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

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

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

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

The energy of the oscillator is

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

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

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

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