\[\frac{d}{d t} \operatorname{Tr}\left[\left|\psi_{S}(t)\right\rangle\left\langle\psi_{S}(t)\right| A_{S}\right]=\frac{d}{d t}\left\langle\psi_{S}(t)\left|A_{S}\right| \psi_{S}(t)\right\rangle=\frac{d...ddtTr[|ψS(t)⟩⟨ψS(t)|AS]=ddt⟨ψS(t)|AS|ψS(t)⟩=ddt⟨ψH|AH(t)|ψH⟩ Using |ψ(t)⟩I=U†0(t)|ψ(t)⟩S with U0(t)=exp(−iH0t/ℏ), calculate the time dependence of an operator in the interaction picture AI(t).
