The √n and √(n+1) factors are a nice place to connect the algebra to the changing wavefunction shapes. I made an animated derivation of both the power-series and ladder-operator routes, if useful alongside this lesson! youtu.be/7_DcdXVh5mc
youtu.be
Quantum Harmonic Oscillator Explained Visually (Full Derivation)
YouTube video by Blake Pi - Physics Videos