I think I finally understand most of the harmonic oscillator. After having learned to derive the classical and quantum HO (first and second quantization, not that it's a meaningful difference) I decided to try to tackle the system from many angles and find as many connections as possible. Things like resonance etc are of course already in the standard literature and not worth mentioning, but I figured out a couple more interesting structures.
>how to reduce the classical equations of motion to a linear first order differential equation using noncanonical transformations
>the connection between ladder operators and said non-canonical coordinates
>an actually well-motivated derivation for the ladder operators as a consequence
>eigenfunctions of the quantum HO from fourier symmetry
>the classical analogue of imaginary time propagation to realize energy minimization/maximization
>complex phase space trajectories (can be interpreted as classically forbidden solutions)
What I know I'm missing is:
>general quantum solutions for V=x^2 for x>0, 0 otherwise
>relativistic oscillator
What else could there be to learn from in this system?