Formula: Commutator Angular momentum operator Momentum operator

$$[ \hat{L}_{\class{red}{i}}, \, \hat{p}_{\class{green}{j}} ] ~=~ \varepsilon_{\class{red}{i}\class{green}{j}\class{blue}{k}} \, \mathrm{i} \, \hbar \, \hat{p}_{\class{blue}{k}}$$

Commutator

$$ [ \hat{L}_{\class{red}{i}}, \, \hat{p}_{\class{green}{j}} ] $$

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

$$ \hat{L}_{\class{red}{i}} $$

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

Momentum operator

$$ \hat{p}_{\class{blue}{j}} $$

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

Levi civita symbol

$$ \varepsilon_{\class{red}{i}\class{green}{j}\class{blue}{k}} $$ 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

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

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

Reduced Planck's constant

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

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} $.