\[\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...\[\frac{d}{d t} \operatorname{Tr}\left[\left|\psi_{S}(t)\right\rangle\left\langle\psi_{S}(t)\right| A_{S}\right]=\frac{d}{d t}\left\langle\psi_{S}(t)\left|A_{S}\right| \psi_{S}(t)\right\rangle=\frac{d}{d t}\left\langle\psi_{H}\left|A_{H}(t)\right| \psi_{H}\right\rangle\tag{3.5}\] Using \(|\psi(t)\rangle_{I}=U_{0}^{\dagger}(t)|\psi(t)\rangle_{S}\) with \(U_{0}(t)=\exp \left(-i H_{0} t / \hbar\right)\), calculate the time dependence of an operator in the interaction picture \(A_{I}(t)\).
The Schrödinger equation describes how the state of a system evolves. Since via experiments we have access to observables and their outcomes, it is interesting to find a differential equation that dir...The Schrödinger equation describes how the state of a system evolves. Since via experiments we have access to observables and their outcomes, it is interesting to find a differential equation that directly gives the evolution of expectation values.
