The equation of motion of a free electron in an electric field is
m((d^2)x)/(dt^2) = -eE (Eq. 1)
If x and E have the time dependence exp(iwt), where w=angular frequency, then
-(w^2)mx = -eE (Eq. 2)
What does the time dependent function whose second derivative equals -(w^2)x look like?
The Attempt at a Solution
I know from my papers that the electromagnetic wave is given by
and I guess the equation I'm after looks similar. The second derivative of x*exp(iwt) would give me the correct answer, but x=x0*exp(iwt) doesn't seem to do, because equation 2 says x and not x0.