Define:
$\hat{p}^{2}= \hat{p}^{2}_{x}+\hat{p}^{2}_{y}+\hat{p}^{2}_{z}= -\hbar^{2} \Delta$
$\hat{L}_{z}= xp_{y}-yp_{x}= -i\hbar \big(x\frac{\partial}{\partial y} - y \frac{\partial}{\partial x} \big)$
consider the Hamiltonian in three dimensions with a potential that depends only on the distance from the origin:
$\hat{H} = \frac{\hat{p}^{2}}{2m} + \hat{V}(r), r=(x^{2}+y^{2}+z^{2})^{1/2}$
Since: $ \hat{p}^{2}\hat{L}_{z} = -\hbar^{2}\Delta\big(-i\hbar(x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x})\big) = i\hbar^{3}(\frac{\partial ^{2}}{\partial x} + \frac{\partial ^{2}}{\partial y} + \frac{\partial ^{2}}{\partial z})(x\frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) = 0 $ and $\hat{L}_{z}\hat{p}^{2}=(-i\hbar (x\frac{\partial}{\partial y}- y\frac{\partial}{\partial x})(-\hbar^{2}\Delta)=i\hbar^{3}(x\frac{\partial}{\partial y} - y \frac{\partial }{\partial x})(\frac{\partial ^{2}}{\partial x} + \frac{\partial^{2}}{\partial y} + \frac{\partial^{2}}{\partial z}) = 0$
$[\hat{p}^{2},\hat{L}_{z}] = \hat{p}^{2}\hat{L}_{z} - \hat{L}_{z}\hat{p}^{2} = 0-0 = 0$
The operator $\hat{L}_{z}$ commutes with $\hat{p}^{2}$.
And $\hat{p}^{2} \hat{V} = -\hbar^{2}(\frac{\partial^{2} \hat{V}}{\partial x} + \frac{\partial^{2} \hat{V}}{\partial y} + \frac{\partial^{2} \hat{V}}{\partial z})$ together ith $\hat{V} \hat{p}^{2} = \hat{V}(-\hbar^{2}\Delta) = -\hbar^{2}(\frac{\partial^{2} \hat{V}}{\partial x} + \frac{\partial^{2} \hat{V}}{\partial y} + \frac{\partial ^{2} \hat{V}}{\partial z})$
$[\hat{p}^{2},\hat{V}]=\hat{p}^{2}\hat{V}- \hat{V}\hat{p}^{2} = 0$
Edit : I tried showing that $\hat{L}_{z}$ commutes with $\hat{V}$ using Norberts method: $(\hat{V}\hat{L}_{z})(f) = V(-i\hbar(x\frac{\partial (f)}{\partial y} - y \frac{\partial (f)}{\partial x}))$ and $(\hat{L}_{z}\hat{V})(f)= \hat{L}_{z}(V(r)(f)) =-i\hbar(x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x})(V(r)f) = -i\hbar((f)x\frac{\partial V}{\partial y} + (V)x\frac{\partial f}{\partial y}-(V)y\frac{\partial (f)}{\partial x} - (f)y \frac{\partial (V)}{\partial x})$ Why are they not the same?
I have never worked with operators before, so can you tell me if this is alright? I will greatly appreciate your advice.