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    • 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}...k[i˙ck(t)eiωktuk(x)+ωck(t)eiωktuk(x)]=k[ck(t)eiωktωkuk(x)+ck(t)eiωktˆV[uk(x)]]
    • https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_(Fowler)/09%3A_Perturbation_Theory/9.05%3A_Time-Dependent_Perturbation_Theory
      We look at a Hamiltonian with some time-dependent perturbation, so now the wavefunction will have perturbation-induced time dependence.
    • https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_(Chong)/02%3A_Resonances/2.04%3A_Fermi's_Golden_Rule
      We can determine f(t) by first studying its Fourier transform, \[F(\omega) \;=\; \int_{-\infty}^\infty dt \; e^{i\omega t}\, f(t) \;=\; \int_0^\infty dt \; e^{i(\omega + i\varepsilon) t} \; \langl...We can determine f(t) by first studying its Fourier transform, F(ω)=dteiωtf(t)=0dtei(ω+iε)tφ|eiˆHt/|φ. Now insert a resolution of the identity, ˆI=n|nn|, where {|n} denotes the exact eigenstates of ˆH (for free states, the sum goes to an integral in the usual way): \[\begin{align} \begin{aligned}F(\…

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