D.36 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 $H\Psi/{\rm i}\hbar$ 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 $-{\rm i}$.