Chapter 7.3, Problem 7.18P

To determine

Show that $p^2$ is hermitian for hydrogen states with $l=0$, $\langle f|p^{2}g\rangle ={-4\pi {\hslash }^2\left(r^{2}f\frac{dg}{dr}-r^{2}gf\frac{df}{dr}\right)|}_0^{\infty }+\langle p^{2}f|g\rangle$. Also, check whether the boundary term vanishes from ${\psi }_{n00}$ for ${\psi }_{n00}\sim \frac{1}{\pi {\left(na\right)}^{5/2}}\exp \left(-r/na\right)$ and show that ${\nabla }^4\left[e^{-kr}\right]=\left(-\frac{4k^3}{r}+k^4\right)e^{-kr}+8\pi k{\delta }^3\left(r\right)$ and confirm that $p^4$ is hermitian.