This solutions for the quantum system of a hydrogen atom extrapolated are orthogonal, they can exist at the same time, and applies to the system simultaneously. This allows us to find the maximum possible number of electrons given \(n\).
For solution of Maximum number of possible orbital electrons
\(n=1\) \(2\)
\(n=2\) \(2+4=6\)
\(n=3\) \(2+4+6=12\)
\(n=4\) \(2+4+6+8=20\)
\(n\) \(n*(n+1)\)
So the maximum number of electrons in orbit given \(n\) is \(n(n+1)\), which is obtained by adding solutions from all lower numbered solutions.
And when an \(n=4\) shell is being filled up.
Other possible solutions to the quantum system (hydrogen atom) show up, and seem to duplicate solutions to lower valued \(n\).
These solutions are not orthogonal, they do not exist at the same time and so does not apply to the quantum system simultaneously.
A energy shell cannot at the same time contain both two electrons and three electrons. But it can contain either two electrons OR three electrons, since both,
(\(n=4, l=0, m=0, s=\pm1/2\)) and
(\(n=4, l=0, m=0, s=\pm1/2\))
(\(n=4, l=1 m=-1\,\, or\,\, 1, s=\pm1/2\))
are valid solutions.
The implication of this interpretation is not that there are no sub-shells, but that they are filled up in time and do not exist simultaneously. The sub-shell evolves as electrons are added,
\(4s\rightarrow4p\rightarrow4d\rightarrow4f\)
When \(n=4\) presents a \(4d\) sub-shell configuration/solution, \(4s\) do not exist!
Just as \(m=-1\) or \(m=1\) but not both for the same electron! One electron can only travel in one direction around its orbit.
Just when you think mistakes on \(_2He\), \(_3Li\) and \(_4Be\) was funny.
