The wave function \( \psi_n(\mathbf{k}) \) can be expanded in a complete set of spherical harmonics, that is $$ \begin{equation} \psi_n(\mathbf{k}) = \sum _{lm} \psi_{nlm}(k)Y_{lm}(\hat{k}) \hspace{1cm} \psi_{nlm}(k) = \int d\hat{k} Y_{lm}^*(\hat{k})\psi_n(\mathbf{k}). , \tag{2} \end{equation} $$ By inserting equation (2) in equation (1), and projecting from the left \( Y_{lm}(\hat{k}) \), the three-dimensional Schroedinger equation (1) is reduced to an infinite set of 1-dimensional angular momentum coupled integral equations, $$ \begin{equation} \left( \frac{\hbar^2}{2\mu} k^2-E_{nlm}\right)\psi_{nlm}(k) = -\sum_{l'm'}\int_{0}^\infty dk' {k'}^2 V_{lm, l'm'}(k,k') \psi_{nl'm'}(k') \tag{3} \end{equation} $$ where the angular momentum projected potential takes the form, $$ \begin{equation} V_{lm, l'm'}(k,k') = \int d{\hat{k}} \int d{\hat{k}'}Y_{lm}^*(\hat{k})V(\mathbf{k}\mathbf{k'})Y_{l'm'}(\hat{k}') \tag{4} \end{equation} $$ here \( d\hat{k} = d\theta\sin(\theta)d\varphi \). Note that we discuss only the orbital momentum, we will include angular momentum and spin later.