Projecting on momentum states, we obtain the momentum space Schroedinger equation as $$ \begin{equation} \frac{\hbar^2}{2\mu}k^2\psi_n(\mathbf{k})+\int d\mathbf{k'}V(\mathbf{k}, \mathbf{k'}) \psi_n(\mathbf{k'})=E_n \psi_n(\mathbf{k}) \tag{1} \end{equation} $$ Here the notation \( \psi_n(\mathbf{k}) =\langle\mathbf{k}\vert\psi_n\rangle \) and \( \langle\mathbf{k}\vert V\vert\mathbf{k}' \rangle =V(\mathbf{k}, \mathbf{k'}) \) has been introduced. The potential in momentum space is given by a double Fourier-transform of the potential in coordinate space, i.e. $$ V(\mathbf{k},\mathbf{k'}) = \left( \frac{1}{2\pi}\right)^3\int d\mathbf{r}\int d\mathbf{r}'\exp{-i\mathbf{kr}}V(\mathbf{r},\mathbf{r}')\exp{i\mathbf{k}'\mathbf{r}'} $$