PX262 - E1b - compatibility

\begin{matrix} [\hat{X}, {\hat{L}}_{x}] = 0 & [\hat{X}, {\hat{L}}_{y}] = + i ℏ \hat{Z} & [\hat{X}, {\hat{L}}_{z}] = - i ℏ \hat{Y} \\ [\hat{Y}, {\hat{L}}_{x}] = - i ℏ \hat{Z} & [\hat{Y}, {\hat{L}}_{y}] = 0 & [\hat{Y}, {\hat{L}}_{z}] = + i ℏ \hat{X} \\ [\hat{Z}, {\hat{L}}_{x}] = + i ℏ \hat{Y} & [\hat{Z}, {\hat{L}}_{y}] = - i ℏ \hat{Y} & [\hat{Z}, {\hat{L}}_{z}] = 0 \end{matrix}

\begin{matrix} [{\hat{P}}_{x}, {\hat{L}}_{x}] = 0 & [{\hat{P}}_{x}, {\hat{L}}_{y}] = + i ℏ {\hat{P}}_{z} & [{\hat{P}}_{x}, {\hat{L}}_{z}] = - i ℏ {\hat{P}}_{y} \\ [{\hat{P}}_{y}, {\hat{L}}_{x}] = - i ℏ {\hat{P}}_{z} & [{\hat{P}}_{y}, {\hat{L}}_{y}] = 0 & [{\hat{P}}_{y}, {\hat{L}}_{z}] = + i ℏ {\hat{P}}_{x} \\ [{\hat{P}}_{z}, {\hat{L}}_{x}] = + i ℏ {\hat{P}}_{y} & [{\hat{P}}_{z}, {\hat{L}}_{y}] = - i ℏ {\hat{P}}_{x} & [{\hat{P}}_{z}, {\hat{L}}_{z}] = 0 \end{matrix}

compatibility between individual components of the angular momentum operator, and between the components and the square:

\begin{aligned} [{\hat{L}}_{x}, {\hat{L}}_{y}] & = i ℏ {\hat{L}}_{z} \\ [{\hat{L}}_{y}, {\hat{L}}_{z}] & = i ℏ {\hat{L}}_{x} \\ [{\hat{L}}_{z}, {\hat{L}}_{x}] & = i ℏ {\hat{L}}_{y} \\ [{\hat{L}}^{2}, {\hat{L}}_{x}] = [{\hat{L}}^{2}, {\hat{L}}_{y}] & = [{\hat{L}}^{2}, {\hat{L}}_{z}] = 0 \end{aligned}

the components are mutually incompatible, but ${\hat{L}}^{2}$ is compatible with all three components
this means that if a system is an eigenstate of one angular momentum component, it cannot simultaneously be in an eigenstate of either of the others
but. the total angular momentum can be measured compatibly with any other component