(a) Show that if $V(r)$ is a two-particle interaction that depends only on the distance $r$ between the particles, the matrix element of the interaction in the k-representation may be reduced to
$$
\left\langle\mathbf{k}_3 \mathbf{k}_4|V| \mathbf{k}_1 \mathbf{k}_2\right\rangle=\delta\left(\mathbf{k}_1+\mathbf{k}_2-\mathbf{k}_3-\mathbf{k}_4\right) \frac{1}{(2 \pi)^3} \int V(r) e^{-t q \cdot \mathbf{r}} d^3 r
$$
where $\hbar \mathbf{q}$ is the momentum transfer $\hbar\left(\mathbf{k}_3-\mathbf{k}_1\right)$.
(b) For this interaction, show that the mutual potential energy operator is
$$
\mathscr{V}=\frac{1}{2} \iiint d^3 k_1 d^3 k_2 d^3 q \boldsymbol{\phi}^{\dagger}\left(\mathbf{k}_1+\mathbf{q}\right) \boldsymbol{\phi}^{\dagger}\left(\mathbf{k}_2-\mathbf{q}\right) \boldsymbol{\phi}\left(\mathbf{k}_2\right) \boldsymbol{\phi}\left(\mathbf{k}_1\right) \boldsymbol{F}(\mathbf{q})
$$
where $F(\mathbf{q})$ is the Fourier transform of the displacement-invariant interaction.