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The angular momentum formalism provides a powerful way to classify atomicstates. Yet, requiring a spherical symmetry from the very first line, thisformalism cannot be used for periodic systems, even though cubic semiconductorstates are commonly classified according to atomic notations. Although nevernoted, it is possible to define the analog of the orbital angular momentum, byonly using the potential felt by the electrons. The spin-orbit interaction forcrystals then takes the $\mathbfcal{\hat{L}}\cdot \hat{\vS}$ form, with$\mathbfcal{\hat{L}}$ reducing to $\hat{\vL}=\vr\times\hat{\vp}$ for sphericalsymmetry. This provides the long-missed support for using the eigenvalues of$\mathbfcal{\hat{L}}$ and $\mathbfcal{\hat{J}}=\mathbfcal{\hat{L}}+\hat{\vS}$,as quantum indices to label cubic semiconductor states. Importantly, thesequantum indices also control the phase factor that relates valence electron tohole operators, in the same way as particle to antiparticle, in spite of thefact that the hole is definitely not the valence-electron antiparticle. Beingassociated with a broader definition, the($\mathbfcal{\hat{L}},\mathbfcal{\hat{J}}$) analogs of the$(\hat{\vL},\hat{\vJ})$ angular momenta, must be distinguished by names: wesuggest "spatial momentum" for $\mathbfcal{\hat{L}}$ that acts in the realspace, and "hybrid momentum" for $\mathbfcal{\hat{J}}$ that also acts on spin,the potential symmetry being specified as "cubic spatial momentum". This wouldcast $\hat{\vJ}$ as a "spherical hybrid momentum", a bit awkward for theconcept is novel.

From spherical to periodic symmetry: the analog of orbital angular momentum for semiconductor crystals