3 Explicit Expressions for the Advanced and Retarded Green's functions

We can solve these equations just like we did for the coulomb Green's functions. For the regions t < t' , and t > t' , the solution is the same as the unforced oscillator: for example a linear combination of sin($\omega_{0}^{}$t) and cos($\omega_{0}^{}$t) . The boundary conditions at t = t' can be obtained either physically or mathematically. The displacement is continuous at t = t' , and integrating across the boundary gives the physical result that an impulsive force gives a discontinuous change in momentum. The retarded Green's function is

 

$$\left \{ \begin{array} {ll} 0 & t< t'\\ \frac{\lambda }{m \omega_0} \sin(\omega_0 [t-t']) & t \gt t'\\ \end{array}\right.$$ (5)

while the advanced Green's function is

 

$$\left \{ \begin{array} {ll} -\frac{\lambda }{m \omega_0} \sin(\omega_0 [t-t']) & t < t'\\ 0 & t \gt t'\\ \end{array}\right.\,$$ (6)

Clearly the sine term is a solution of the undriven harmonic oscillator, with the displacement zero both before and after the impulse. Calculating the momentum before and after the collision it is easy to see that it changes by $\lambda$ as required. It is very important to ignore the choice of t and t' , since these are both dummy variables. The important thing to realize is that the second variable, t' in Eqs. 5 and 6 is the variable that we take the derivatives of in the adjoint equation.