Given a wave packet:
$\psi({\it x,t}) = \frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty} exp[i(px - \frac{p^{2}}{2m}t)/\hbar]\phi(p){\it dp} $$$\psi({\it x,t}) = \frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty} \exp\left[\frac{i}{\hbar}\left(px - \frac{p^{2}}{2m}t\right)\right]\phi(p){\it dp} $$
we are asked to find that $\langle p \rangle$ does not change with time. So it's basically asking us to show $\frac{d}{dt}\langle p\rangle=0$. I know that:
$\frac{d}{dt}\langle p\rangle= -\langle \frac{\partial V}{\partial x}\rangle$$$\frac{d}{dt}\langle p\rangle= -\left< \frac{\partial V}{\partial x}\right>$$
But I'm unsure how to solve for V in this case. Additionally, I've been advised to use the formula:
$\langle p_{x}\rangle = \int \psi^{*}({\bf r},t)(-i\hbar \frac{\partial}{\partial x})\psi ({\bf r},t) d{\bf r}$$$\langle p_{x}\rangle = \int \psi^{*}({\bf r},t)\left(-i\hbar \frac{\partial}{\partial x}\right)\psi ({\bf r},t) d{\bf r}$$
and that I will be using a delta function. I'm just kind of lost and trying to figure out which relevant equations to use for this problem and would appreciate some advice on how proceed.
Edit: The triple integral from the textbook my classmate suggested was:
$\langle p_{x}\rangle = (2\pi\hbar)^{-3}\int d{\bf p} \int d{\bf r} \int d{\bf r'} e^{i{\bf p}\cdot{\bf r'}/\hbar}\Psi ^{*}({\bf r'},t)(i\hbar\frac{\partial}{\partial x}e^{-i{\bf p}\cdot{\bf r}/\hbar})\Psi({\bf r},t) $$$\langle p_{x}\rangle = (2\pi\hbar)^{-3}\int d{\bf p} \int d{\bf r} \int d{\bf r'} e^{i{\bf p}\cdot{\bf r'}/\hbar}\Psi ^{*}({\bf r'},t)\left(i\hbar\frac{\partial}{\partial x}e^{-i{\bf p}\cdot{\bf r}/\hbar}\right)\Psi({\bf r},t) $$
I believe he then used the delta to achieve an exponent value of 1.
