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

$$ L_{\text x} $$

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

Angular momentum operator

$$ L_{\text y} $$

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

Imaginary unit

$$ \mathrm{i} $$

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

Planck constant

$$ \hbar $$

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

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