[NOTES/QM-18011] Green Function for Perturbative Solution of Scattering

For the scattering problem we need to set up an integral equation for the Schrodinger equation. \begin{equation}\label{EQ34} -\frac{\hbar^2}{2\mu} \nabla^2 \psi(\vec{r}) + V(\vec{r})\psi(\vec{r}) = E\psi(\vec{r}). \end{equation} We write the Schrodinger equation as \begin{equation}\label{EQ35} \big( \nabla^2 \psi(\vec{r}) + k^2 \psi(\vec{r})\big) = \frac{2\mu}{\hbar^2} V(\vec{r})\psi(\vec{r}) , \end{equation} where \(k^2= \sqrt{\frac{2\mu E}{\hbar^2}}\), and introduce the Green function as solution of the equation \begin{equation}\label{EQ36} \big( \nabla^2 \psi(\vec{r}) + k^2 \psi(\vec{r})\big)G(\vec{r}) = -\delta(\vec{r}). \end{equation} The solutions for the \MkGBox{Green function can be obtained}{qm}{18002} using Fourier transform method and are given by \begin{equation}\label{EQ37} G_\pm(\vec{r}) = \frac{e^{\pm i kr}}{4\pi r}. \end{equation} Note that the Green function introduced here is for {\tt free particle Schrodinger equation} \eqref{EQ35}. This is because we wish to set up an integral equation for perturbative solution of the scattering problem. \MkGBox[4727]{It turns out that the Green function}{qm}{18003} \(G_+(\vec{r})\) gives solution obeying correct boundary condition for scattering problem.