# Formula: Commutator Angular momentum operator    Momentum operator

## Commutator

The commutator represents the part that must be added when the two operators $$\hat{L}_{\class{red}{i}} \, \hat{p}_{\class{green}{j}}$$ are swapped: $$\hat{p}_{\class{green}{j}} \, \hat{L}_{\class{red}{i}}$$.

Here: $$\class{red}{i}, \class{green}{j}, \class{blue}{k} \in \{1,2,3\}$$.

## Angular momentum operator

It is the $$\class{red}{i}$$-th component of the angular momentum vector operator $$\hat{\boldsymbol{L}}$$.

## Momentum operator

It is the $$\class{blue}{j}$$-th component of the linear momentum vector operator $$\hat{\boldsymbol{p}}$$.

## Levi civita symbol

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

## Imaginary unit

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

## 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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