Why does the possible orbitals equal n squared?

[1MileCrash]

If i take any integer, x

write out all integers from 0 to x - 1

write out all numbers between negative and positive values of those numbers

I get x squared total values. why? what is the mathematical logoc behind this?

x = 3
0 to x - 3 = 0, 1, 2

0
-1, 0, 1
-2, -1, 0, 1, 2

9 total values, x squared total values, but why does this work? how is that process related to squaring?

 

[Mute]

For a given value of the quantum number [itex]\ell[/itex], the magnetic quantum number [itex]m[/itex] can run from [itex]-\ell[/itex] to [itex]\ell[/itex], which means there are [itex]2\ell + 1[/itex] values of m for each [itex]\ell[/itex]. The quantum number [itex]\ell[/itex] can run from 0 to [itex]n-1[/itex], where n is the principal quantum number. You can fill up orbitals for each [itex]\ell[/itex] until you hit this max, so the total number of orbitals is

[tex]\sum_{\ell = 0}^{n-1} (2\ell + 1) = 2\left(\sum_{\ell = 0}^{n-1}\ell\right) + n[/tex]

It is a know result that

[tex]\sum_{k=1}^{N} k = \frac{N(N+1)}{2}[/tex]
hence, we get

[tex]2\left(\sum_{\ell = 0}^{n-1}\ell\right) + n-1 = 2\frac{(n-1)n}{2} + n = n(n-1 + 1) = n^2.[/tex]

 

[1MileCrash]

Thank you, I just couldn't find that connection and needed to know out of pure curiosity..

 

[gb7nash]

Thank you, I just couldn't find that connection and needed to know out of pure curiosity..

You may not realize it, but that's a great thing that you discovered that relationship without knowing about the equation. There's a lot of cool relationships you'll discover in math.

 

[CellCoree]

The reason why the possible orbitals equal n squared is due to the quantum mechanical principle known as the Aufbau principle. This principle states that electrons occupy the lowest available energy levels before filling higher levels. The number of electrons that can fill a particular energy level is determined by the quantum number n, which represents the principal energy level.

For example, if n = 3, there are 3 possible orbitals (s, p, and d) that can be filled with a maximum of 2 electrons each. This gives a total of 6 possible electrons. In general, the number of possible orbitals is equal to n squared.

The mathematical logic behind this can be explained by the formula for the number of orbitals in a particular energy level, which is given by 2n^2. This formula takes into account the different sublevels (s, p, d, etc.) that can exist within an energy level. When we take any integer, x, and square it, we are essentially calculating the maximum number of electrons that can exist in that energy level.

In the example given, x = 3, and when we square it, we get 9. This means that there can be a maximum of 9 electrons in the third energy level, which is consistent with the Aufbau principle and the 2n^2 formula.

In terms of writing out all integers from 0 to x-1, and then writing out all numbers between negative and positive values of those numbers, this process is related to squaring because it represents the different possible combinations of sublevels and electrons within a particular energy level. By writing out all integers and their negative and positive values, we are essentially calculating the maximum number of electrons that can exist in that energy level, which is determined by the quantum number n. This is why the number of possible orbitals equals n squared.