# Formula: Commutator for All Angular Momentum Operators Angular momentum component operator    Levi Civita Symbol

## Angular momentum component operator

This is an angular momentum operator, namely the $$\class{red}{i}$$-th component of the angular momentum vector operator $$\boldsymbol{L}$$, i.e. $$L_1$$, $$L_2$$ or $$L_3$$.

## Levi Civita Symbol

Unit
The indices $$\class{red}{i}, \class{green}{j},\class{blue}{k}$$ can take on values from 1 to 3. Depending on how their combination is, the Levi-Civita symbol yields either 1, -1 or 0. $$\varepsilon_{\class{red}{i}\class{green}{j}\class{blue}{k}}$$ is 1 if all indices are interchanged (even permutation). $$\varepsilon_{\class{red}{i}\class{green}{j}\class{blue}{k}}$$ is -1 if only two of the indices are permuted (odd permutation). And, if at least two indices are equal, $$\varepsilon_{\class{red}{i}\class{green}{j}\class{blue}{k}} = 0$$.

## Imaginary unit

Imaginary unit is a complex number with the property: $$\mathrm{i} ~=~ \sqrt{-1}$$.

## Reduced Planck's constant

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

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