Find the two-body potential energy


 * 1) Quantise the classic potential,\(V=\int d\mathbf{r}\,d\mathbf{r}\prime V(\mathbf{r}-\mathbf{r}\prime)\rho(\mathbf{r})\rho(\mathbf{r}\prime)\): \(V=\int d\mathbf{r}\,d\mathbf{r}\prime V(\mathbf{r}-\mathbf{r}\prime)\phi^{\dagger}(\mathbf{r})\phi(\mathbf{r})\phi^{\dagger}(\mathbf{r}\prime)\phi(\mathbf{r}\prime)\)
 * 2) Second quantise by writing the potential in terms of operators: \(V=\int d\mathbf{r}\,d\mathbf{r}\prime V(\mathbf{r}-\mathbf{r}\prime)\hat{\phi}^{\dagger}(\mathbf{r})\hat{\phi}(\mathbf{r})\hat{\phi}^{\dagger}(\mathbf{r}\prime)\hat{\phi}(\mathbf{r}\prime)\)
 * 3) Normal order the integral.
 * 4) Use the commutator/anticommutator of the ladder operators to rearrange the operators.
 * 5) Find \([\hat{a}^{\dagger}(\mathbf{r}),\hat{a}(\mathbf{r}\prime)]_{\pm}\) (assuming \(\mathbf{r} \neq \mathbf{r}\prime\)): \([\hat{a}^{\dagger}(\mathbf{r}),\hat{a}(\mathbf{r}\prime)]_{\pm} = \delta(\mathbf{r} - \mathbf{r}\prime)\) \(\hat{a}^{\dagger}(\mathbf{r})\hat{a}(\mathbf{r}\prime) = \pm \hat{a}^{\dagger}(\mathbf{r})\hat{a}(\mathbf{r}\prime)\)
 * 6) Substitute this into the potential: \(V=\pm \int d\mathbf{r}\,d\mathbf{r}\prime V(\mathbf{r}-\mathbf{r}\prime)\rho(\mathbf{r})\rho(\mathbf{r}\prime)\): \(V=\int d\mathbf{r}\,d\mathbf{r}\prime V(\mathbf{r}-\mathbf{r}\prime)\hat{\phi}^{\dagger}(\mathbf{r})\hat{\phi}^{\dagger}(\mathbf{r}\prime)\hat{\phi}(\mathbf{r})\hat{\phi}(\mathbf{r}\prime)\)
 * 7) Find \([\hat{a}^{\dagger}(\mathbf{r}),\hat{a}^{\dagger}(\mathbf{r}\prime)]_{\pm}\): \([\hat{a}^{\dagger}(\mathbf{r}),\hat{a}^{\dagger}(\mathbf{r}\prime)]_{\pm} = \delta(\mathbf{r} - \mathbf{r}\prime)\) \(\hat{a}^{\dagger}(\mathbf{r})\hat{a}^{\dagger}(\mathbf{r}\prime) = \pm \hat{a}^{\dagger}(\mathbf{r}\prime)\hat{a}^{\dagger}(\mathbf{r})\)
 * 8) Substitute this into the potential: \(V=\int d\mathbf{r}\,d\mathbf{r}\prime V(\mathbf{r}-\mathbf{r}\prime)\hat{\phi}^{\dagger}(\mathbf{r}\prime)\hat{\phi}^{\dagger}(\mathbf{r})\hat{\phi}(\mathbf{r})\hat{\phi}(\mathbf{r}\prime)\)