I have a nice derivation for the quantum harmonic oscillator I have not yet seen
in the literature. It's quite simple, and motivates generators fairly quickly.
You just use the Hamiltonian's symmetry with respect to Fourier transforms.
So, we start with
$$\hat H=-\frac{\hbar^2}{2m}\partial_x^2 + \frac12 m\omega^2x^2$$
We now nicen up the units as is common practice: $q=\sqrt{\frac{m\omega}{\hbar}}\cdot x$
Consequently,we get
$$\hat H(-i\partial_q,q)=\frac{\hbar\omega}{2}\left(-\partial_q^2 + q^2\right)$$
Choosing natural units of energy/time, we can now set $\hbar \omega=1$.
It now becomes rather apparent that this expression is invariant under the Fourier transform. (cont.)