# Momentum in Quantum Mechanics

For a particle in state $\Psi$, the expectation value of $x$ is $\langle x\rangle=\int_{-\infty}^{+\infty}x|\Psi(x,t)|^2\,dx$.

$\langle p\rangle=m\frac{d\langle x\rangle}{dt}=-i\hbar\int_{-\infty}^{+\infty}\left(\Psi^*\frac{\partial\Psi}{\partial x}\right)\,dx$.

In general, $\langle Q(x,p)\rangle=\int_{-\infty}^{+\infty}\Psi^*\,Q\left(x,\frac{\hbar}{i}\frac{\partial}{\partial x}\right)\Psi\,dx$.

For example, $T=\frac12mv^2=\frac{p^2}{2m}$, so $\langle T\rangle=-\frac{\hbar^2}{2m}\int_{-\infty}^{+\infty}\Psi^*\frac{\partial^2\Psi}{\partial x^2}\,dx$.