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 scattering matrix relation can then be re-written as \[\begin{align} c^+_\mu &= c^+_{i,\mu} + c^+_{s,\mu} = \sum_{\mu\nu} S_{\mu\nu} c^-_{i,\nu} \\ \Rightarrow \;\;\; c^+_{s,\ell m} &= 2 \pi \sum_...The scattering matrix relation can then be re-written as c+μ=c+i,μ+c+s,μ=∑μνSμνc−i,ν⇒c+s,ℓm=2π∑ℓ′m′(Sℓm,ℓ′m′−δℓℓ′δmm′)eiℓ′π/2Y∗ℓ′m′(ˆki)Ψi. Using this, the scattered wavefunction can be written as \[\begin{align}\begin{aligned}\psi_s(\mathbf{r}) &= \sum_{\ell m} c^+_{s,\ell m} h_{\ell}…