Consider a solution $\psi(x, t)$ of the time-dependent Schrödinger equation for a particle of mass $m$ in a potential $V(x)$. The expectation value of an operator $\mathcal{O}$ is defined as

$$\langle\mathcal{O}\rangle=\int d x \psi^{*}(x, t) \mathcal{O} \psi(x, t)$$

Show that

$$\frac{d}{d t}\langle x\rangle=\frac{\langle p\rangle}{m},$$

where

$$p=\frac{\hbar}{i} \frac{\partial}{\partial x},$$

and that

$$\frac{d}{d t}\langle p\rangle=\left\langle-\frac{\partial V}{\partial x}(x)\right\rangle$$

[You may assume that $\psi(x, t)$ vanishes as $x \rightarrow \pm \infty .]$