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Corrector for the Ehrenfets equation

  1. Jun 27, 2014 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations
    The Ehrenfest Theorem
    [itex]m\frac{d^2}{dt^2}=-\left \langle \frac{d\hat{V}}{dx} \right \rangle[/itex]

    3. 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.
     
  2. jcsd
  3. Jun 27, 2014 #2

    BvU

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    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.
     
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