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.

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