D.35 The evolution of expectation values

To verify the stated formulae for the evolution of expectation values, just write the definition of expectation value, $\langle\Psi\vert A\Psi\rangle$, differentiate to get

\begin{displaymath}
\langle\Psi_t\vert A\Psi\rangle+
\langle\Psi\vert A\Psi_t\rangle+
\langle\Psi\vert A_t\Psi\rangle
\end{displaymath}

and replace $\Psi_t$ by $H\Psi$$\raisebox{.5pt}{$/$}$${\rm i}\hbar$ on account of the Schrö­din­ger equation. Note that in the first inner product, the ${\rm i}$ appears in the left part, hence comes out as its complex conjugate $\vphantom0\raisebox{1.5pt}{$-$}$${\rm i}$.