Ciro Santilli @cirosantilli 37

If Ciro Santilli ever becomes rich, he's going to solve this with: website front-end for a mathematical formal proof system, promise.

Those guys are really good, Ciro Santilli especially enjoyed their quantum mechanics playlist: www.youtube.com/playlist?list=PL193BC0532FE7B02C

The quantum electrodynamics one was a bit too slow paced for Ciro unfortunately, too much groundwork and too little results.

Accompanying website with a tiny little bit of code: viascience.org/what.html

TODO: authors and their affiliation.

Videos licensed as CC BY-SA, those guys are so good.

This equation is a subcase of Equation "Schrödinger equation for a one dimensional particle" with $V(x)=x^2$.

We get the time-independent Schrödinger equation by substituting this $V$ into Equation "time-independent Schrödinger equation for a one dimensional particle":

Now, there are two ways to go about this.

The first is the stupid "here's a guess" + "hey this family of solutions forms a complete basis"! This is exactly how we solved the problem at Section "Solving partial differential equations with the Fourier series", except that now the complete basis are the Hermite functions.

The second is the much celebrated ladder operator method.