18.1: More General Energy Exchange

We’ll derive a formula for the energy fed into an oscillator by an arbitrary time-dependent external force.

The equation of motion can be written

$$\dfrac{d}{d t}(\dot{x}+i \omega x)-i \omega(\dot{x}+i \omega x)=\dfrac{1}{m} F(t)$$

and defining $\xi=\dot{x}+i \omega x$, this is

$$(d \xi / d t-i \omega \xi=F(t) / m$$

This first-order equation integrates to

$$\xi(t)=e^{i \omega t}\left(\int_{0}^{t} \dfrac{1}{m} F\left(t^{\prime}\right) e^{-i \omega t^{\prime}} d t^{\prime}+\xi_{0}\right)$$

The energy of the oscillator is

$$E=\dfrac{1}{2} m\left(\dot{x}^{2}+\omega^{2} x^{2}\right)=\dfrac{1}{2} m|\xi|^{2}$$

So if we drive the oscillator over all time, with beginning energy zero,

$$E=\dfrac{1}{2 m}\left|\int_{-\infty}^{\infty} F(t) e^{-i \omega t} d t\right|^{2}$$

This is equivalent to the quantum mechanical time-dependent perturbation theory result: $\xi, \xi^{*}$ are equivalent to the annihilation and creation operators.