The principal quantum number, n, is the main energy level quantum number. If you look at a periodic table, the rows correspond to the principal quantum number. You count from the lowest main energy level up. 1, 2, 3, 4,... just good ol' counting numbers. You actually will finish the entire current periodic table when you reach n = 7. So you don't even need to count that far.
The next quantum number is the angular momentum quantum number. This gets the symbol ℓ (I prefer showing the cursive ell for this). This quantum number is related to the "shape" of the wavefunction / orbital.
ℓ is an integer and can have any value starting at zero and going up to n-1. That means that ℓ is never larger than n-1.
Since the ℓ values are also related to some historical experiments and spectroscopy terms, they have a letter that is used in place of the number when writing out shorthand electron configurations. Orbitals where ℓ=0 are called s or s-orbitals, ℓ=1 are p, ℓ=2 are d, ℓ=3 are f, ℓ=4 are g, ℓ=5 are h....and so on.
For a given n and ℓ, there are a number of degenerate (same energy) solutions to the Schrodinger equation. We say that the number of degenerate states matches up with the number of orientations of that orbital shape. The number of degenerate levels is equal to 2ℓ+1. There are three p solutions ℓ=1, 2(1)+1 = 3. There are seven f solutions ℓ=3, 2(3)+1 = 7.
Each of these solutions gets a unique quantum number mℓ
mℓ is also an integer and can range from mℓ = -ℓ,...0...,+ℓ
So if \(\ell=4\) the nine possible \(m_\ell\) values are -4,-3,-2,-1,0,+1,+2,+3,+4.
It was discovered much later because electrons were found to have another property that wasn't accounted for originally. For lack of a better term, "spin" was the winner for the name of this property. This is the easiest quantum number. It is either +½ or –½. That's it. One thing or the other. In electron energy diagrams we use and up arrow or better, a "harpoon" ↿ to depict a +½ spin state and a down harpoon ⇂ to depict a –½ spin state. When they are paired in an orbital you show both together as a pair, ⥮.
As stated in the list above, we tend to just say "orbital" when referring to a specific set of quantum numbers. And really, just the first two are needed to identify the right KIND of orbital and energy level. So just n and ℓ are used.
In this notation we simply state the principal quantum number n as a number. A letter is used to denote the ℓ term as letters s,p,d, and f.
So let's imagine a hydrogen atom in its lowest energy state. This is the ground state. The quantum numbers for this are n=1, ℓ=0, mℓ=0, and ms=+1/2. Just using the n and ℓ we call this "orbital" a 1s orbital. Notice we don't even mention mℓ or ms. So just use the orbital designation for quantum number shorthand.
|Principal||n||1,2, ...||main energy level||size|
|Angular Momentum||ℓ||0,1, ..., n-1||sub-level||shape|
|Magnetic||mℓ||–ℓ,..0..,+ℓ||orbitals in the subshell||orientation|
|Spin||ms||+1/2, -1/2||spin state||spin direction|
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