What I’m about to tell you is not something new. These ideas have been around for many years already (since classical mechanics was still a thing). Moreover, I have been bothering my friends, students, teachers and some angry chemists with this for the last four or five years, ever since a student in San Luis Potosi made the mistake of saying the word “Hamiltonian” in front of me. This let another professor to reveal me one of the most guarded treasures for the physicist: the all-mighty Lagrangian.
This one I wanted to do for a while. I like very much the math that can be derived from the Quantum Harmonic Oscillator, but there’s also a lot of stuff that I personally don’t like about the approach that is taken in some quantum physics books. I prefer a rather algebraic analysis of this system since I think that it is pretty much cleaner and easier to justify; then again I’m not pretending to be absolutely formal, just enough for it not to look like some kind of magic trick in the middle of the thing.