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6.4: Fermi’s Golden Rule
https://phys.libretexts.org/Bookshelves/Nuclear_and_Particle_Physics/Introduction_to_Applied_Nuclear_Physics_(Cappellaro)/06%3A_Time_Evolution_in_Quantum_Mechanics/6.04%3A_Fermis_Golden_Rule
\[\sum_{k}\left[i \hbar \dot{c}_{k}(t) e^{-i \omega_{k} t} u_{k}(x)\right. \left.+\hbar \omega c_{k}(t) e^{-i \omega_{k} t} u_{k}(x)\right]= \sum_{k}\left[c_{k}(t) e^{-i \omega_{k} t} \hbar \omega_{k}... $\sum_{k}\left[i \hbar \dot{c}_{k}(t) e^{-i \omega_{k} t} u_{k}(x)\right. \left.+\hbar \omega c_{k}(t) e^{-i \omega_{k} t} u_{k}(x)\right]= \sum_{k}\left[c_{k}(t) e^{-i \omega_{k} t} \hbar \omega_{k} u_{k}(x)+\right. \left.c_{k}(t) e^{-i \omega_{k} t} \hat{V}\left[u_{k}(x)\right]\right] \nonumber$

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