# Why the oscillator would vibrate without a driving force?

I am going through "Optics" by E. Hecht and I think I am misnderstanding the following paragraph:

$x(t) = \frac{(q_e/m_e)}{(w_0^2-w^2)}E(t)$

This is the relative displacement between the negative cloud and the positive nucleus. [...]

Withouth a driving force (no incident wave), the oscillator will vibrate at its resonance frequency $w_0$.

So, I understand how the frequency of the incident wave would influence the oscillation, but how a particle is oscillating without any driving force?

From the equation: if $E(t)=0$, then why there is an oscillation at $w_0$?

• -1 Not clear what you are asking. You are surely familiar with a mass on a spring which continues oscillating after an applied force has been removed? ... The equation does not make sense because the displacement can be out of phase with the applied field. Please can you post an image of the surrounding text? – sammy gerbil Dec 1 '17 at 16:13

The equation describing a driven (undamped) oscillator is: $$\ddot x +\omega_0^2x=f(t)$$ where $f(t)$ is the driving force, and $\omega_0$ the natural frequency of the oscillator. If $f(t)=0$, the solution is given by: $$x(t)=A\cos(\omega_0 t)+B\sin(\omega_0 t)$$ so the system oscillates nonetheless. If you add the force, the exact solution depends on the force that you apply. However, if the driving force is "oscillating", then it will be of the form $f(t) = C\sin{\omega t}$, where $\omega$ is the frequency of the oscillations of this force. If $\omega=\omega_0$ things are a bit more complicated, because you get resonant behaviour. In the easier case $\omega \neq \omega_0$, a solution will be given by: $$x(t)=\frac{1}{\omega_0^2-\omega^2} f(t)$$ By substitution, it's easy to check that this is a solution. However, nothing prevents you from adding the previous, undriven solution. After all, it gives zero! In other words, the general solution is given by:
$$x(t)=A\cos(\omega_0 t)+B\sin(\omega_0 t)+\frac{1}{\omega_0^2-\omega^2} f(t)$$
So as expected if $f(t)=0$ you recover the undriven solution, which is oscillating.