The Schrödinger equation
In order to describe the quantum mechanics of a particular system, we know that we should solve the equation of motion for the \(\require{physics} N\)-particle wave function \(\Psi(t, \vb{r}_1, \cdots, \vb{r}_N)\): \[\begin{equation} i \hbar \pdv{t} \Psi(t, \vb{r}_1, \cdots, \vb{r}_N) = % \mathcal{H}^c \thinspace \Psi(t, \vb{r}_1, \cdots, \vb{r}_N) % \thinspace , \end{equation}\] which is called the Schrödinger equation and in which \(\mathcal{H}^c\) represents the Schrödinger Hamiltonian in coordinate representation. For a time-independent Hamiltonian, we can factorize the wave function and thus the stationary (time-independent) Schrödinger equation \[\begin{equation} \mathcal{H}^c \thinspace \Psi(\vb{r}_1, \cdots, \vb{r}_N) = % E \thinspace \Psi(\vb{r}_1, \cdots, \vb{r}_N) % \thinspace , \end{equation}\] is sufficient to obtain a characterization of the system with energy \(E\).
This means that, on the one hand, we are looking for the valid description of the stationary states \(\Psi(\vb{r}_1, \cdots, \vb{r}_N)\) and on the other hand we should find correct expressions for the Hamiltonian \(\mathcal{H}^c\), because it includes the physics of the interactions that are present in this system.
Readers who are unfamiliar with quantum mechanics are pointed to the excellent works by Bransden and Joachain (Bransden and Joachain 2000) and Dickhoff and Van Neck (Dickhoff and Van Neck 2008).