Putting the Pieces of Schrödinger’s Theory Together

To put all of the ideas from Schrödinger’s theory together we need to organize our thoughts. Let’s start with the mathematical ideas:

• n = energy level 
• n-1 = number of nodes 
• ℓ = number of angular nodes. Can have any value from 0 up to (n-1) 
• (n-1) - ℓ = number of spherical nodes

That means that for just the first 4 energy levels we have the following orbitals 

It is worth noting a mathematical relationship here. We know that n = energy level, but notice that n also tells us the number of orbital types on that energy level. So, level 1 has only the s orbital, while level 2 has s and p orbitals, and level 3 has s, p and d orbitals.

In addition, n2 is equal to the total number of orbitals on that energy level. So level 1 has only the 1s orbital, but level 2 has the 2s, 2px, 2py and 2pz orbitals (total of 4).

The same table made of images would look like this:

These images are taken from “The Orbitron: a gallery of atomic orbitals and molecular orbitals” published by the University of Sheffield.

The difficulty that many people have is remembering that ALL of these orbitals are arranged around the SAME central nucleus. This becomes a VERY confusing picture. For instance, if we take JUST the 3 different p orbitals on level 2, like this...

And put them around the same nucleus, we get this

Now if we add in the 2s, we get

And if we add the 1s

and the level 3 orbitals, we have a mess.

So, how do we make sense of all of this?

We remember that, although they occupy the same physical space, they occupy different energy levels. Let's explore that idea.