Introducing the classical harmonic oscillator and the transition to the quantum model.

Setting up the time-independent Schrödinger equation for the parabolic potential.

Simplifying the physics using dimensionless variables for cleaner mathematics.

Deriving the differential equation that governs the system's wave functions.

How energy quantization emerges from the requirement of well-behaved wave functions.

The significance of the ground state energy and why it isn't zero.

Visualizing the first few eigenfunctions and their physical characteristics.