Formula: Commutator Angular momentum operator    Momentum operator   

Formula: Commutator
Lz of the Quantum Mechanical Angular Momentum Precisely Determined But Lx and Ly Not


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

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

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

Reduced Planck's constant

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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