Formula: Angular momentum commutator (Lx and Ly)

$$[ L_{\text x}, L_{\text y} ] ~=~ \mathrm{i} \, \hbar \, L_{\text z}$$

Commutator

$$ [L_{\text x}, L_{\text y}] $$

This commutator for angular momentum is the term you get (in this case: $\mathrm{i} \, \hbar \, L_{\text z}$) when you swap the two angular momentum components $L_{\text x}$ and $L_{\text y}$. As you can see, the commutator is NOT zero.

Angular momentum operator

$$ L_{\text x} $$ Unit $$ \mathrm{Js} $$

This is the $x$-th component of the angular momentum vector operator $ \boldsymbol{L} $.

Angular momentum operator

$$ L_{\text y} $$ Unit $$ \mathrm{Js} $$

This is the $y$-th component of the angular momentum vector operator $ \boldsymbol{L} $.

Imaginary unit

$$ \mathrm{i} $$ Unit $$ - $$

Imaginary unit is a complex number for which is true: $ \mathrm{i} ~=~ \sqrt{-1} $.

Planck constant

$$ \hbar $$ Unit $$ \mathrm{Js} = \frac{ \mathrm{kg} \, \mathrm{m}^2 }{ \mathrm{s} } $$

Planck constant is a natural constant (of quantum mechanics) and has the value: $$ \hbar ~=~ \frac{h}{2\pi} ~=~ 1.054 \cdot 10^{-34} \, \text{Js} $$