In 1885, Balmer discovered Balmer’s spectrum line of Hydrogen,
$$ \lambda_{n} = \left(\frac{bn^2}{n^2-4}\right) $$
where $b = 3645.6 \r{A}$.
Rydberg generalized Balmer’s formula in 1885,
$$ \lambda_{n_2, n_1} = \frac{1}{C}\left(\frac{n_1^2n_2^2}{n_2^2-n_1^2}\right) \\
\frac{1}{\lambda_{n_2, n_1}} = -C \left(\frac{1}{n_2^2}-\frac{1}{n_1^2}\right) $$
Rydberg formula XXX
Rutherford nuclear model:
(1) orbits of electron are stable against radiation
(2) Energy differences between two levels follow Planck’s law, $E=\hbar \omega$
Newtonian mechanics, centrifugal force = Coulomb’s force
$$ \frac{mv^2}{r} = \frac{e^2}{4\pi\epsilon_0 r^2} : \quad (mrv)^2 = \frac{e^2}{4\pi\epsilon_0} mr $$
Angular momentum is quantized
$$ L = r(mv) = \hbar n $$
where $n$ is quantum number.
$$ r = \left(\frac{4\pi\epsilon_0\hbar^2}{me^2}\right) n^2 = a_0 n^2 $$
where $a_0$ is Bohr radius.