Quantized Lz Component of the Angular Momentum

Quantized Lz Component of the Angular Momentum
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The \( L_{\text z} \) component of the total orbital angular momentum \( \boldsymbol{L} \) of a quantum mechanical system (e.g. H-atom) is quantized in a fixed direction (here the \(z\)-direction). In this illustration, the angular momentum quantum number \( l = 2\) (a \(d\)-state) was chosen as an example. Then, the magnetic quantum number \(m_l\) can take the values: -2, -1, 0, 1, and 2.

The orbital angular momentum \(L_{\text z}\) is a multiple of the reduced Planck constant \(\hbar\):

  • An electron with \(\class{violet}{m} = 0 \) has no angular momentum in the \(z\) direction: \(L_{\text z} = 0\).
  • A state with \(\class{violet}{m} = 1 \) has angular momentum in \(z\) direction: \(L_{\text z} = 1\).
  • A state with \(\class{violet}{m}= 2 \) has angular momentum in \(z\) direction: \(L_{\text z} = 2\hbar\).
Analogous for the negative magnetic quantum number. In this case, the orbital angular momentum \( \boldsymbol{L} \) points in the opposite direction.

Note that the \(L_{\text y}\) and \(L_{\text x}\) components of the total angular momentum \(\boldsymbol{L}\) are indeterminate due to Heisenberg's uncertainty principle. The exemplarily drawn angular momentum \(\boldsymbol{L}\) could lie somewhere rotationally symmetric around the \(L_{\text z}\) axis and have the same \(L_{\text z}\) component.

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