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\begin{document}

Perturbation theory is divided into:

\begin{itemize}

\item time independent perturbation theory (TIPT)

\item degenerate perturbation theory (DPT)

\item WKB and variational approximation 

\item time dependent perturbation theory (TDPT)

\end{itemize}

Perturbation theory is used for problems which are too difficult.
\( H|n>=E_{n}|n> \). Assume H is dividable into a solvable part: \( H=H_{0}+H_{1} \) with \( H_{0}>>H_{1} \)and \( H_{0}|n^{(0)}>=E_{n}|n^{(0)}> \)

Example: Helium \( H=\frac{p_{1}^{2}}{2m}+\frac{p_{2}^{2}}{2m}-\frac{Ze}{r_{1}}-\frac{Ze}{r_{2}}+\frac{e^{2}}{|r^{i}_{1}-r_{2}^{i}|} \) The last term is \( H_{1} \), the other terms are just the
solution for the hydrogen twice in a tensor product. Solution of \( H_{0} \),
\( <r_{1}^{i}r_{2}^{j}|n_{1}l_{1}m_{1}n_{2}l_{2}m_{2}>=R_{nl_{1}}(r_{1})R_{nl_{2}}(r)Y_{l_{1}m_{1}}(r_{1}^{i})Y_{l_{2}m_{2}}(r_{2}^{i}) \). Then \( E_{n}^{(0)}=-Z^{2}R_{y}(\frac{1}{n_{1}^{2}}+\frac{1}{n_{2}^{2}}) \).

In matrix language: \( H|n>=E_{n}|n> \) so \( <p^{(0)}|H|m^{(0)}><m^{(0)}|n>=E_{n}<p^{(0)}|n> \). Unfortunately, this is not easily solvable.

Method: start with one \( |n^{(0)}> \) and see how it changes as \( H_{1} \)is turned on.
\( H=H_{0}+\lambda H_{1} \) for \( \lambda  \) very small.

Theory developed in perturbation expansion.

\end{document}
