of the ground state $|\psi_{o}\rangle$. [*Note that the phase factors cancel out and do not appear in the exciton eigenvalue equation*]{}. The exciton wavefunctions are orthonormal and complete in the sense [@Kira12], $$\int \frac{d^{2}\vec{k}}{(2\pi)^{2}} \phi^{*}_{\alpha}(\vec{k}) \phi_{\beta}(\vec{k}) = \delta_{\alpha,\beta} \label{eq:orth}$$ $$\sum_{\alpha} \phi_{\alpha}(\vec{k}) \phi^{*}_{\alpha}(\vec{k}') = (2\pi)^{2} \delta^{2}(\vec{k}-\vec{k}') \label{eq:comp}$$ The sum over $\alpha$ above includes all the discrete bound exciton states as well as the continuum of ionized exciton states. Finally, the probability of finding an electron and a hole at a distance $\vec{r}$ in the exciton state $| \psi_{\alpha,\vec{Q}}(\vec{k}) \rangle$ can be computed by destroying an electron and a hole using the real-space field destruction operators and then taking the overlap of the resulting state with the ground state $|\psi_{o}\rangle$. The result is $|\phi_{\alpha}(\vec{r})|^{2}$ where $\phi_{\alpha}(\vec{r})$ is the Fourier transform of $\phi_{\alpha}(\vec{k})$. Note that $\phi_{\alpha}(\vec{r})$ is not the Fourier transform of $\psi_{\alpha,\vec{Q}}(\vec{k})$, which also includes extra phase factors (see (\[eq:psiphi\]).

Exciton Basis {#sec:ex_basis}
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In what follows, we will use the exciton basis. The exciton creation operator $B^{\dagger}_{s,s',\alpha}(\vec{Q})$ can be defined as, $$B^{\dagger}_{s,s',\alpha}(\vec{Q}) =  \frac{1}{\sqrt{A}}\sum_{\vec{k}} \psi_{\alpha,\vec{Q}}(\vec{k}) c_{\vec{k} + \frac{m_{e}}{m_{ex}}\vec{Q},s}^{\dagger} b_{\vec{k}-\frac{m_{h}}{m_{ex}}\vec{Q},s'}$$ Using the completeness and the orthogonality of the exciton wavefunctions given in (\[eq:comp\]) and (\[eq:orth\]), we get, $$c_{\vec{k},s}^{\dagger} b_{\vec{k}',s'} = \frac{1}{\sqrt{A}}\sum_{\alpha} \psi^{*}_{\alpha,\vec{Q}}(\vec{k}_{r}) 