对价电子和空穴算符之间的相位因子缺乏理解
Missing understanding of the phase factor between valence-electron and hole operators
摘要：
本文提供了长期缺失的基础，连接半导体和原子符号，并支持不正确的结果，如果半导体电子拥有轨道角动量。本文证明了价电子破坏算符和空穴产生算符之间的相位因子与量子相对论中粒子和反粒子之间的相位因子相同，即$\hat{a}_{m}=(-1)^{j-m} \hat{b}^†_{-m}$，只要$m=(j,j-1\cdots,-j)$标记了$(2j+1)$折叠电子能级的简并态.这个结果是值得注意的，因为空穴肯定不是一个天真的反粒子，因为剩余的价电子; $(ii)$不像原子电子在一个中心的潜力，半导体电子在一个周期性晶体不具有轨道角动量$\textbf{L}=\textbf{r}\wedge\textbf{p}$也没有角动量$\textbf{J}=\textbf{L}+\textbf{S}$。因此，半导体电子的$(j,m)$仅仅是标记简并能级状态的方便符号。为了说明物理意义，我们讨论光子和半导体之间的带间耦合，在价电子和空穴：相位因子是至关重要的，以建立明亮的激子是在自旋单态。
英文摘要：
This paper provides the long-missing foundation to connect semiconductor and atomic notations and to support results incorrectly obtained by doing as if semiconductor electrons possessed an orbital angular momentum. We here show that the phase factor between valence-electron destruction operator and hole creation operator is the same as the one between particle and antiparticle in quantum relativity, namely $\hat{a}_{m}=(-1)^{j-m} \hat{b}^†_{-m}$ provided that $m=(j,j-1\cdots,-j)$ labels the degenerate states of the $(2j+1)$-fold electron level at hand. This result is remarkable because $(i)$ the hole is definitely not a naive antiparticle due to the remaining valence electrons; $(ii)$ unlike atomic electrons in a central potential, semiconductor electrons in a periodic crystal do not have orbital angular momentum $\textbf{L}=\textbf{r}\wedge\textbf{p}$ nor angular momentum $\textbf{J}=\textbf{L}+\textbf{S}$. Consequently, $(j,m)$ for semiconductor electrons merely are convenient notations to label the states of a degenerate level. To illustrate the physical implications, we discuss the interband couplings between photons and semiconductor, in terms of valence electrons and of holes: the phase factor is crucial to establish that bright excitons are in a spin-singlet state.
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