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\begin{document}


\section{time dependent perturbation theory}

\begin{itemize}

\item \( H=H_{0}+H'(t) \) 

\item \( H_{0}|n>=E_{n}|n> \)

\end{itemize}

at t=0, \( \psi =|n> \)

Turn on perturbation \( H'(t) \) and at a later time, what is the probability
of a transition from \( |n>\rightarrow |m> \)?

\( \psi (t)=\sum _{n}C_{n}(t)e^{-iE_{n}t/\hbar }|n> \) with \( C_{n}(t)=e^{iE_{n}t/\hbar }<n|\psi (t)> \) then p(transition) = \( |C_{m}(t)|^{2} \).

Use time dependent schrodinger equation to get: 

\( \sum _{n}i\hbar c_{n}'e^{-iE_{n}t/\hbar }|n>=\sum _{n}c_{n}e^{-iE_{n}t/\hbar }H'|n> \). Projecting out, you get: \( i\hbar c_{m}'e^{-iE_{m}t/\hbar }=\sum _{n}c_{n}e^{-iE_{n}t/\hbar }<m|H'|n> \). \( i\hbar c_{m}'=\sum _{n}c_{n}e^{i(w_{mn}=(E_{m}-E_{n})/\hbar )t}H^{'}_{mn} \).

The time dependent perturbation expansion gives: 

\( c_{m}(t)=c_{m}(t=0)-\frac{i}{\hbar }\sum _{n}\int _{0}^{t}H_{I_{mn}}'(t)C_{n}(t)dt' \). Do a series expansion. This gives:

\begin{itemize}

\item 0th order: \( C_{m}(t)=C_{m}(0) \)

\item 1st order: \( C_{m}(t)=C_{m}(0)-\frac{i}{\hbar }\sum _{n}\int _{0}^{t}H_{I_{mn}}'(t)C_{n}(0)dt \)

\item 2nd order: \( C_{m}(t)=C_{m}(0)-\frac{i}{\hbar }\sum _{n}\int _{0}^{t}H_{I_{mn}}'(t)C_{n}(0)dt \) \( +\frac{-1}{\hbar ^{2}}\sum _{n,l}\int _{0}^{t}dt'\int _{0}^{t'}dt''H_{I_{ml}}'(t')H_{I_{ln}}(t'')C_{n}(0)dt \)

\item Exactly: \( C_{m}(t)=\sum _{n}T*exp(\frac{-i}{\hbar }\int _{0}^{t}H_{I_{mn}}'(t')dt')c_{n}(0) \) is the exact dyson sum.

\end{itemize}

Example:

\begin{itemize}

\item oscillating magnetic field

\end{itemize}

\end{document}
