(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that [itex] <p> = m \frac{d<x>}{dt} [/itex]

2. Relevant equations

Schrödinger Equation: [itex]i\hbar[/itex] [itex]\frac{\partial \Psi} {\partial x}[/itex] = -[itex] \frac{\hbar^2}{2m}[/itex] [itex]\frac{\partial^2 \Psi}{\partial x^2}[/itex] + [itex]V{} \Psi[/itex]

Respective complex conjugate from equation above

Expectation Position: <x> = [itex]\int_{-\infty}^{+\infty} x\Psi {\Psi}^*[/itex] dx

3. The attempt at a solution

Derive <x> with respect to t... with V real, we know that V = V*, and after some basic steps we get:

[itex]\frac {d<x>}{dt}[/itex] = [itex]\frac{i \hbar}{2m}[/itex] [itex]\int[/itex] [itex]dx[/itex] [itex]x[/itex][[itex]\Psi^*[/itex][itex](\frac{\partial^2 \Psi}{\partial x^2}[/itex]) - [itex]\Psi[/itex] [itex](\frac{\partial^2 \Psi^*}{\partial x^2})[/itex]]

Then my problem is with the integration by parts... for

[itex] \int_{a}^{b}[/itex] [itex] f \frac{dg}{dx} dx [/itex] = [itex] fg [/itex] [itex]{|}^{b}_{a}[/itex] - [itex] \int_{a}^{b}[/itex] [itex] g \frac{df}{dx} dx [/itex]

I'm choosing [itex]f = x\Psi^*[/itex] and [itex] g = \frac{\partial \Psi}{\partial x}[/itex], but I think I'm not getting right these limits considerations... any sugestions or enlightenments?

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EDIT ([itex]\frac{\partial \Psi}{\partial}[/itex] with respect to time, not position)

Schrödinger Equation: [itex]i\hbar[/itex] [itex]\frac{\partial \Psi} {\partial t}[/itex] = -[itex] \frac{\hbar^2}{2m}[/itex] [itex]\frac{\partial^2 \Psi}{\partial x^2}[/itex] + [itex]V{} \Psi[/itex]

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# Homework Help: (Quantum Mechanics) Prove that <p> = m (d<x>/dt)

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