PX262 - E1a - angular momentum operators

in classical physics, the angular momentum is defined as: $\vec{L} = \vec{r} \times \vec{p}$
therefore, using postulate 3, the angular momentum operator is defined as: $\hat{\vec{L}} = \hat{\vec{R}} \times \hat{\vec{P}}$
therefore, operators representing the components of angular momentum in cartesian coordinates are:

\begin{matrix} {\hat{L}}_{x} = \hat{Y} {\hat{P}}_{z} - \hat{Z} {\hat{P}}_{y}, & {\hat{L}}_{y} = \hat{Z} {\hat{P}}_{x} - \hat{X} {\hat{P}}_{z}, & {\hat{L}}_{z} = \hat{X} {\hat{P}}_{y} - \hat{Y} {\hat{P}}_{x} . \end{matrix}

{\hat{L}}^{2} = {\hat{L}}_{x}^{2} + {\hat{L}}_{y}^{2} + {\hat{L}}_{z}^{2}