and that state will be of a particular size. (I say "heuristically" because there are quite a few technicalities involved, and the original Heisenberg uncertainty principle turns out not to be the right formulation for this problem.)

A good discussion is here:

https://arxiv.org/pdf/1111.0170.pdf

 

[arupel]

Thanks for the information. It is interesting that to note solids do not penetrate
each other requires a more detailed explanation than "electrical repulsion."
In a sense this is otherwise cheating students with an over simplified response.Just a note. I already knew the "clitch" classical physics had in handling rotating electrons about an atom. The history how this evolved into the wave equation and probability is a detailed and interesting one.