Corrector for the Ehrenfets equation

[carllacan]

Homework Statement

The potential V(x) in the equation
m\frac{d^2}{dt^2}=-\left \langle \frac{d\hat{V}}{dx} \right \rangle
changes very slowly for the typical wavelength wavefunction. Calculate the lowest corrector for the classical equation of motion.

Homework Equations

The Ehrenfest Theorem
m\frac{d^2}{dt^2}=-\left \langle \frac{d\hat{V}}{dx} \right \rangle

The Attempt at a Solution

I don't understand the question. I can't find in any book a mention of a corrector for the Ehrenfest equation. And what does it mean with the wavelentgh of the wavefunction?

Thank you for your time.

 

[BvU]

Funny, your only relevant equation is also a given in this exercise. My (very old ) QM book has ${d\over dt}<{\bf p}> = -\int \Psi^* (\nabla V)\Psi\, d\tau = -<\nabla V> = <{\bf F}>$as Ehrenfest's theorem; with the comment: "this is simply Newton's law, but now for expectation values".

Must say my book is easier to understand for me than your rendering of he exercise: $m\frac{d^2}{dt^2}\,$ looks like an operator to me, not an expectation value like $-\left \langle \frac{d\hat{V}}{dx} \right \rangle\,$.

So I am on your side in "not understanding the question". I need some reassurance this really is exactly how the exercise was formulated...

This link, by prof. Fitzpatrick, Texas university Austin, sides with Eugen Merzbacher. It makes me think a <x> fell by the wayside somewhere...

Wavelength of the wave function generally has a $\hbar$ in it somewhere, making the wavelength real small compared to change in V.