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I'm trying to solve problem 11.1 form Pathria R. K. & Beale P. D. - Statistical mechanics book (the hyperlink will get you straight to the page of the problem). The point (b) is to show the equivalence of the two equations \begin{equation} \left(-\frac{\hbar^2}{2m}\sum_{i =1}^N\nabla^2_i + \sum_{i<j} U_{ij}\right)\Phi_{NE}(\vec{r}_1,\dots, \vec{r}_N) = E\Phi_{NE}(\vec{r}_1,\dots, \vec{r}_N), \end{equation} where all the quantities are defined in previous pages, and the equation \begin{equation} \frac1{\sqrt{N!}}\langle0| \hat{\Psi}(\vec{r}_1)\cdots\hat{\Psi}(\vec{r}_N)\hat{H}|\Phi_{NE}\rangle = E\Phi_{NE}(\vec{r}_1,\dots, \vec{r}_N), \end{equation} by making use of the commutation relation \begin{equation} [\hat{\Psi}(\vec{r}_j), \hat{H}] = \left(-\frac{\hbar^2}{2m}\nabla^2_j + \int\mathrm{d}\vec{r}\hat{\Psi}^\dagger(\vec{r})U(\vec{r}, \vec{r}_j)\hat{\Psi}(\vec{r})\right)\hat{\Psi}(\vec{r}_j). \end{equation} I was able to carry out the kinetic term $-\frac{\hbar^2}{2m}\sum_{i =1}^N\nabla^2_i$, but I can't figure out how to leave only the term $U_{ij}$ (which I presume being $U(r_i, r_j)$) because I get some other terms of the form $\int\mathrm{d}\vec{r}\hat{\Psi}^\dagger(\vec{r})U(\vec{r}, \vec{r}_j)\hat{\Psi}(\vec{r})$, which are non vanishing. Some ideas?

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Let's consider one-particle operators of the form $$ \hat{A} = \int d\vec{r}\ \hat{\Psi}^\dagger(\vec{r}) a(\vec{r}) \hat{\Psi}(\vec{r}). $$ Using commutation relations and vacuum vector properties it is straightforward to obtain following property of this operators $$ \left<0\right|\hat{\Psi}(\vec{r}_1)\cdots\hat{\Psi}(\vec{r}_{j-1})\hat{A} = \sum_{i=1}^{j-1} a(\vec{r}_i) \left<0\right|\hat{\Psi}(\vec{r}_1)\cdots\hat{\Psi}(\vec{r}_{j-1}).\qquad (*) $$ If we take $a(\vec{r}) = U(\vec{r},\vec{r}_j)$, then it is not difficult to get from $(*)$ to the equality $$ \left<0\right|\hat{\Psi}(\vec{r}_1)\cdots\hat{\Psi}(\vec{r}_{j-1})\int d\vec{r}\ \hat{\Psi}^\dagger(\vec{r})U(\vec{r},\vec{r}_j)\hat{\Psi}(\vec{r})\hat{\Psi}(\vec{r}_j) \cdots\hat{\Psi}(\vec{r}_N)\left|\Phi_{NE}\right>\ = $$ $$ =\ \sum_{i=1}^{j-1} U(\vec{r}_i,\vec{r}_j) \left<0\right|\hat{\Psi}(\vec{r}_1)\cdots\hat{\Psi}(\vec{r}_N)\left|\Phi_{NE}\right> $$ Now it is easy to finish the solution.

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