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## A short derivation of the time dependent pertubation result.

Posted by peeterjoot on December 9, 2011

[Click here for a PDF of this post with nicer formatting and figures if the post had any (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

# Guts

A super short derivation of the time dependent pertubation result. With

\begin{aligned}{\left\lvert {\psi{t}} \right\rangle} = \sum_k c_k(t) e^{-i\omega_k t} {\left\lvert {k} \right\rangle}\end{aligned} \hspace{\stretch{1}}(1.1)

\begin{aligned}0&=\left( H_0 + H' - i\hbar \frac{d}{dt} \right){\left\lvert {\psi{t}} \right\rangle} \\ &=\left( H_0 + H' - i\hbar \frac{d}{dt} \right)\sum_k c_k e^{-i\omega_k t} {\left\lvert {k} \right\rangle} \\ &=\sum_k e^{-i\omega_k t} \left(\not{{c_k E_k}} + H' c_k - \not{{i\hbar (-i \omega_k) c_k}} -i\hbar c_k'\right){\left\lvert {k} \right\rangle}\end{aligned}

Bra with ${\left\langle {m} \right\rvert}$

\begin{aligned}\sum_k e^{-i\omega_k t} H'_{mk} c_k =i\hbar e^{-i\omega_m t} c_m',\end{aligned} \hspace{\stretch{1}}(1.2)

or

\begin{aligned}c_m'=\frac{1}{{i\hbar}}\sum_k e^{-i\omega_{km} t} H'_{mk} c_k \end{aligned} \hspace{\stretch{1}}(1.3)

Now we can make the assumptions about the initial state and away we go.