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\begin{document}

DPT = degenerate perturbation theory.

Do perturbation theory around a 0th order degenerate state.

Then \( |E_{n}^{(1)},K>=|E_{n}^{(0)},K>+\frac{Q}{E}V|E_{n}^{(0)},K>+... \)

If \( H_{1} \)partially splits the degeneracy, then the particular basis you
choose in the degenerate subspace is important.

Define projectors: 

\( P=\sum _{k=1}^{g}|E_{n}^{(0)}k><E_{n}^{(0)}k| \), \( Q=1-P \) P is the projector of the degenerate subspace. In the degenerate
subspace: \( H_{1}=PH_{1}P+PH_{1}Q+QH_{1}P+QH_{1}Q \). 

Define \( H_{0}^{'}=H_{0}+PH_{1}P \), \( H_{1}'=PH_{1}Q+QH_{1}P+QH_{1}Q \). Solving \( H_{0}' \) exactly is first order degenerate perturbation
theory. Solving \( H_{1}' \) is second order.

Solutions of \( H_{0}'=|E_{n}^{(1)},K> \) which are less degenerate... Then higher order approximations
don't have to worry about degeneracy.

\( |E_{n}^{(2)},K>=|E_{n}^{(1)},K>-\frac{(Q_{k}=1-|E_{n}^{(1)},K><E_{n}^{(1)},K|)}{H_{0}-E_{nK}^{(1)}}(V'=\triangle ^{(1)}-H_{1}')|E_{n}^{(1)}K> \)

\( =|E_{n}^{(1)},K>-\frac{QH_{1}}{H_{0}-E_{nK}^{(1)}}|E_{n}^{(1)}K> \)

And \( E_{nk}^{(2)}=E_{nk}^{(1)}-\sum _{p\neq n}\frac{|<p|H_{1}|nk^{(1)}>|^{2}}{E_{p}^{(2)}-E_{n}^{(1)}} \)

applications:

\begin{itemize}

\item explicitly construct gxg matrix and diagonalize

\item coupling angular momentum via clebsch-gordon gives diagonal basis

\end{itemize}

\end{document}
