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

\langle\Psi_t\vert A\Psi\rangle+
\langle\Psi\vert A\Psi_t\rangle+
\langle\Psi\vert A_t\Psi\rangle

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 $\vphantom{0}\raisebox{1.5pt}{$-$}$${\rm i}$.