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\begin{document}

\( (H_{0}+H_{1})|n>=(E_{n}^{(0)}+\triangle n)|n> \)

Projecting onto \( <n^{(0)}| \) you get: \( \triangle n=\frac{<n^{(0)}|H_{1}|n>}{<n^{(0)}|n>} \). if \( |n> \)is known to \( O(H_{1}^{p}) \) then \( \triangle n \) is known to
\( O(H_{1}^{p+1}) \).

Expansion for \( |n> \): \( (H_{0}-E_{n}^{(0)})|n>=(\triangle n-H_{1})|n>=V|n> \)(defining V)

\( |n>=|n^{(0)}>+\sum _{p\neq n}|p^{(0)}><p^{(0)}|\frac{V}{H_{0}-E_{n}^{(0)}}|n> \). Note that \( |n> \) is not normalized.

Define \( Q=|p^{(0)}><p^{(0)}| \). Then \( |n>=|n^{(0)}>+\frac{Q}{H_{0}-E_{n}^{(0)}}V|n> \) . Note that \( [Q,H_{0}]=0 \) and \( Q^{2}=Q \), \( Q|n^{(0)}>=0 \), and \( Q^{\dagger }=Q \). normalization: \( <n^{(0)}|n>=1 \).
Define \( e=H_{0}-E_{n}^{(0)} \). Then \( |n>=|n^{(0)}>+\frac{Q}{e}V|n>= \) \( |n^{(0)}>+\frac{Q}{e}V|n^{(0)}>+\frac{QV}{e}\frac{QV}{e}|n^{(0)}>+...=\frac{1}{1-\frac{QV}{e}}|n^{(0)}> \) . Then \( \triangle n=<n^{(0)}|H_{1}\frac{1}{1-\frac{Q}{e}V}|n^{(0)}> \)

\begin{enumerate}

\item zeroth order: \( \triangle n^{(0)}=0 \), \( |n^{(0)}>=|n^{(0)}> \)

\item 1st order: \( \triangle n^{(1)}=<n^{(0)}|H_{1}|n^{(0)}> \), \( |n^{(1)}>=|n^{(0)}>+\frac{Q}{e}V|n^{(0)}> \) \( = \) \( |n^{(1)}>=|n^{(0)}>-\frac{\sum _{p}|p^{(0)}><p^{0}|H_{1}|n^{0}>}{E_{p}^{(0)}-E_{n}^{(0)}} \) which breaks down for degeneracy.

\item 2nd order: \( \triangle n^{(2)}=<n^{(0)}|H_{1}|n^{(1)}> \) \( =<n^{(0)}|H_{1}|n^{(0)}>-\sum _{p\neq n}\frac{|H_{1_{np}}|^{2}}{E_{p}^{(0)}-E_{n}^{(0)}} \), \( |n^{(2)}>=|n^{(0)}>+\frac{Q}{e}H_{1}|n^{(0)}>+\frac{Q}{e}H_{1}\frac{Q}{e}H_{1}|n^{(0)}> \) \( -\triangle n^{(1)}\frac{Q^{2}}{e^{2}}H_{1}|n^{(0)}> \) 

\item higher orders: \( |n^{(p)}>=|n^{(0)}>+\frac{Q}{e}V|n^{(p-1)}> \), and \( \triangle n^{(p)}=<n^{(0)}|H^{1}|n^{p-1}> \)

\end{enumerate}

In general, corrections get smaller.

\end{document}
