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

Formula: Commutator for All Angular Momentum Operators
Angular Momentum Ladder Operators: Raising and Lowering Operator
Quantized Lz Component of the Angular Momentum

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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