In second quantization, a field operator of e.g. an electron is given by $\Psi(\vec{r})=\sum_n \varphi_n(\vec{r}) \hat{c}_n$ with the operator $\hat{c}_n^+$ creating an electron within the state/wavefunction $\varphi_n(\vec{r})$.

In Solid-State-Physics, the idea of an electron-hole is to replace the electron creation/anihilation operators $\hat{c}_n^+$/$\hat{c}_n$ for all states below the Fermi-energy by hole anihilation/creation operators $\hat{d}_n$/$\hat{d}_n^+$, thus \begin{equation} \hat{c}_n^+ \to \hat{d}_n \text{ and } \hat{c}_n \to \hat{d}_n^+ \end{equation} and to redefine the vacuum-state to a state with all energy levels occupied up to the Fermi-energy.

Lets say, I know the wavefunction of a certain valence-band electron. How can I derive the wavefunction of the associated hole? It is my motivation to calculate the symmetry properties of excitons.

Is there a relation to time-reversal-symmetry?



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