The Quantum Harmonic Oscillator (QHO) was a key topic in a course on Quantum Physics I took (definitely one of the most interesting lectures I ever took). We solved it using the algebraic approach, which involves the factorization of the Hamiltonian using ladder operators. While this method is elegant, the derivation of the ladder operators in the lecture felt somewhat arbitrary to me, and I struggled to grasp the deeper motivation behind choosing this particular approach to solve the problem.

I stumbled across Dynamical systems in the context of chaos theory and stability. Examples of such systems include the Van der Pol oscillator and the Lorenz attractor. However, even simpler structures like the logistic map seem to offer quite interesting behavior. This article also briefly touches upon the concept of Lyapunov exponents, which measure the sensitivity to initial conditions and provide insights into system predictability. I have not taken any course and am just exploring these topics out of random interest, because I find the visualizations and just the nature of this stuff existing quite interesting - I have no academic background in mathematics of physics up to this point.