Electrons exposed to time-dependent force

[M-Speezy]

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!

 

[Orodruin]

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).

 

[M-Speezy]

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).

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.

 

[Orodruin]

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.

 

[M-Speezy]

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.