Riley hobson bence · Problem 17.15

Eigenfunction methods for differential equations

Mathematics — Methods · greens-function-born-approximation-scattering

Problem

In the quantum-mechanical study of the scattering of a particle by a potential, a Born-approximation solution can be obtained in terms of a function

y(\mathbf{r})

that satisfies an equation of the form

(-\nabla^2 - K^2)y(\mathbf{r}) = F(\mathbf{r}).

Assuming that

y_{\mathbf{k}}(\mathbf{r}) = (2\pi)^{-3/2} \exp(i\mathbf{k}\cdot\mathbf{r})

is a suitably normalised eigenfunction of

-\nabla^2

corresponding to eigenvalue

k^2

, find a suitable Green's function

G_K(\mathbf{r}, \mathbf{r}')

. By taking the direction of the vector

\mathbf{r} - \mathbf{r}'

as the polar axis for a

\mathbf{k}

-space integration, show that

G_K(\mathbf{r}, \mathbf{r}')

can be reduced to

\frac{1}{4\pi^2|\mathbf{r} - \mathbf{r}'|} \int_{-\infty}^{\infty} \frac{w \sin w}{w^2 - w_0^2}\, dw,

where

w_0 = K|\mathbf{r} - \mathbf{r}'|

. [This integral can be evaluated using a contour integration (chapter 24) to give

(4\pi|\mathbf{r} - \mathbf{r}'|)^{-1} \exp(iK|\mathbf{r} - \mathbf{r}'|)