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Electrons exposed to time-dependent force

  1. Dec 23, 2015 #1
    I have begun studying Ashcroft + Mermin on my own, and am trying to follow the math in the text. They suggest that an electron in a metal with some momentum p(t) and exposed to a force f(t) will at some time later (t+dt) have a contribution to the momentum on the order of f(t)dt plus another term on the order of dt*dt. My question is where does this dt^2 term enter? My instinct is to say that F=dp/dt, and that the change in momentum can then be given (very simplistically?) by f(t)dt. Obviously, a form of f(t) and an integral is in order, but I cannot see the logic of what is stated in the text.

    Any and all help or guidance on the matter would be greatly appreciated!
     
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  3. Dec 23, 2015 #2

    Orodruin

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    The term of order dt^2 comes from the fact that f(t) is not necessarily constant in time. It is related to the derivative of f(t).
     
  4. Dec 23, 2015 #3
    Why does this matter, though? I would think Newton's 2nd law would be used, and then a change in the momentum would simply be given by f(t)dt. I'm not sure what else should be done to lead to anything else.
     
  5. Dec 23, 2015 #4

    Orodruin

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    It is just using Newton's second law, but Newton's second law is just the limit when dt goes to zero and so agrees with your result.
     
  6. Dec 23, 2015 #5
    I figured it out I think. If the force at t+dt is instead expressed using a first-order approximated taylor series, then the extra dt comes out.
     
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