I'm in a quantum mechanics class, and it is given in the book that the operators $\hat{L^{2}}$ and $\hat{H}$ commute for the 3D Harmonic Oscillator, but no definite mathematical proof is given, and I'm having a hard time proving it myself, and conceptualizing why this must be true.
I've been trying to use spherical coordinate to prove it, and I know that in spherical coordinates $$\hat{L^{2}}= \frac{-h^{2}}{4\pi^{2}}(\frac{1}{\sin\theta}\frac{d}{d\theta}(\sin\theta \frac{d}{d\theta})+\frac{1}{\sin^{2}\theta}\frac{d^{2}}{d\phi^{2}})$$
And $$\hat{H}=\frac{-h^{2}}{4\pi^{2}(2m)}\Delta+\frac{1}{2}kr^{2}.$$
I've been trying to prove it using the very basic $[\hat{L^{2}},\hat{H}]f= \hat{L^{2}}\hat{H}f-\hat{H}\hat{L^{2}}f$ method of showing the commutation relationship. My first thought was that by applying $\hat{L^{2}}$ to $\hat{H}$ all the terms that are dependent on r or $\frac{r}{dr}$ would disappear, but if some arbitrary function f had cross terms, this isn't necessarily true, and the algebra got pretty messy after that. Is there a better way to prove this?