Let us do this for just the first two terms in the series: \[\begin{align} \begin{aligned}f(\mathbf{k}_i\rightarrow \mathbf{k}_f) &\approx - \frac{2m}{\hbar^2} \; 2\pi^2 \Bigg[\int d^3r_1\; \frac{\exp...Let us do this for just the first two terms in the series: \[\begin{align} \begin{aligned}f(\mathbf{k}_i\rightarrow \mathbf{k}_f) &\approx - \frac{2m}{\hbar^2} \; 2\pi^2 \Bigg[\int d^3r_1\; \frac{\exp(-i\mathbf{k}_f \cdot \mathbf{r}_1)}{(2\pi)^{3/2}} \, V(\mathbf{r}_1) \, \frac{\exp(i\mathbf{k}_i \cdot \mathbf{r}_1)}{(2\pi)^{3/2}} \\&\qquad\qquad\quad + \int d^3r_1 \!\! \int d^3r_2 \; \frac{\exp(-i\mathbf{k}_f \cdot \mathbf{r}_2)}{(2\pi)^{3/2}} \, V(\mathbf{r}_2) \, \langle\mathbf{r}_2|\hat{G}_…
The coefficients of the successive power of r are the Legendre polynomials; the coefficient of rl, which is Pl(x), is the Legendre polynomial of order l, and it is a polynomial in \(...The coefficients of the successive power of r are the Legendre polynomials; the coefficient of rl, which is Pl(x), is the Legendre polynomial of order l, and it is a polynomial in x including terms as high as xl.
