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, \[F(\omega) \;=\; \int_{-\infty}^\infty dt \; e^{i\omega t}\, f(t) \;=\; \int_0^\infty dt \; e^{i(\omega + i\varepsilon) t} \; \langle\varphi|e^{-i\hat{H}t/\hbar}|\varphi\rangle.\] Now insert a resolution of the identity, \(\hat{I} = \sum_n |n\rangle\langle n|\), where \(\{|n\rangle\}\) denotes the exact eigenstates of \(\hat{H}\) (for free states, the sum goes to an integral in the usual way): \[\begin{align} \begin{aligned}F(\…
