I have some difficulties when I am reading time-independent perturbation theory of Sakurai's book "Modern Quantum Mechanics" p.290. Let me introduce the background of this question first. Originally, the unperturbed Schrödinger's equation is:
\begin{equation} H_{0}|n^{0}> = E^{0}_{n}|n^{0}> \tag{1} \end{equation} where $|n^{0}>$ is the unperturbed eigenstates. However, if there is a small perturbation that perturbes the system, the Schrödinger's equation becomes the follows: \begin{equation} (H_{0}+\lambda V)|n> = E_{n}|n> \tag{2} \end{equation} $\lambda V$ is the perturbation of the system and $\lambda$ is a real parameter to control the perturbation.
We denote the energy difference of the perturbed energy and the unperturbed energy as $\Delta_{n} = E_{n} - E^{0}_{n}$
Then, Sakurai showed the eigenstate $|n> $ as follows: \begin{equation} |n> = |n^{0}> + \frac{1}{E^{0}_{n} - H_{0}}\phi_{n}(\lambda V- \Delta_{n})|n> \tag{3} \end{equation} And $\phi_{n}$ is the complementary projective operator \begin{equation} \phi_{n} = 1 - |n^{0}><n^{0}| = \Sigma_{k \neq n} |k^{0}><k^{0}| \tag{4} \end{equation}
The difficulties that I encountered is the following, Sakurai said that it was customary to write: \begin{equation} \frac{1}{E^{0}_{n} - H_{0}}\phi_{n} \rightarrow \frac{\phi_{n}}{E^{0}_{n} - H_{0}} \tag{5} \end{equation}
and similarly
\begin{equation} \frac{1}{E^{0}_{n} - H_{0}}\phi_{n} = \phi_{n}\frac{1}{E^{0}_{n} - H_{0}}= \phi_{n}\frac{1}{E^{0}_{n} - H_{0}}\phi_{n} \tag{6} \end{equation}
The order of the operator is important as operators are not commute in general. Therefore, I do not know why Sakurai can "customary" to write equation(5) . Besides, Why Sakurai can change the order of $\phi_{n}$ and $\frac{1}{E^{0}_{n}-H_{0}}$, and multiply $\phi_{n}$ at both left and right of $\frac{1}{E^{0}_{n}-H_{0}}$ in equation(6)$. Can anyone help me to solve my problem?
By "customary" he means this is a "shorthand" for eqn. (5.1.24). We are supposed to understand this as:
1) Consider the restriction of ${E^{0}_{n} - H_{0}}$ to the complement space as an operator (call it $G$) in that space.
2) $G$ is strictly positive so it has an inverse as an operator in that space: $G^{-1}$.
3) Form an operator on the full space equal to $G^{-1}$ on the complement space and $x$ on the $n^0$ space. $x$ can be any number, it doesn't matter since the full operator will always be be used with a projection operator on one side.