Check that Equation 11.65 satisfies Equation 11.52, by direct substitution. Hint:${\mathbf{\nabla }}^{\mathbf{2}}\left(1/r\right)\mathbf{=}\mathbf{-}\mathbf{4}\mathit{\pi }{\mathit{\delta }}^{\mathbf{3}}\mathbf{\left(}\mathit{r}\mathbf{\right)}$

Equation 11.65is given by:

$G=\frac{e^{ikr}}{4\pi r}$ …….. (1)

We need to check that satisfy equation which is given by:

$\left({\nabla }^2+k^2\right)G\left(r\right)={\delta }^3\left(r\right)$ ……… (2)

We have:

$\nabla G=-\frac{1}{4\pi }\left(\frac{1}{r}\nabla e^{ikr}+e^{ikr}\nabla \frac{1}{r}\right)$

So, the first term of equation (2) is:

role="math" localid="1658314246252" ${\nabla }^{2}G=\nabla .\left(\nabla G\right).......\left(3\right)$

role="math" localid="1658314207228" ${\nabla }^{2}G=-\frac{1}{4\pi }\left[2\left(\nabla \frac{1}{r}\right).\left(\nabla e^{ikr}\right)+\frac{1}{r}{\nabla }^2\left(e^{ikr}\right)+e^{ikr}{\nabla }^2\left(\frac{1}{r}\right)\right]......\left(4\right)$

But,

role="math" localid="1658314351283" $$\begin{gathered} \nabla \frac{1}{r}=-\frac{1}{r^2}\hat{r}\nabla \left(e^{ikr}\right) \\ \nabla \frac{1}{r}=ik{e}^{ikr}\hat{r} \end{gathered}$$

And,

$$\begin{gathered} {\nabla }^2{e}^{ikr}=ik\nabla .\left(e^{ikr}\hat{r}\right) \\ =ik\frac{1}{r^2}\frac{d}{dr}\left(r^2{e}^{ikr}\right) \\ =\frac{ik}{r^2}\left(2r{e}^{ikr}+ik{r}^2{e}^{ikr}\right) \\ =ik{e}^{ikr}\left(\frac{2}{r}+ik\right) \end{gathered}$$

Plug all these values into equation (3), so we get (note that ${\nabla }^2\left(1/r\right)=-4\pi {\delta }^3\left(r\right)$):

$$\begin{gathered} {\nabla }^{2}G=-\frac{1}{4\pi }\left[2\left(-\frac{1}{r^2}\hat{r}\right).\left(ik{e}^{ikr}\hat{r}\right)+\left(\frac{2}{r}+ik\right)-4\pi e^{ikr}{\delta }^3\left(r\right)\right] \\ ={\delta }^3\left(r\right)-\frac{1}{4\pi }{e}^{ikr}\left[-\frac{2ik}{r^2}+\frac{2ik}{r^2}-\frac{k^2}{r}\right] \\ ={\delta }^3\left(r\right)+k^2\frac{e^{ikr}}{4\pi r} \\ ={\delta }^3\left(r\right)-k^{2}G \end{gathered}$$

Note that we used$e^{ikr}{\delta }^3\left(r\right)={\delta }^3\left(r\right)$in the second line. Thus:

$\left({\nabla }^2+k^2\right)G\left(r\right)={\delta }^3\left(r\right)$.