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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(t)
and
cos(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

G^{ (R)}(t,t') =
 (5)

while the advanced Green's function is

G^{ (A)}(t,t') = .
 (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 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.
Next: 4 Solution using Green's
Up: Green's functions for the
Previous: 2 The Ideal Driven