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##\left(\nabla ^2+k^2\right)\psi =V \psi##

Of which there are two solutions, the homogeneous solution which tends to the incident plane wave, and a particular solution which includes the scattering potential.

It then says that the general solution to this equation is

## \psi(r) = \phi_{\text{inc}} (r) + \frac{2 \mu}{\hbar^2} \int G(r-r') V(r') d^3 r ##

Then that the Green's function is obtained by solving the point source equation

## (\nabla^2 + k^2) G(r-r') = \delta (r-r') ##

I have some questions

1. Why does the homogeneous solutions tend to the incident plane wave? Does this just mean at large r the interaction potential is basically not felt?

2. How is the solution obtained with the Green's function? When will you know when to use a Green's function, why is this the time?

3. The solution to the point source equation is quite easy to get but I don't understand why we have asserted that that is the equation which we should solve