This equation forms the starting point for all perturbative derivations. However, as it stands it represents nothing but a mere formal rewriting of Schroedinger's equation and is not of much practical use. The exact wave function \( \vert \Psi_0\rangle \) is unknown. In order to obtain a perturbative expansion, we need to expand the exact wave function in terms of the interaction \( \hat{H}_I \).

Here we have assumed that our model space defined by the operator \( \hat{P} \) is one-dimensional, meaning that $$ \hat{P}= \vert \Phi_0\rangle \langle \Phi_0\vert , $$ and $$ \hat{Q}=\sum_{m=1}^{\infty}\vert \Phi_m\rangle \langle \Phi_m\vert . $$