Momentum in Quantum Mechanics

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

langle prangle=mfrac{dlangle xrangle}{dt}=-ihbarint_{-infty}^{+infty}left(Psi^*frac{partialPsi}{partial x}right),dx.

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

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

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Author: Evgeny Adamenkov

Programmer/Owner at Eugene AI Studio

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