Rydberg Atoms: A highly excited atom or ion has an electron with a

large principal quantum number $n.$ The electron $($or the atom$)$ is said

to be in a 'high Rydberg state' and a highly excited atom is sometimes

simply called a Rydberg atom. In this problem, I want you to employ

atomic units. Bohr's model predicts, the energy levels of a one electron

atom$/$ion to be

$E_n=-\frac{Z^2}{2n^2}$

$($a$)$ Show that the energy of the transitions between two shells with prin$-$

cipal quantum numbers $n$ and $n+1$ is proportional to $\frac{1}{n^3}$ for large

$n.$

$($b$)$ Calculate the frequency of the transition between the $n=51$ and

$n=50$ shells of a neutral atom. This emission frequency belongs

to which band of electromagnetic spectrum? $($Use conversion factors

computed in Q$1)$