Angular momentum $\vec{L} = \vec{r}\times \vec{p}$

$$ \vec{L} = \vec{r} \times (-i\hbar \nabla) \\

=\underbrace{-i\hbar(y\partial_z - z\partial_y)}_{\hat{L}x} \hat{x} \underbrace{-i\hbar( z\partial_x-x\partial_z)}{\hat{L}y} \hat{y}\underbrace{-i\hbar(x\partial_y - y\partial_x)}{\hat{L}_z} \hat{z}

$$

The relation between Cartesian and spherical coordinate is

$$ x = r \sin\theta \cos\phi \\ y = r \sin\theta \sin\phi \\ z = r \cos\theta

$$

$$ \hat{r} = \hat{x}\sin\theta\cos\phi + \hat{y}\sin\theta\sin\phi + \hat{z}\cos\theta \\

\hat{\theta} = \hat{x}\cos\theta\cos\phi + \hat{y}\cos\theta\sin\phi -\hat{z}\sin\theta \\

\hat{\phi} = -\sin\phi\hat{x} + \cos\phi \hat{y}

$$

$$ \nabla = \hat{r} \partial_r + \hat{\theta} \frac{\partial_\theta}{r} +\hat{\phi} \frac{\partial_\phi}{r \sin \theta} \\ =(\hat{x}\sin\theta\cos\phi + \hat{y}\sin\theta\sin\phi + \hat{z}\cos\theta)\partial_r \\ +(\hat{x}\cos\theta\cos\phi + \hat{y}\cos\theta\sin\phi -\hat{z}\sin\theta )\frac{\partial_\theta}{r}\\ +(-\sin\phi\hat{x} + \cos\phi \hat{y})\frac{\partial_\phi}{r \sin \theta} \\

=\underbrace{\left(\sin\theta\cos\phi\partial_r+\frac{\cos\theta\cos\phi}{r}\partial_\theta -\frac{\sin\phi}{r \sin \theta}\partial_\phi \right)}{\partial_x}\hat{x} \\ +\underbrace{\left(\sin\theta\sin\phi\partial_r+\frac{\cos\theta\sin\phi}{r}\partial\theta +\frac{\cos\phi}{r \sin \theta}\partial_\phi \right)}{\partial_y}\hat{y} \\ +\underbrace{\left(\cos\theta\partial_r-\frac{\sin\theta}{r}\partial\theta\right)}_{\partial_z}\hat{z}

$$

$$ \hat{L}x=-i\hbar(y\partial_z - z\partial_y) =-i\hbar(-\sin\phi \partial\theta - \cot\theta\cos\phi\partial_\phi) \\ \hat{L}y=-i\hbar( z\partial_x-x\partial_z) = -i\hbar (\cos\phi \partial\theta - \cot\theta\sin\phi\partial_\phi)\\ \hat{L}z=-i\hbar(x\partial_y - y\partial_x) = -i\hbar \partial\phi

$$

$$ \vec{L} = \vec{r}\times \vec{p} = \vec{r} \times (-i\hbar \nabla) \\

= -i\hbar \left[(r\hat{r})\times \left( \hat{r} \partial_r + \hat{\theta} \frac{\partial_\theta}{r} +\hat{\phi} \frac{\partial_\phi}{r \sin \theta}\right) \right] \\

=-i\hbar \left[ \hat{\phi}\partial_\theta -\hat{\theta}\frac{\partial_\phi}{\sin\theta} \right] $$

$$ \hat{L}z = -i\hbar \partial\phi \\

\hat{L}^2 = -\hbar^2\left[\frac{\partial_\theta(\sin\theta\partial_\theta)}{\sin\theta}+\frac{\partial_\phi^2}{\sin^2\theta}\right] $$

$$ \hat{L}_z = [\vec{r}\times\vec{p}]_z = xp_y-yp_x \\

=-i\hbar (x\partial_y- y\partial_x) $$