Chapter 4, Problem 4.55P

(a)

To determine

Show that the radial wave function takes the form $R_{n\left(n-1\right)}=N_n{r}^{n-1}{e}^{-r/na}$, when $l=n-1$ using the recursion formula and determine the normalization constant $N_n$ by direct integration.

(b)

To determine

The expectation value of $\langle r\rangle$ and $\langle r^2\rangle$ for states of the form ${\psi }_{n\left(n-1\right)m}$.

(c)

To determine

Show that the “uncertainty” in $r\left({\sigma }_r\right)$ is $\langle r\rangle /\sqrt{2n+1}$ for such states. Also sketch the radial wave functions for several values of $n$.