$(30$ points$)$ Quantum defects in alkali atoms The effective quantum number $n^{**}$ is equal to

$n-{}_{nl},$ where ${}_{nl}$ is the quantum defect. Using the numerical method you implemented in

Problem $1,$ you will now calculate the quantum defects in sodium.

$($a$)(5$ points$)$ The potential of an alkali atom can be approximated by the Coulomb potential

with central charge $+Ze$ for $r_c+er>r_c{r}_{c}V(r)tilde(V)(x)x=\frac{r}{a_0}$tilde$(V)(x)=\{\begin{array}{ccc}-\frac{2Z}{x}+\frac{l(l+1)}{x^2}+\frac{(Z-1)2a_0}{r_c}& \text{f}\text{o}\text{r}& r{r}_c\\ -\frac{2}{x}+\frac{l(l+1)}{x^2}& \text{f}\text{o}\text{r}& r>r_c\end{array}r,$ where $r_{c}is$ the mean radius $of$ the highest closed

electron shell, and the Coulomb potential with charge $+e$ for $r>r_c.An$ offset constant $is$

added $to$ the potential inside $r_{c}to$ ensure continuity $of$ the potential. Justify why the

aforementioned approximation $is$ valid for alkali atoms.

$(b)(5$ points$)$ Write down the expressions for $V(r)$ and tilde$(V)(x),$ based $on$ the radial equation

shown $in$ lecture and using $x=\frac{r}{a_0}.$ Show that

tilde$(V)(x)=\{\begin{array}{ccc}-\frac{2Z}{x}+\frac{l(l+1)}{x^2}+\frac{(Z-1)2a_0}{r_c}& \text{f}\text{o}\text{r}& r{r}_c\\ -\frac{2}{x}+\frac{l(l+1)}{x^2}& \text{f}\text{o}\text{r}& r>r_c\end{array}$