两体密度矩阵
$$\begin{align} \rho(\mathbf{r}, \mathbf{r}'; \mathbf{r}_1) \equiv \langle \Psi | \psi^{\dagger}(\mathbf{r}) \psi^{\dagger}(\mathbf{r}_1) \psi(\mathbf{r}_1) \psi(\mathbf{r'}) | \Psi \rangle = \phi^*(\mathbf{r}, \mathbf{r}_1)\phi(\mathbf{r}_1, \mathbf{r}') \end{align}$$其中 $|\Psi\rangle$ 是两体态
$$\begin{align} | \Psi \rangle = \int \mathrm{d} \mathbf{r}_3 \int \mathrm{d} \mathbf{r}_4\cdot \phi (\mathbf{r}_3, \mathbf{r}_4)\cdot \psi^{\dagger}(\mathbf{r}_3) \psi^{\dagger}(\mathbf{r}_4) | 0 \rangle \end{align}$$ $$\begin{align} \langle \Psi| = \int \mathrm{d} \mathbf{r}_5 \int \mathrm{d} \mathbf{r}_6\cdot \phi^{*} (\mathbf{r}_5, \mathbf{r}_6)\cdot \langle 0 |\psi(\mathbf{r}_6) \psi(\mathbf{r}_5) \end{align}$$所以
$$\begin{align} \rho(\mathbf{r}, \mathbf{r}'; \mathbf{r}_1) = &\int \mathrm{d} \mathbf{r}_3 \int \mathrm{d} \mathbf{r}_4 \int \mathrm{d} \mathbf{r}_5\int \mathrm{d} \mathbf{r}_6\cdot \phi^{*} (\mathbf{r}_5, \mathbf{r}_6)\phi (\mathbf{r}_3, \mathbf{r}_4)\\ &\langle 0 | \psi(\mathbf{r}_6) \psi(\mathbf{r}_5)\cdot \psi^{\dagger}(\mathbf{r}) \psi^{\dagger}(\mathbf{r}_1) \psi(\mathbf{r}_1) \psi(\mathbf{r'})\cdot \psi^{\dagger}(\mathbf{r}_3) \psi^{\dagger}(\mathbf{r}_4) |0\rangle \\ =&\int \mathrm{d} \mathbf{r}_3 \int \mathrm{d} \mathbf{r}_4 \int \mathrm{d} \mathbf{r}_5\int \mathrm{d} \mathbf{r}_6\cdot \phi^{*} (\mathbf{r}_5, \mathbf{r}_6)\phi (\mathbf{r}_3, \mathbf{r}_4)\\ &\left[ \delta (\mathrm{r} - \mathrm{r}_{5}) \delta(\mathrm{r}_1 - \mathrm{r}_{6}) -\delta (\mathrm{r} - \mathrm{r}_6) \delta(\mathrm{r}_1 - \mathrm{r}_5) \right]\cdot \left[ \delta (\mathrm{r}' - \mathrm{r}_3) \delta(\mathrm{r}_1 - \mathrm{r}_4) -\delta (\mathrm{r}' - \mathrm{r}_4) \delta(\mathrm{r}_1 - \mathrm{r}_3) \right] \end{align}$$so
$$\begin{align} \rho(\mathbf{r}, \mathbf{r}'; \mathbf{r}_1) = \sharp \phi^*(\mathbf{r}, \mathbf{r}_1)\phi(\mathbf{r}_1, \mathbf{r}') \end{align}$$Reference
- Harald Siegfried Friedrich, Theoretical Atomic Physics-Springer (2005) Chap 1.4.1