Consider a system of fermions created by the field $\psi$ $^{\dag }$ $\vec{(r)}$ interacting under the Yukawa potential

\begin{displaymath}
V(r) = \frac{Ae^{-\lambda r}}{4 \pi r}.
\end{displaymath}

(i) Write the Hamiltonian in second quantized form, using the position basis.

(ii) Write the Hamiltonian in second quantized notation in the momentum basis, where

\begin{displaymath}
c_{\vec{k}} \ ^{\dagger} = \int d^{3}r \psi ^{\dagger}(\vec{r})e^{i \vec{k}. \vec{r}}.
\end{displaymath}

You will find it helpful to derive the Fourier representation

\begin{displaymath}
V(r) = \int \frac{d^{3}q} {(2\pi)^{3}}
e^{i\vec{q}.\vec{r} } \frac{A}{(q^{2} + \lambda^{2})}.
\end{displaymath}