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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