Algebraic solution of the Quantum Harmonic Oscillator

The Quantum Harmonic Oscillator (QHO) was a key topic in a course on Quantum Physics I 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…