Solving Schrödinger equation
The Hamiltonian of the \(N\)-particle system is given by
\begin{equation} \hat{H} = \sum_{i=1}^N \biggl[-\frac{\hbar^2}{2m} \nabla_i^2 + V_{\text{mean}}(\mathbf{r}_i) + V_{\text{SO}}(\mathbf{r}_i) (\mathbf{l}_i \cdot \mathbf{s}_i)\biggr] + \sum_{i < j=2}^N V_{ij}(\mathbf{r}_{ij}) \end{equation}1. Particle in a spherically symmetric potential
A particle of mass \(M\) moving in a mean spherically symmetric potential \(V(r)\) are determined by the Schrödinger equation
\begin{equation} \biggl[\nabla^2 + \frac{2M}{\hbar^2} \bigl(E - V(r)\bigr)\biggr] \psi(r) = 0 \end{equation}where \(E\) is the energy eigenvalue. The solution is of the form
\begin{equation} \psi_{nlm}(r) = u_{nl}(r) Y_{lm}(\theta,\phi) \end{equation}where \(Y_{lm}(\theta,\phi)\) are the spherical harmonics and \(u_{nl}\) is the radial function.
2. Independent/individual particle model
The actual wave function \(\Psi\) of the nucleus is obtained by solving the Schrödinger equation
\begin{equation} \hat{H}\Psi = E\Psi \end{equation}We expand \(\ket{\Psi}\) in terms of a set of normalized many-body basis functions
\begin{equation} \Psi = \sum_m^M c_{mn} \psi_m \end{equation}Inserting this expansion on both sides of the TISE and then taking the inner product with an arbitrary basis function \(\psi_l\), we obtain the TISE in matrix form
\begin{equation} \begin{bmatrix} H_{11} & \ldots & H_{1M} \\ \vdots & & \vdots \\ H_{M1} & \ldots & H_{MM} \end{bmatrix} \begin{bmatrix} c_{1n} \\ \vdots \\ c_{Mn} \end{bmatrix} = E_n \begin{bmatrix} c_{1n} \\ \vdots \\ c_{Mn} \end{bmatrix} \end{equation}where
\begin{equation} H_{mn} = \bra{\varphi_m} \hat{H} \ket{\varphi_n} \end{equation}then we can solve for the eigenvalues \(E_0, E_1, \ldots, E_M\).
Second quantization gives
\begin{equation} \hat{H} = \sum_m \epsilon_m \hat{a}^{\dagger}_m \hat{a}_m \end{equation}