Particles’ wave function, $\Psi(x, t)$, by solving Schrodinger equation
\[i\hbar \frac{\partial\Psi}{\partial t} = - \frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V \Psi\]$\left| \Psi(x, t) \right|^2$ is the probability density.
\[\int_{-\infty}^{+\infty} \left| \Psi(x, t) \right|^2 dx = 1\]Average position and momentum
\[\left< x \right> = \int_{-\infty}^{+\infty} x \left | \Psi(x, t) \right |^2 dx\] \[\left< p \right> = m \frac{d \left< x \right>}{dt} = -i\hbar \int \left( \Psi^* \frac{\partial \Psi}{\partial x} \right)dx\]As operator
\[\left< Q(x, p) \right> = \int \Psi^* \left[ Q\left(x, -i\hbar \frac{\partial}{\partial x}\right) \right]\Psi dx\]