Is Pauli's Exclusion Principle a restatement of what experiments have shown? So were studying the configuration of electrons in an atom and one thing that popped up was Pauli's Exclusion Principle. In our class, as well as our textbook, it was stated as the fact that two electrons in an orbital cannot have the same set of quantum numbers.
For example, if we consider the 4s orbital, then the electrons will have the sets:-

*

*4, 0, 0, +1/2

*4, 0, 0, -1/2

My question - Is this principle an observational one? That is, it is simply a restatement of what experiments have shown, or if not, then what exactly is it? Why can't the electrons have the same set of quantum numbers?
Request - Please explain in the level of a high schooler, if possible.
 A: You have to understand that quantum numbers aren't pertaining to a single electron, but rather the distribution of electron density in an atom.
In 1926 (after Bohr gave us an understanding of the atom that today remains unchallenged and that I will not rehash here), Erwin Schrödinger comes along and states that the behaviour and energy of submicroscopic particles can be described probabilistically by the wave function (ψ (x,y,z)), a mathematical function of spatial coordinates (x,y,z) which is found by solving the equation using Advanced Calculus. (It’s srsly messed up, but gorgeous at the same time. This is the one-dimensional time-dependent equation here, just so you can see it. He also developed it for the Hydrogen atom.)
$\frac{-h^2}{2m} \frac{\partial^2\psi(x)}{\partial m^2}+U(x)\psi(x, t) = i \hbar  \frac{\psi  (x, t)}{\partial t}$
This equation has as many solutions as the atoms have quantum states. <--Very important to recall in a minute and which has direct bearing on the quantum numbers and their physical significance.
The function is analagous to Newton’s Laws of Motion for macroscopic objects. It was unique in that it incorporates both particle behaviour in terms of mass, m, and wave behaviour in terms of the wave function  ψ  (psi, pronounced sigh).
The solutions are indexed by four quantum numbers:  $n, l, m_{l}, m_{s}$
The wave function  ψ  has no physical meaning, but the probability of finding the electron in a certain region in space is proportional to the square of the wave function  $ψ^2$.
The idea of relating  $ψ^2$  to probability stemmed from wave theory, where the probability of finding a photon is where the intensity of the light is the greatest.
This launches a new field called Quantum Mechanics, and while it doesn’t allow us to specify the exact location of an electron in an atom, it does define the region where the electron is most likely to be found at any given time, as well as how it should behave.
In the Bohr model of the H atom, only one quantum number was necessary to describe it: n.
But in quantum mechanics, three numbers are required to (and this is very important) describe the distribution of electron density in an atom. There’s your significance. Again, these quantum numbers are derived from the mathematical solutions to Schrödinger’s Wave Equation for the hydrogen atom.
The numbers and their meanings are as follows:

*

*Principal Quantum Number (n)—this designates the size of the orbital, also called the shell. The larger the value for n, the greater the average distance of an electron in the orbital from the nucleus and therefore the larger the orbital.


*Angular Momentum Quantum Number (l)—this describes the shape of the orbital, but again, you can relate it back to the periodic table. The values of l are integers that depend on the value of the principal quantum number, n. These values of l have letter designations: s, p, d, and f. So if l = 0 we have an s orbital; if l = 1, the orbital is p; if l = 2, the orbital is d, and if l = 3, the orbital is f.
Since this is the shape quantum number, use those letter designations to help you remember the electron density shapes. s is spherical; p looks like a peanut; d reminds me of a daisy, and f is generally harder to predict.


*Magnetic Quantum Number ($m_l$)—this describes the orientation of the orbital in space, or the direction along the axes in which it faces. Within a subshell, the value of this depends on the value of l. For a certain value of l, there are (2l + 1) integral values, or -l, … 0, … +l.


*Electron Spin Quantum Number ($m_s$)—where three quantum numbers are sufficient to describe an atomic orbital, an additional number becomes necessary to describe an electron that occupies the orbital. And for any general orbital, two electrons occupy it. When doing experiments on the emission spectra of hydrogen and sodium atoms, the application of a magnetic field would split the emission line into two lines. This led physicists to conclude that electrons behave like magnets (and remember that magnets have 2 opposing poles). If they’re thought of as spinning on their own axes, like Earth, these magnetic properties can be accounted for. Electromagnetic theory states that a spinning charge generates a magnetic field. For me personally, this is enough to show that electrons do have physical spin.
There’s two possible spin directions that are opposite one another, so therefore $m_s$ can have two values: $+\frac{1}{2}$, or  $−\frac{1}{2}$. Conclusive proof of an electron’s spin was established by German physicists Otto Stern and Walther Gerlach in 1924. But Quantum Mechanics has developed the word "spin" to distinguish the fact that an electron can have 2 different energies. One physics camp suggests that electrons do not have a physical spin, and rather refer to their internal angular momentum. Yet another camp suggests that some experiments have actually detected true, physical spin. Personally, I tend to wonder if being able to actually "see" a physical spin wouldn't come under the topic of Heisenberg. I found a great article you may want to read about scientists who tried "watching" an electron stream, and the very act of watching it caused it to behave differently.
https://www.sciencedaily.com/releases/1998/02/980227055013.htm
Cheers.
A: I'll take a swing at this, from the standpoint of a puzzled high schooler, which I myself was 52 years ago.
Experiments showed that there were orbitals surrounding nuclei which contained not one but two electrons in residence there. This conflicted with the simplest form of quantum mechanical rules which said that two particles with the same quantum numbers could not occupy the same spot in space.
Pauli then invented a new quantum number called spin whereby two electrons could occupy the same orbital by assigning one of them the spin quantum number +1/2 and the other -1/2. Not only did this solve the conceptual problem, but the mathematics behind this move had huge consequences for advancing QM and explaining a bunch of other experimental observations which up until then were a mystery.
So, it would seem to have been a combination of observations and some really coool mathematics which gave us the thing we call spin- which is a bit of a misnomer, because the electron isn't physically "spinning" the way a top or a baseball might.
A: Fermions are antisymmetric wave functions and bosons are symmetric wave functions.
Due to this, in quantum mechanics you can show that two fermion states with the same spin tend to separate their "mean position", while the bosons with same spin tend to "come together" (all of this is more complex since they are actually distributions in space).
This separation for fermions with same spins, means they will not share the same level, since that would imply being "pretty close", so same spin particles go to different levels. Then a fermion with different spin can go there, but again only 1 since the next ones will already see a fermion with same spin there.
A: Identical particles can be$^1$ indistinguishable in quantum mechanics. That implies that the state$^2$ of the system won't change under permutation of particles. In simpler words, you stop caring about which particle is in a specific state and you only care about how many particles are in a state.
It can be proved that the states of a system of identical particles must be either antisymmetric or symmetric under permutation of particles. In the first case, particles are called fermions, in the second case, bosons. These antisymmetric/symmetric states are built from linear combinations of the products of single particle states.
Electrons are fermions, then the state of a system made up of electrons must be antisymmetric under permutations. You can prove that an antisymmetric combination with two electrons in the same state is always zero, yielding a non normalizable state, that is not acceptable. This is known as Pauli's exclusion principle.

$^1$And in the case of electrons in an atom they are.
$^2$If this terminology is not familiar to you, consider a state as a set of quantum numbers. Then, having the same quantum numbers means occupying the same state.
