Joint Probabilities

The goal of this section is to show that the joint probability $g_{ij}$ that an electron is in state $i$ and another electron of the same spin is in state $j$ is

$$g_{ij} = \overline{n_i n_j} - \delta_{ij} \overline{n_i},$$ (230)

where $\overline{n_i}$ is the average population of state $i$. Note that the average $\overline{n_i n_j}$ thus provides information about correlations between different particles. Consider the population of state $i$, $n_i$, in terms of the sum of occupation variables $n_i^{(\alpha)}$ over all electrons $\alpha$,

$$n_i = \sum_{\alpha} n_i^{(\alpha)},$$ (231)

where $n_i^{(\alpha)}=1, 0$. Therefore, the probability that states $i$ and $j$ are populated is

$$\overline{n_i n_j} = \sum_{\beta} \sum_{\alpha} \overline{n_i^{(\... ...\alpha} \overline{n_i^{(\alpha)}n_j^{(\beta)}}. }_{g_{ij} \text{by definition}}$$ (232)

Note that

$$\sum_{\alpha} \overline{n_i^{(\alpha)} n_j^{(\alpha)}} = \sum_{\alpha} \overline{(n_i^{(\alpha)})^2} \delta_{ij},$$ (233)

and that

$$\overline{n_i^2} = \sum_{\alpha} \sum_{\beta} \overline{n_i^{(\al... ...um_{\alpha} \sum_{\beta \neq \alpha} \overline{n_i^{(\alpha)} n_i^{(\beta)}} ,$$ (234)

where the double sum with $\beta \neq \alpha$ in Eq. (234) is equal to 0 because it corresponds to the joint probability that both particles $\alpha$ and $\beta$ are in state $i$. Substituting Eqs. (233) and (234) into Eq. (232), we obtain

$$\overline{n_i n_j} = \overline{n_i^2} \delta_{ij} + g_{ij}.$$ (235)

Eq. (235) is identical to Eq. (230) because $\overline{n_i^2}=\overline{n_i}$ when $n_i=0,1$. Finally, note that according to Eq. (143),

$$\overline{n_i n_j} = \frac{\partial^2 \Xi}{\partial(-\beta \epsil... ...partial(-\beta \epsilon_i)} \Bigg)_{V,T} + \overline{n_j}\Bigg) \overline{n_i},$$ (236)

Therefore,

$$g_{ij} = \Bigg (\delta_{ij} \frac{\partial }{\partial(-\beta \epsilon_i)} \Bigg)_{V,T} + \overline{n_j} - \delta_{ij} \Bigg) \overline{n_i},$$ (237)