Schrodinger's Math - the BASICS

It is WAY beyond the scope of this text (and the author’s talents and understanding) to attempt to teach Schrödinger’s math. However, we can get a sense of what his math does and the types of “answers” we get from the math. 

If you aren't sure why, here is a SIMPLIFIED version of the equation.

So, let’s start with what the answers to the equation are.

Anyone reading this text has, almost certainly, taken some basic algebra. Which means that you have seen the equation 

y=mx+b 

You should recognize this equation as the point slope form of a linear equation. In other words, if I asked you what this equation “gave” as an answer, you would say “a line.”

If I then asked you “WHAT line?” you would tell me that it depended on the values for m (the slope) and b (the y intercept).

There are 2 really important ideas here. 

1. The answer to an equation can be a graph - not a number 
2. What that graph looks like can depend on the values put into the variable for that equation.

So, the “answers” to Schrödinger’s equation are graphs, whose appearance depends on the numerical values of the variable in the equation.  There is a dizzying array of these graphs, as seen here:

Now, let’s talk about what those pictures show.

The graphs that are generated by Schrödinger’s equation are called by two names: probability plots and orbitals.  

Probability plots show “the area, within which, there is a probability of measuring the electron.” In other words, the equation creates a graphical shape that describes an electron wave around the nucleus. It is important to remember that, according to Schrödinger’s work, the electron is NOT a little particle. Rather, the electron is a wave that is centered around the nucleus. His equation then, is describing the shape of that wave. If you haven’t read this description of waves yet, you need to.

These probability plots vary in size and shape and are generally called orbitals.

To further understand these orbitals, we need to know little bit about: 

1. The variables in Schrödinger’s equation and what they mean 
2. How energy level changes orbitals 
3. How the orbitals fit together 
4. How electrons fit into the picture that the equation has created.