# Proving equivalence of first and second quantisation (Pathria's way)

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 $$\begin{equation} \left(-\frac{\hbar^2}{2m}\sum_{i =1}^N\nabla^2_i + \sum_{i where all the quantities are defined in previous pages, and the equation $$\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?

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.